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Introduction

Common perspectives in numerical analysis

Applications, computer software.

Historical background
Numerical linear and nonlinear algebra
Approximation theory
Solving differential and integral equations
Effects of computer hardware

numerical analysis

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Mathematics LibreTexts - Numerical Methods - Introduction
Table Of Contents

numerical analysis , area of mathematics and computer science that creates, analyzes, and implements algorithms for obtaining numerical solutions to problems involving continuous variables. Such problems arise throughout the natural sciences, social sciences, engineering, medicine, and business. Since the mid 20th century, the growth in power and availability of digital computers has led to an increasing use of realistic mathematical models in science and engineering, and numerical analysis of increasing sophistication is needed to solve these more detailed models of the world. The formal academic area of numerical analysis ranges from quite theoretical mathematical studies to computer science issues.

With the increasing availability of computers, the new discipline of scientific computing, or computational science, emerged during the 1980s and 1990s. The discipline combines numerical analysis, symbolic mathematical computations, computer graphics , and other areas of computer science to make it easier to set up, solve, and interpret complicated mathematical models of the real world.

Numerical analysis is concerned with all aspects of the numerical solution of a problem, from the theoretical development and understanding of numerical methods to their practical implementation as reliable and efficient computer programs. Most numerical analysts specialize in small subfields, but they share some common concerns, perspectives, and mathematical methods of analysis. These include the following:

When presented with a problem that cannot be solved directly, they try to replace it with a “nearby problem” that can be solved more easily. Examples are the use of interpolation in developing numerical integration methods and root-finding methods.
There is widespread use of the language and results of linear algebra , real analysis , and functional analysis (with its simplifying notation of norms, vector spaces , and operators).
There is a fundamental concern with error , its size, and its analytic form. When approximating a problem, it is prudent to understand the nature of the error in the computed solution. Moreover, understanding the form of the error allows creation of extrapolation processes to improve the convergence behaviour of the numerical method.
Numerical analysts are concerned with stability , a concept referring to the sensitivity of the solution of a problem to small changes in the data or the parameters of the problem. Consider the following example. The polynomial p ( x ) = ( x − 1)( x − 2)( x − 3)( x − 4)( x − 5)( x − 6)( x − 7), or expanded, p ( x ) = x 7 − 28 x 6 + 322 x 5 − 1,960 x 4 − 6,769 x 3 − 13,132 x 2 + 13,068 x − 5,040 has roots that are very sensitive to small changes in the coefficients. If the coefficient of x 6 is changed to −28.002, then the original roots 5 and 6 are perturbed to the complex numbers 5.459 0.540 i —a very significant change in values. Such a polynomial p ( x ) is called unstable or ill-conditioned with respect to the root-finding problem. Numerical methods for solving problems should be no more sensitive to changes in the data than the original problem to be solved. Moreover, the formulation of the original problem should be stable or well-conditioned.
Numerical analysts are very interested in the effects of using finite precision computer arithmetic . This is especially important in numerical linear algebra, as large problems contain many rounding errors .
Numerical analysts are generally interested in measuring the efficiency (or “cost”) of an algorithm . For example, the use of Gaussian elimination to solve a linear system A x = b containing n equations will require approximately 2 n 3 / 3 arithmetic operations. Numerical analysts would want to know how this method compares with other methods for solving the problem.

Modern applications and computer software

Numerical analysis and mathematical modeling are essential in many areas of modern life. Sophisticated numerical analysis software is commonly embedded in popular software packages (e.g., spreadsheet programs) and allows fairly detailed models to be evaluated, even when the user is unaware of the underlying mathematics. Attaining this level of user transparency requires reliable, efficient, and accurate numerical analysis software, and it requires problem-solving environments (PSE) in which it is relatively easy to model a given situation. PSEs are usually based on excellent theoretical mathematical models, made available to the user through a convenient graphical user interface .

Computer-aided engineering (CAE) is an important subject within engineering, and some quite sophisticated PSEs have been developed for this field. A wide variety of numerical analysis techniques is involved in solving such mathematical models. The models follow the basic Newtonian laws of mechanics, but there is a variety of possible specific models, and research continues on their design. One important CAE topic is that of modeling the dynamics of moving mechanical systems, a technique that involves both ordinary differential equations and algebraic equations (generally nonlinear). The numerical analysis of these mixed systems, called differential-algebraic systems, is quite difficult but necessary in order to model moving mechanical systems. Building simulators for cars, planes, and other vehicles requires solving differential-algebraic systems in real time.

Another important application is atmospheric modeling. In addition to improving weather forecasts, such models are crucial for understanding the possible effects of human activities on the Earth’s climate. In order to create a useful model, many variables must be introduced. Fundamental among these are the velocity V ( x , y , z , t ), pressure P ( x , y , z , t ), and temperature T ( x , y , z , t ), all given at position ( x , y , z ) and time t . In addition, various chemicals exist in the atmosphere, including ozone, certain chemical pollutants, carbon dioxide , and other gases and particulates, and their interactions have to be considered. The underlying equations for studying V ( x , y , z , t ), P ( x , y , z , t ), and T ( x , y , z , t ) are partial differential equations; and the interactions of the various chemicals are described using some quite difficult ordinary differential equations. Many types of numerical analysis procedures are used in atmospheric modeling, including computational fluid mechanics and the numerical solution of differential equations. Researchers strive to include ever finer detail in atmospheric models, primarily by incorporating data over smaller and smaller local regions in the atmosphere and implementing their models on highly parallel supercomputers.

Modern businesses rely on optimization methods to decide how to allocate resources most efficiently. For example, optimization methods are used for inventory control, scheduling, determining the best location for manufacturing and storage facilities, and investment strategies.

Software to implement common numerical analysis procedures must be reliable, accurate, and efficient. Moreover, it must be written so as to be easily portable between different computer systems. Since about 1970, a number of government-sponsored research efforts have produced specialized, high-quality numerical analysis software.

The most popular programming language for implementing numerical analysis methods is Fortran, a language developed in the 1950s that continues to be updated to meet changing needs. Other languages, such as C, C++, and Java, are also used for numerical analysis. Another approach for basic problems involves creating higher level PSEs, which often contain quite sophisticated numerical analysis, programming , and graphical tools. Best known of these PSEs is MATLAB, a commercial package that is arguably the most popular way to do numerical computing. Two popular computer programs for handling algebraic-analytic mathematics (manipulating and displaying formulas) are Maple and Mathematica.

Numerical Analysis

Numerical Analysis deals with the process of getting the numerical solution to complex problems. The majority of mathematical problems in science and engineering are difficult to answer precisely, and in some cases it is impossible. To make a tough Mathematical problem easier to solve, an approximation is essential. Numerical approximation has become more popular as a result of tremendous advances in computational technology. As a result, a great deal of scientific software is being developed to solve more complex challenges quickly and easily. Let us go through the definition of numerical analysis as well as the various concepts included, such as errors, interpolation and so on in this article.

Introduction to Numerical Analysis

Numerical analysis is a discipline of mathematics concerned with the development of efficient methods for getting numerical solutions to complex mathematical problems. There are three sections to the numerical analysis. The first section of the subject deals with the creation of a problem-solving approach. The analysis of methods, which includes error analysis and efficiency analysis, is covered in the second section. The efficiency analysis shows us how fast we can compute the result, while the error analysis informs us how correct the result will be if we utilize the approach. The construction of an efficient algorithm to implement the approach as a computer code is the subject’s third part. All three elements must be familiar to have a thorough understanding of the numerical analysis.

Meanwhile, there are at least three reasons to learn the theoretical foundations of numerical methods:

Learning various numerical methods and analyzing them will familiarize a person with the process of inventing new numerical methods. When the existing approaches are insufficient or inefficient to handle a certain problem, this is critical.
In many cases, there are multiple solutions to a problem. As a result, using the right procedure is critical for getting a precise answer in less time.
With a solid foundation, one can effectively apply methods (especially when a technique has its own restrictions and/or drawbacks in certain instances) and, more significantly, analyze what went wrong when results did not meet expectations.

Let’s have a look at some of the key topics in numerical analysis.

Different Types of Errors

The disparity between the approximate representation of a real number and the actual value is termed an error .

Error = True Value – Approximate Value , is the formula for calculating the error in a computed amount.

The absolute error is defined as the absolute value of the error defined above.

Relative Error = Error / True Value is a measurement of the error in respect to the magnitude of the true value.

The relative error is multiplied by 100 to get the percentage error .

The phrase “ truncation error ” refers to the error that occurs when a smooth function is approximated by reducing its Taylor series representation to a limited number of terms.

Significant Digits

If x A is an approximation to x, so we can conclude that x A approximates x to r significant β-digits if |x − x A | ≤ (½)β s−r+1 with “s” the greatest integer such that β s ≤ |x|.

As an example, the approximate value x A = 0.333 includes three significant digits for x = ⅓, since |x − x A | ≈ .00033 < 0.0005 = 0.5 × 10 −3 .

But 10−1 < 0.333 · · · = x.

Hence, in this case s = −1 and and therefore r = 3.

Propagation of Errors

When an error is committed, it has an impact on subsequent outcomes because it propagates through subsequent calculations. We’ll look at how utilizing approximate numbers rather than actual numbers affects the outcomes before moving on to function evaluation. We’ll now explore how error propagates in four basic arithmetic operations .

In addition and subtraction, the total of the error bounds for the terms provides an error bound for the results .
In multiplication and division, The sum of the bounds for the relative errors of the given integers gives a limitation for the relative error of the results.

Finite Difference Operators

Now, let us discuss the various finite difference operators in brief.

Forward Operator

Assume that “h” be the finite difference, then

Δf(x) = f(x+h) – f(x)

Δ 2 f(x) = f(x+2h)-2f(x+h) + f(x)

Δ 3 f(x)= f(x+3h) – 3f(x+2h) + 2f(x+h) – f(x)

Shift Operator

Assume that h be the finite difference.

Then, E f(x) = f(x+h)

E n f(x) = f(x+nh)

Backward Difference

Suppose h be the finite difference.

Central Difference Operator

Averaging operator, factorial notation, relation between different finite operators.

Relationship Between Δ and E

E ≡ 1 + Δ and Δ ≡ E-1

Hence, E n ≡ (1+Δ) n and Δ n ≡ (E-1) n

Interpolation

Interpolation is the process of determining the approximate value of a function f(x) for an x between multiple x values x 0, x 1 , …, x n for which the value of f(x) is known.

I.e., f(x i ) = f i (i = 0, 1, 2, …, n)

If the real-valued function f(x) has (n+1) different values, then x 0 x 1 , ..x n . A polynomial of degree n or less is P n (x i ) = f(x). It indicates that there can only be one polynomial with a degree less than or equal to n that interpolates f(x) at (n+1) unique points x 0 , x 1 , x 2 , …x n .

Solved Example on Numerical Analysis

Show that μ 4 = μ 3 + Δμ 2 + Δ 2 μ 1 + Δ 3 μ 1

As we know that

Δμ x = μ x+h – μ x

Hence, μ 4 – μ 3 = Δμ 3

μ 3 – μ 2 = Δμ 2

μ 2 – μ 1 = Δμ 1

μ 4 = μ 3 + Δμ 3

μ 4 = μ 3 + Δμ 2 – Δ 2 μ 2

μ 4 = μ 3 + Δμ 2 + Δ 2 μ 1 + Δ 3 μ 1

Hence, proved.

Stay tuned to BYJU’S – The Learning App and download the app all the Maths-related concepts easily by exploring more videos.

Frequently Asked Questions on Numerical Analysis

What is numerical analysis.

Numerical analysis is a branch of mathematics concerned with the development of efficient methods for solving complicated mathematical problems numerically.

What are the different types of numerical analysis?

The different types of numerical analysis are finite difference methods, propagation of errors, interpolation methods, and so on.

Is calculus required for learning numerical analysis?

Yes, calculus is required for learning numerical analysis, as we should know differential integration.

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1.8: Solving Problems in Physics

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Learning Objectives

Describe the process for developing a problem-solving strategy.
Explain how to find the numerical solution to a problem.
Summarize the process for assessing the significance of the numerical solution to a problem.

Problem-solving skills are clearly essential to success in a quantitative course in physics. More important, the ability to apply broad physical principles—usually represented by equations—to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physics in everyday life.

A photograph of a student’s hand, working on a problem with an open textbook, a calculator, and an eraser.

As you are probably well aware, a certain amount of creativity and insight is required to solve problems. No rigid procedure works every time. Creativity and insight grow with experience. With practice, the basics of problem solving become almost automatic. One way to get practice is to work out the text’s examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and then progressing to the more difficult. After you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text.

Although there is no simple step-by-step method that works for every problem, the following three-stage process facilitates problem solving and makes it more meaningful. The three stages are strategy, solution, and significance. This process is used in examples throughout the book. Here, we look at each stage of the process in turn.

Strategy is the beginning stage of solving a problem. The idea is to figure out exactly what the problem is and then develop a strategy for solving it. Some general advice for this stage is as follows:

Examine the situation to determine which physical principles are involved . It often helps to draw a simple sketch at the outset. You often need to decide which direction is positive and note that on your sketch. When you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless.
Make a list of what is given or can be inferred from the problem as stated (identify the “knowns”) . Many problems are stated very succinctly and require some inspection to determine what is known. Drawing a sketch be very useful at this point as well. Formally identifying the knowns is of particular importance in applying physics to real-world situations. For example, the word stopped means the velocity is zero at that instant. Also, we can often take initial time and position as zero by the appropriate choice of coordinate system.
Identify exactly what needs to be determined in the problem (identify the unknowns). In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help identify the unknowns.
Determine which physical principles can help you solve the problem . Since physical principles tend to be expressed in the form of mathematical equations, a list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all the other variables are known—so you can solve for the unknown easily. If the equation contains more than one unknown, then additional equations are needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer.

The solution stage is when you do the math. Substitute the knowns (along with their units) into the appropriate equation and obtain numerical solutions complete with units . That is, do the algebra, calculus, geometry, or arithmetic necessary to find the unknown from the knowns, being sure to carry the units through the calculations. This step is clearly important because it produces the numerical answer, along with its units. Notice, however, that this stage is only one-third of the overall problem-solving process.

Significance

After having done the math in the solution stage of problem solving, it is tempting to think you are done. But, always remember that physics is not math. Rather, in doing physics, we use mathematics as a tool to help us understand nature. So, after you obtain a numerical answer, you should always assess its significance:

Check your units . If the units of the answer are incorrect, then an error has been made and you should go back over your previous steps to find it. One way to find the mistake is to check all the equations you derived for dimensional consistency. However, be warned that correct units do not guarantee the numerical part of the answer is also correct.
Check the answer to see whether it is reasonable. Does it make sense? This step is extremely important: –the goal of physics is to describe nature accurately. To determine whether the answer is reasonable, check both its magnitude and its sign, in addition to its units. The magnitude should be consistent with a rough estimate of what it should be. It should also compare reasonably with magnitudes of other quantities of the same type. The sign usually tells you about direction and should be consistent with your prior expectations. Your judgment will improve as you solve more physics problems, and it will become possible for you to make finer judgments regarding whether nature is described adequately by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to solve a problem mechanically.
Check to see whether the answer tells you something interesting. What does it mean? This is the flip side of the question: Does it make sense? Ultimately, physics is about understanding nature, and we solve physics problems to learn a little something about how nature operates. Therefore, assuming the answer does make sense, you should always take a moment to see if it tells you something about the world that you find interesting. Even if the answer to this particular problem is not very interesting to you, what about the method you used to solve it? Could the method be adapted to answer a question that you do find interesting? In many ways, it is in answering questions such as these science that progresses.

Browse Course Material

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Prof. Arthur Mattuck
Prof. Haynes Miller
Dr. Jeremy Orloff
Dr. John Lewis

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Mathematics

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Differential Equations
Linear Algebra

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Numerical methods.

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Session Overview

Many differential equations cannot be solved exactly. For these DE’s we can use numerical methods to get approximate solutions. In the previous session the computer used numerical methods to draw the integral curves.

We will start with Euler’s method. This is the simplest numerical method, akin to approximating integrals using rectangles, but it contains the basic idea common to all the numerical methods we will look at. We will also discuss more sophisticated methods that give better approximations.

Session Activities

Read the course notes:

Numerical Methods: Introduction (PDF)

Watch the lecture video clip:

Euler’s Method
Motivation and Implementation of Euler’s Method (PDF)
Example of Euler’s Method

Complete the practice problem:

An Example of Euler’s Method (PDF)
Exercise Solution (PDF)

Learn from the Mathlet materials:

Watch the video Exploration of the Euler’s Method Applet
Read about how to work with the Euler’s Method Applet (PDF)
Work with the Euler’s Method Applet
Errors in Euler’s Method (PDF)
Better Methods
Further Numerical Methods (PDF)

Watch the problem solving video:

Complete the practice problems:

Practice Problems 3 (PDF)
Practice Problems 3 Solutions (PDF)

Check Yourself

Take the quiz:

Problem Set Part I Problems (PDF)

Problem Set Part I Solutions (PDF)

Problem Set Part II Problems (PDF)

Problem Set Part II Solutions (PDF)

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Numerical Analysis

An Introduction to Numerical Analysis

Numerical Analysis is the Mathematics branch responsible for designing effective ways to find numerical solutions to complex Mathematical problems. Most Mathematical problems from science and engineering are very complex and sometimes cannot be solved directly. Therefore, measuring a complex Mathematical problem is very important to make it easier to solve. Due to the great advances in computational technology, numeracy has become very popular and is a modern tool for scientists and engineers. As a result many software programs are being developed such as Matlab, Mathematica, Maple etc. the most difficult problems in an effective and simple way. These softwares contain functions that use standard numeric methods, in which the user can bypass the required parameters and obtain the results in a single command without knowing the numerical details.

The Numerical Analysis method is mainly used in the area of Mathematics and Computer Science that creates, analyzes, and implements algorithms for solving numerical problems of continuous Mathematics. Such types of problems generally originate from real-world applications of algebra, geometry and calculus, and they also involve variables that vary continuously. These problems occur throughout the natural sciences, social sciences, engineering, medicine, and the field of business. Introduction of Numerical Analysis during the past half-century, the growth in power and availability of digital computers has led to the increasing use of realistic Mathematical models in science and engineering. Here we will learn more about numerical method and analysis of numerical methods.

Numerical Method

Numerical methods are techniques that are used to approximate Mathematical procedures. We need approximations because we either cannot solve the procedure analytically or because the analytical method is intractable (an example is solving a set of a thousand simultaneous linear equations for a thousand unknowns).

Different Types of Numerical Methods

The numerical analysts and Mathematicians used have a variety of tools that they use to develop numerical methods for solving Mathematical problems. The most important idea, mentioned earlier, that cuts across all sorts of Mathematical problems is that of changing a given problem with a 'near problem' that can be easily solved. There are other ideas that differ on the type of Mathematical problem solved.

An Introduction to Numerical Methods for Solving Common Division Problems Given Below:

Euler method - the most basic way to solve ODE

Clear and vague methods - vague methods need to solve the problem in every step

The Euler Back Road - the obvious variation of the Euler method

Trapezoidal law - the direct method of the second system

Runge-Kutta Methods - one of the two main categories of problems of the first value .

Numerical Methods

Newton method

Some calculations cannot be solved using algebra or other Mathematical methods. For this we need to use numerical methods. Newton's method is one such method and allows us to calculate the solution of f (x) = 0.

Simpson Law

The other important ones cannot be assessed in terms of integration rules or basic functions. Simpson's law is a numerical method that calculates the numerical value of a direct combination.

Trapezoidal law

A trapezoidal rule is a numerical method that calculates the numerical value of a direct combination. The other important ones cannot be assessed in terms of integration rules or basic functions.

Numerical Computation

The term “numerical computations” means to use computers for solving problems involving real numbers. In this process of problem-solving, we can distinguish several more or less distinct phases. The first phase is formulation. While formulating a Mathematical model of a physical situation, scientists should take into account the fact that they expect to solve a problem on a computer. Therefore they will provide for specific objectives, proper input data, adequate checks, and for the type and amount of output.

Once a problem has been formulated, then the numerical methods, together with preliminary error analysis, must be devised for solving the problem. A numerical method that can be used to solve a problem is called an algorithm. An algorithm is a complete and unambiguous set of procedures that are used to find the solution to a Mathematical problem. The selection or construction of appropriate algorithms is done with the help of Numerical Analysis. We have to decide on a specific algorithm or set of algorithms for solving the problem, numerical analysts should also consider all the sources of error that may affect the results. They should consider how much accuracy is required. To estimate the magnitude of the round-off and discretization errors, and determine an appropriate step size or the number of iterations required.

The programmer should transform the suggested algorithm into a set of unambiguous that is followed by step-by-step instructions to the computer. The flow chart is the first step in this procedure. A flow chart is simply a set of procedures, that are usually written in logical block form, which the computer will follow. The complexity of the flow will depend upon the complexity of the problem and the amount of detail included. However, it should be possible for someone else other than the programmer to follow the flow of information from the chart. The flow chart is an effective aid to the programmer, they must translate its major functions into a program. And, at the same time, it is an effective means of communication to others who wish to understand what the program does.

Numerical Computing Characteristics

Accuracy: Every numerical method introduces errors. It may be due to the use of the proper Mathematical process or due to accurate representation and change of numbers on the computer.

Efficiency: Another consideration in choosing a numerical method for a Mathematical model solution efficiency Means the amount of effort required by both people and computers to use the method.

Numerical instability: Another problem presented by a numerical method is numerical instability. Errors included in the calculation, from any source, increase in different ways. In some cases, these errors are usually rapid, resulting in catastrophic results.

Numerical Computing Process

Construction of a Mathematical model.

Construction of an appropriate numerical system.

Implementation of a solution.

Verification of the solution.

Trapezoidal Law

In Mathematics, trapezoidal law, also known as trapezoid law or trapezium law, is the most important measure of direct equity in Numerical Analysis. Trapezoidal law is a coupling law used to calculate the area under a curve by dividing the curve into a small trapezoid. The combination of all the small trapezoid areas will provide space under the curve. Let's understand the trapezoidal law formula and its evidence using examples in future sections.

Numerical and Statistical Methods

Numerical methods, as said above, are techniques to approximate Mathematical procedures. On the other hand, statistics is the study and manipulation of data, including ways to gather, review, analyze, and draw conclusions from the given data. Thus we can say, statistical methods are Mathematical formulas, models, and techniques that are used in the statistical analysis of raw research data. The application of statistical methods extracts information from research data and provides different methods to assess the robustness of research outputs. Some common statistical tools and procedures are given below :

Descriptive

Mean (average)

Inferential

Linear regression analysis

Analysis of variance

Null hypothesis testing

Introduction to Finite Element Method

The various laws of physics related to space and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). If we have the vast majority of geometries and problems, these PDEs cannot be solved using analytical methods. Instead of that, we have created an approximation of the equations, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. Thus, the solution to the numerical model equations is, in turn, an approximation of the real solution to the PDEs. The finite element method is used to compute such approximations.

The finite element method is a numerical technique that is used for solving problems that are described by partial differential equations or can be formulated as functional minimization. A domain of interest is represented by the assembly of finite elements. Approximating functions in finite elements are determined in terms of nodal values of a physical field. A continuous physical problem is transformed into a discretized finite element problem with the help of unknown nodal values. For a linear problem, a system of linear algebraic equations must be solved. We can recover values inside finite elements using the nodal values.

Two Features of the Fem are Mentioned below:

Piecewise approximation of physical fields on finite elements provides good precision even with simple approximating functions (i.e. increasing the number of elements we can achieve any precision).

Locality of approximation leads to sparse equation systems that are mainly used for a discretized problem. With the help of this, we can solve problems with a very large number of nodal unknowns.

Typical Classes of Engineering Problems That Can be Solved Using Fem are:

Structural mechanics

Heat transfer

Electromagnetics

Finite Element Method MATLAB

Finite element analysis is a computational method for analyzing the behaviour of physical products under loads and boundary conditions. A typical FEA workflow in MATLAB includes

Importing or creating geometry.

Generating mesh.

Defining physics of the problem with the help of load, boundary and initial conditions.

Solving and visualizing results.

The design of experiments or optimization techniques can be used along with FEA to perform trade-off studies or to design an optimal product for specific applications.

MATLAB is Very Useful Software and is Very Easy to Apply Finite Element Analysis Using MATLAB. It Helps Us in Applying Fem in Several Ways:

Partial differential equations (PDEs) can be solved using the inbuilt Partial Differential Equation Toolbox.

In MATLAB, with the help of Statistics and Machine Learning Toolbox, we can apply the design of experiments and other statistics and machine learning techniques with finite element analysis.

Also, the optimization techniques can be applied to FEM simulations to come up with an optimum design with Optimization Toolbox.

Parallel Computing Toolbox speeds up the analysis by distributing multiple Finite element analysis simulations to run in parallel.

FAQs on Numerical Analysis

1. What's the Trapezoidal Rule?

Trapezoidal Rule is an integration rule, in Calculus, that evaluates the location beneath the curves via dividing the total location into smaller trapezoids in preference to using rectangles.

2. Why is the guideline named after a trapezoid?

The call trapezoidal is because whilst the location under the curve is evaluated, then the full vicinity is divided into small trapezoids rather than rectangles. Then we find the region of these small trapezoids in a definite c program language period.

3. What is the use of Numerical techniques?

Numerical strategies are used in Mathematics and computer technological know-how that creates, analyzes, and implements algorithms to acquire the numerical answers to problems using non-stop variables. Such troubles rise up in the course of the herbal sciences, social sciences, engineering, medicine, and also in commercial enterprise.

4. What are the basics of the Finite detail method?

The finite element approach is a Mathematical method used to calculate approximate answers to differential equations. The intention of this method is to convert the differential equations into hard and fast linear equations that can then be solved by the computer in a routine manner.

5. What is the distinction between the Trapezoidal Rule and Riemann Sums rule?

In the Trapezoidal Rule, we use trapezoids to approximate the region under the curve while in Riemann sums we use rectangles to discover areas below the curve, in case of integration.

6. Define the Trapezoid Rule of Numerical Analysis.

The trapezoidal rule is used to find the exact value of a definite integral using a numerical method. This rule is based on the concept of the Newton-Cotes formula which states that we can find the exact value of the integral as the nth order polynomial.

7. What is the Use of Numerical Methods?

Numerical methods are used in Mathematics and Computer Science that creates, analyzes, and implements algorithms to obtain the numerical solutions to problems using continuous variables. Such

8. What are the Basics of the Finite Element Method?

The finite element method is a Mathematical procedure used to calculate approximate solutions to differential equations. The goal of this method is to transform the differential equations into a set of linear equations that can then be solved by the computer in a routine manner.

9. Why is the guideline named after a trapezoid?

10. What is the use of Numerical techniques?

11. What are the basics of the Finite detail method?

12. What is the distinction between the Trapezoidal Rule and Riemann Sums rule?

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3: Numerical Solutions

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Russell Herman
University of North Carolina Wilmington

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"The laws of mathematics are not merely human inventions or creations. They simply ’are; they exist quite independently of the human intellect." - M. C. Escher (1898-1972)

SO FAR WE HAVE SEEN SOME OF THE STANDARD METHODS for solving first and second order differential equations. However, we have had to restrict ourselves to special cases in order to get nice analytical solutions to initial value problems. While these are not the only equations for which we can get exact results, there are many cases in which exact solutions are not possible. In such cases we have to rely on approximation techniques, including the numerical solution of the equation at hand.

The use of numerical methods to obtain approximate solutions of differential equations and systems of differential equations has been known for some time. However, with the advent of powerful computers and desktop computers, we can now solve many of these problems with relative ease. The simple ideas used to solve first order differential equations can be extended to the solutions of more complicated systems of partial differential equations, such as the large scale problems of modeling ocean dynamics, weather systems and even cosmological problems stemming from general relativity.

3.1: Euler’s Method In this section we will look at the simplest method for solving first order equations, Euler’s Method. While it is not the most efficient method, it does provide us with a picture of how one proceeds and can be improved by introducing better techniques, which are typically covered in a numerical analysis text.
3.2: Implementation of Numerical Packages One can use Matlab to obtain solutions and plots of solutions of differential equations. This can be done either symbolically, using dsolve, or numerically, using numerical solvers like ode45. In this section we will provide examples of using these to solve first order differential equations. We will end with the code for drawing direction fields, which are useful for looking at the general behavior of solutions of first order equations without explicitly finding the solutions.
3.3: Higher Order Taylor Methods Euler’s method for solving differential equations is easy to understand but is not efficient in the sense that it is what is called a first order method. The error at each step, the local truncation error, is of order Δx , for x the independent variable. The accumulation of the local truncation errors results in what is called the global error. In order to generalize Euler’s Method, we need to rederive it.
3.4: Runge-Kutta Methods In this section we will find approximations of to solutions that avoid the need for computing the derivatives.
3.5.1: The Nonlinear Pendulum
3.5.2: Extreme Sky Diving
3.5.3: The Flight of Sports Balls
3.5.4: Falling Raindrops
3.5.5: The Two-body Problem
3.5.6: The Expanding Universe
3.5.7: The Coefficient of Drag
3.6: Problems

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From divvying up a restaurant bill among your friends, telling the time on a 12-hr clock, to tougher activities such as mixing the right proportions of different ingredients, mathematics is an essential tool in our everyday lives. This notion alone is enough to give many the willies, especially for those who associate their poor mathematical abilities with the very well-known label that says “I am just not a math person”.

However, this building block of engineering and sciences is not just for cracking onerous physical problems like quantum gravity but is also a skill that allows us to solve many of the practical problems that we stumble upon in our daily lives.

Numerical skills refer to one’s ability to grasp the fundamental concepts of basic maths operations, such as addition , subtraction , division , and multiplication . More advanced mathematical skills may cover the understanding of algebraic and calculus concepts, and even the ability to draw on statistical knowledge to solve real-world problems .

The Importance of Numerical Skills in Today’s Era

The COVID-19 pandemic brought with it a tidal wave of financial troubles across the globe, dealing a heavy blow to the finances of millions or possibly billions of households and individuals. As the pandemic drags on, reports on job losses, closures of stores, and even bankruptcies have frequently hit the headlines; the labor market has been getting increasingly competitive, with some sectors coping with an excessive supply of labor.

That being the case, gearing yourself up with knowledge and skills that will remain relevant across various industries is more vital than ever, and numerical skills are also a reflection of your critical thinking and problem solving abilities. Big companies often rely upon big data to better guide their decision-making; because of this, you can spruce up your resume by just demonstrating a good level of mathematical skills, giving yourself a higher chance in the competitive job market.

Besides, it has been well proven that there is a correlation between financial literacy and numeracy. People who lack confidence in numbers may have a hard time managing their finances, and find living through this pandemic more challenging than those with strong numeracy skills. Even simple financial activities like managing your daily cash inflows and outflows, making price comparisons, and even developing a weekly budget plan demand a certain level of mathematical competency from you.

Three Basic Ways to Improve Your Numerical Skills

We have all become accustomed to hearing the myth of “I am just not a math person”, and many have deeply instilled in themselves the idea that being good at maths is a natural-born talent, and so they are unable to improve their numerical ability. This mentality has been confronting teachers and researchers for many years, and the truth is that all of us can essentially be math people. If your thoughts tell you otherwise, you have probably been helping nurture a myth that has been considered one of the most destructive ideas in the US.

With all that said, everyone, including you, has the ability to succeed in learning mathematics. With effort, patience, and the right strategies, you can definitely witness an improvement in your numeracy skills. Here are some of the steps that you can take to enhance your own mathematical ability.

Don’t Memorise

With maths, it is not all about memorizing countless mathematical laws and principles, and in fact, memorizing facts and procedures is what you should avoid at all times in the learning process. Many arithmetic operations like addition, subtraction, division and multiplication, and even operations on square roots carry their own intuitive meanings that we can easily understand. Aside from these, you should also be familiar with the properties of numbers, such as the commutative, associative, distributive, and identity. Learn to recognize the patterns behind these operations and understand them.

Tougher mathematical concepts like trigonometry , algebra , and calculus require considerably more time to learn; however, the benefits of learning these are tremendous. They have a wide range of applications in real life, equipping you with the skills to unravel varying puzzling problems you may encounter either at work or in your personal life. A very simple yet practical question can involve the use of all of these concepts - that is, for instance, what is the maximum area of a rectangular we can have given the length of the perimeter is restricted to a maximum of 10 meters?

Practice, Practice, and Practice - Don’t Be Afraid of Making Mistakes

Many of us may think that making mistakes is wrong. However, it is actually part of the learning journey of mathematics. Recent neurological research has shown that making a mistake is not a bad thing; as we reflect upon the mistakes we have made, it is a great time for our brains to grow.

You should learn to embrace whatever mistakes and faults you have committed and turn them into a learning opportunity. When it comes to mastering mathematics, there are no magic tricks besides patience and effort. As you practice, your performance will improve so will your understanding. The more problems you solve, the better you will get at handling those types of problems.

Frequent practice of applying mathematical concepts and techniques helps you develop mastery of various mathematical skills and operations. You will be capable of adopting various mathematical techniques to solve problems in business, science, or daily life.

Don’t Be Hesitant to Look for Help

The internet is rife with resources and free-to-download applications that may guide you through the learning process, or you can always recur to an online math calculator, such as an arithmetic sequence solver , when you're in need of clear explanations and step-by-step examples to help with your math needs. For instance, if you are trying to come up with a mathematical function or model that calculates the savings you will get from leasing a car as opposed to purchasing one, Orix has a calculator on their website that you can use to determine the correctness and accuracy of your calculations. However, if you're interested in learning math fundamentals, such as the difference between even and odd numbers , you can check out instructional videos on Smartick. This will help you better understand the number system and better prepare you for whole number operations.

Lastly, in addition to the numeracy skills pages on Skills You Need, there are lots of other websites offering free online courses in mathematics, some even award certificates at an affordable price. These courses cover simple mathematical subjects to more advanced ones like linear algebra and discrete mathematics, and it is never too late to embark on a lifelong journey of learning in the field of mathematics to master numeracy skills.

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What’s the difference between analytical and numerical approaches to problems?

I don't have much (good) math education beyond some basic university-level calculus.

What do "analytical" and "numerical" mean? How are they different?

numerical-methods

23 $\begingroup$ For some reason, I'm irritated that convention has settled on "analytic" instead of "symbolic." "Numerical" usually indicates an approximate solution obtained by methods of numerical analysis . "Analytical" solutions are exact and obtained by methods of symbolic manipulation, derived using analysis . The methods of numerical analysis are themselves derived using (symbolic) analysis. "Analytical" really fails to convey the intended distinction for me, since both approaches seem analytical. $\endgroup$ – Michael E2 Commented Dec 15, 2015 at 18:50
1 $\begingroup$ The answers are mostly correct but ... when you do a "numerical solution" you are generally only getting one answer. Whereas analytic/symbolic solutions gives you answers to a whole set of problems. In other words: for every set of parameters the numerical approach has to be recalculated and the analytic approach allows you to have all (well some) solutions are your fingertips. Generically numerical approaches don't give you deep insight but analytic approaches can. Paraphrasing, having a hammer doesn't make everything a nail. $\endgroup$ – rrogers Commented Jun 11, 2019 at 21:07

7 Answers 7

Analytical approach example:

Find the root of $f(x)=x-5$ .

Analytical solution: $f(x)=x-5=0$ , add $+5$ to both sides to get the answer $x=5$

Numerical solution:

let's guess $x=1$ : $f(1)=1-5=-4$ . A negative number. Let's guess $x=6$ : $f(6)=6-5=1$ . A positive number.

The answer must be between them. Let's try $x=\frac{6+1}{2}$ : $f(\frac{7}{2})<0$

So it must be between $\frac{7}{2}$ and $6$ ...etc.

This is called bisection method.

Numerical solutions are extremely abundant. The main reason is that sometimes we either don't have an analytical approach (try to solve $x^6-4x^5+\sin (x)-e^x+7-\frac{1}{x} =0$ ) or that the analytical solution is too slow and instead of computing for 15 hours and getting an exact solution, we rather compute for 15 seconds and get a good approximation.

$\begingroup$ Would an analytic solution be an exact form or maybe an integral/infinite sum? $\endgroup$ – Тyma Gaidash Commented Jun 4, 2023 at 14:32
$\begingroup$ @mvw What do you mean by that? Is there an example that I can read what all comes under analytic. $\endgroup$ – piepi Commented Jun 22, 2023 at 10:29
$\begingroup$ @piepi The term "analytical solution" seems a bit less sharp than "numerical solution". If you stumble on this term you should find out what the authors or the community mean. $\endgroup$ – mvw Commented Jun 23, 2023 at 13:04

The simplest breakdown would be this:

Analytical solutions can be obtained exactly with pencil and paper;
Numerical solutions cannot be obtained exactly in finite time and typically cannot be solved using pencil and paper.

These distinctions, however, can vary. There are increasingly many theorems and equations that can only be solved using a computer; however, the computer doesn't do any approximations, it simply can do more steps than any human can ever hope to do without error. This is the realm of "symbolic computation" and its cousin, "automatic theorem proving." There is substantial debate as to the validity of these solutions -- checking them is difficult, and one cannot always be sure the source code is error-free. Some folks argue that computer-assisted proofs should not be accepted.

Nevertheless, symbolic computing differs from numerical computing. In numerical computing, we specify a problem, and then shove numbers down its throat in a very well-defined, carefully-constructed order. If we are very careful about the way in which we shove numbers down the problem's throat, we can guarantee that the result is only a little bit inaccurate, and usually close enough for whatever purposes we need.

Numerical solutions very rarely can contribute to proofs of new ideas. Analytic solutions are generally considered to be "stronger". The thinking goes that if we can get an analytic solution, it is exact, and then if we need a number at the end of the day, we can just shove numbers into the analytic solution. Therefore, there is always great interest in discovering methods for analytic solutions. However, even if analytic solutions can be found, they might not be able to be computed quickly. As a result, numerical approximation will never go away, and both approaches contribute holistically to the fields of mathematics and quantitative sciences.

Analytical is exact; numerical is approximate.

For example, some differential equations cannot be solved exactly (analytic or closed form solution) and we must rely on numerical techniques to solve them.

2 $\begingroup$ You write: some differential equations cannot be solved exactly . I think that must be: most differential equations cannot be solved exactly. And we must rely on numerical techniques to solve them. $\endgroup$ – Han de Bruijn Commented Apr 4, 2019 at 19:44
6 $\begingroup$ Decent point but “Some” just means an unspecific amount. It doesn’t mean “few” or less than the majority. I actually would be careful with "most" - though surely from a practical, applied perspective this is true. But are the cardinality of the solution sets of closed form vs not different? We’d want to define “closed form” more precisely in this context - but, for example, we know that the set of elementary functions with elementary anti-derivatives are the same as those without so I wouldn’t throw out “most” without a bit more care personally. $\endgroup$ – user115411 Commented Apr 4, 2019 at 23:29

Numerical methods use exact algorithms to present numerical solutions to mathematical problems.

Analytic methods use exact theorems to present formulas that can be used to present numerical solutions to mathematical problems with or without the use of numerical methods.

Analytical method gives exact solutions, more time consuming and sometimes impossible. Whereas numerical methods give approximate solution with allowable tolerance, less time and possible for most cases

2 $\begingroup$ You have to elaborate on what you mean by "more time" and "less time". Analytical methods can be harder to derive but if derived are typically faster to compute than their computational counterparts. Examples would be solving the heat equation in a homogeneous cylindrical shell. $\endgroup$ – Frenzy Li Commented Aug 28, 2016 at 10:24

Analytical Method

When a problem is solved by means of analytical method its solution may be exact.
it doesn't follow any algorithm to solve a problem
This method provides exact solution to a problem
These problems are easy to solve and can be solved with pen and paper
When a problem is solved by mean of numerical method its solution may give an approximate number to a solution
It is the subject concerned with the construction, analysis and use of algorithms to solve a probme
It provides estimates that are very close to exact solution
This method is prone to erro

3 $\begingroup$ Many of your statements are wrong. 1) By definition the solution of a problem is exact. 2) We use an algorithm to compute the exact solution of, say, linear differential equations of 2nd order. 3) This statement contradicts 1). 4) On the contrary, many problems which admit a close form expression are not easy to solve. 7) No, at best the appropriate numerical method can be very accurate. 8) What do you mean? 9) No, Newton's method for computing the square root of 2 can be done by with pen and paper. $\endgroup$ – Carl Christian Commented Oct 6, 2019 at 20:46

The easiest way to understand analytical and numerical approaches is given below: pi=22/7 is the approximate value which is numerical 1/2=0.5 is the exact value means analytic.

$\begingroup$ This is the same as the previous answer. $\endgroup$ – Tengu Commented Oct 22, 2017 at 23:39

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Physics Numericals for Class 11: Mastering Concepts with Practical Problems

Kanwal Hafeez
July 21, 2023

1. introduction to physics numericals.

Physics is a branch of science that deals with the study of matter, energy, and their interactions. Numerical problem-solving plays a crucial role in mastering physics concepts and understanding their practical applications. By solving physics numericals, students can deepen their understanding of theories, equations, and principles while developing problem-solving skills. In this section, we will explore some fundamental concepts and units commonly encountered in physics numericals.

1.1 Basic Concepts and Units

Displacement: Displacement is the change in position of an object in a particular direction. It is a vector quantity and is measured in units of length, such as meters (m) in the International System of Units (SI).
Velocity: Velocity is the rate of change of displacement with time. It is also a vector quantity and is expressed in units of distance per time, such as meters per second (m/s).
Acceleration: Acceleration is the rate of change of velocity with time. It is a vector quantity and is measured in units of distance per time squared, like meters per second squared (m/s²).
Newton’s Second Law of Motion: This law states that the force acting on an object is directly proportional to the mass of the object and the acceleration produced. Mathematically, F = m * a, where F is the force in Newtons (N), m is the mass in kilograms (kg), and a is the acceleration in meters per second squared (m/s²).

Example Numerical:

A car of mass 1000 kg is accelerating uniformly at 2 m/s². Calculate the force acting on the car. Solution: Given: Mass (m) = 1000 kg, Acceleration (a) = 2 m/s² Using Newton’s Second Law of Motion: F = m * a F = 1000 kg * 2 m/s² = 2000 N

1.2 Measurement and Error Analysis

Precision and Accuracy: Precision refers to the closeness of measurements to each other, while accuracy is the closeness of a measurement to the true value. High precision means less variability among repeated measurements, while high accuracy implies less systematic error.
Absolute Error: The absolute error is the difference between the measured value and the true value of a quantity.
Relative Error: The relative error is the ratio of the absolute error to the true value, usually expressed as a percentage.
Significant Figures: Significant figures in a measurement are the digits that carry meaningful information. They include all the certain digits plus one uncertain or estimated digit.

A student measures the length of a pencil as 18.2 cm using a ruler with millimeter markings. The true length of the pencil is 18.0 cm. Calculate the absolute and relative error in the measurement. Solution: Given: Measured length = 18.2 cm, True length = 18.0 cm Absolute Error = |Measured length – True length| = |18.2 cm – 18.0 cm| = 0.2 cm Relative Error = (Absolute Error / True length) * 100 Relative Error = (0.2 cm / 18.0 cm) * 100 ≈ 1.11%

By mastering these basic concepts and units, and understanding measurement and error analysis, students can confidently tackle more complex physics numericals and gain a deeper appreciation for the practical aspects of the subject. Regular practice with solved numericals will undoubtedly enhance their problem-solving skills and pave the way for success in physics examinations and beyond.

2. Kinematics Numericals

2.1 displacement, velocity, and acceleration calculations.

In kinematics, we study the motion of objects without considering the forces causing that motion. This section focuses on problems related to displacement, velocity, and acceleration.

Numerical 1:

A car starts from rest and accelerates uniformly at 3 m/s² for 8 seconds. Calculate its final velocity and displacement during this time. Solution: Given: Initial velocity (u) = 0 m/s, Acceleration (a) = 3 m/s², Time (t) = 8 seconds Using the equation v = u + at: Final velocity (v) = 0 + (3 m/s² * 8 s) = 24 m/s Using the equation s = ut + (1/2)at²: Displacement (s) = (0 m/s * 8 s) + (0.5 * 3 m/s² * (8 s)²) = 96 m

Numerical 2:

An object is moving with a constant velocity of 5 m/s. Calculate the displacement of the object after 10 seconds. Solution: Given: Velocity (v) = 5 m/s, Time (t) = 10 seconds Using the equation s = vt: Displacement (s) = 5 m/s * 10 s = 50 m

Numerical 3:

A train decelerates uniformly at 2 m/s² until it comes to rest. If its initial velocity was 20 m/s, calculate the time taken for the train to stop. Solution: Given: Initial velocity (u) = 20 m/s, Acceleration (a) = -2 m/s² (negative as it’s deceleration), Final velocity (v) = 0 m/s Using the equation v = u + at: 0 m/s = 20 m/s + (-2 m/s² * t) Solving for t: t = 20 m/s / 2 m/s² = 10 seconds

Numerical 4:

An athlete runs along a straight track. He covers the first 40 meters in 5 seconds and then stops for 2 seconds. After that, he runs at a constant velocity of 6 m/s for 8 seconds. Calculate the total distance covered by the athlete. Solution: Total distance covered = Distance in the first 5 seconds + Distance during constant velocity Distance in the first 5 seconds = 40 meters (given) Distance during constant velocity = 6 m/s * 8 s = 48 meters Total distance covered = 40 meters + 48 meters = 88 meters

Numerical 5:

A stone is dropped from the top of a tower 100 meters high. Calculate its velocity when it hits the ground. Solution: Given: Initial velocity (u) = 0 m/s (since it’s dropped), Displacement (s) = 100 meters, Acceleration (a) = 9.8 m/s² (acceleration due to gravity, downwards) Using the equation v² = u² + 2as: Final velocity (v) = √(0 m/s)² + 2 * 9.8 m/s² * 100 m ≈ 44.29 m/s (rounded to two decimal places)

2.2 Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, under the influence of gravity. In this section, we will solve numerical problems related to projectile motion.

A ball is thrown horizontally from the top of a cliff 80 meters high with an initial velocity of 20 m/s. Calculate the time it takes for the ball to reach the ground and the horizontal distance traveled. Solution: Given: Initial vertical velocity (uy) = 0 m/s (thrown horizontally), Displacement in the y-direction (sy) = -80 meters (negative as it’s downward), Acceleration in the y-direction (ay) = -9.8 m/s² (acceleration due to gravity, downward), Horizontal velocity (ux) = 20 m/s Time to reach the ground can be calculated using the equation: t = (2 * |sy| / |ay|)^(1/2) t = (2 * 80 m / 9.8 m/s²)^(1/2) ≈ 4.04 seconds (rounded to two decimal places) The horizontal distance (dx) can be calculated using the equation: dx = ux * t dx = 20 m/s * 4.04 s ≈ 80.8 meters (rounded to one decimal place)

A soccer player kicks a ball at an angle of 30 degrees above the horizontal with a velocity of 15 m/s. Determine the maximum height reached by the ball and the total time of flight. Solution: Given: Launch angle (θ) = 30 degrees, Launch velocity (v) = 15 m/s, Acceleration due to gravity (g) = 9.8 m/s² The time taken to reach the maximum height can be calculated using the equation: t = uy / g uy = v * sin(θ) = 15 m/s * sin(30 degrees) ≈ 7.5 m/s t = 7.5 m/s / 9.8 m/s² ≈ 0.77 seconds (rounded to two decimal places) The maximum height (H) can be calculated using the equation: H = (uy²) / (2 * g) H = (7.5 m/s)² / (2 * 9.8 m/s²) ≈ 2.29 meters (rounded to two decimal places) The total time of flight can be calculated using: Total time = 2 * t Total time ≈ 2 * 0.77 s ≈ 1.54 seconds (rounded to two decimal places)

A stone is thrown horizontally from the top of a cliff with a velocity of 12 m/s. Calculate the horizontal distance it travels before hitting the ground. Solution: Given: Initial horizontal velocity (ux) = 12 m/s, Time of flight (t) = ? The horizontal distance (dx) can be calculated using the equation: dx = ux * t We need to find the time of flight first. Since the stone is thrown horizontally, the vertical component of velocity (uy) is 0 m/s. The time of flight (t) can be calculated using: t = 2 * (vertical displacement) / g Vertical displacement = 0 m (since the stone starts and ends at the same height) t = 2 * 0 m / 9.8 m/s² = 0 seconds Now, the horizontal distance can be calculated: dx = 12 m/s * 0 s = 0 meters

A football is kicked from the ground with an initial velocity of 25 m/s at an angle of 45 degrees above the horizontal. Calculate the range of the football. Solution: Given: Launch angle (θ) = 45 degrees, Launch velocity (v) = 25 m/s, Acceleration due to gravity (g) = 9.8 m/s² The horizontal distance (range) can be calculated using the equation: Range (R) = (v² * sin(2θ)) / g Range (R) = (25 m/s)² * sin(2 * 45 degrees) / 9.8 m/s² ≈ 31.88 meters (rounded to two decimal places)

A ball is thrown vertically upward with an initial velocity of 30 m/s. Calculate the maximum height reached by the ball and the time taken to reach it. Solution : Given: Initial velocity (u) = 30 m/s (upward), Acceleration due to gravity (g) = 9.8 m/s² The maximum height (H) can be calculated using the equation: H = (u²) / (2 * g) H = (30 m/s)² / (2 * 9.8 m/s²) ≈ 45.92 meters (rounded to two decimal places) The time taken to reach the maximum height (t) can be calculated using the equation: t = u / g t = 30 m/s / 9.8 m/s² ≈ 3.06 seconds (rounded to two decimal places)

2.3 Circular Motion Problems

Circular motion involves an object moving in a circular path. In this section, we will explore numerical problems related to circular motion.

A car moves around a circular track with a radius of 50 meters at a constant speed of 20 m/s. Calculate its centripetal acceleration. Solution: Given: Radius of the circular track (r) = 50 meters, Speed of the car (v) = 20 m/s Centripetal acceleration (a) can be calculated using the equation: a = v² / r a = (20 m/s)² / 50 m ≈ 8 m/s²

A stone tied to a string is whirled around in a horizontal circle with a constant speed of 6 m/s. If the radius of the circle is 2 meters, calculate the centripetal force acting on the stone. Solution: Given: Speed of the stone (v) = 6 m/s, Radius of the circular path (r) = 2 meters, Mass of the stone (m) = ? Centripetal force (F) can be calculated using the equation: F = m * a Centripetal acceleration (a) = v² / r = (6 m/s)² / 2 m = 18 m/s² Since F = m * a, we need to find the mass (m) of the stone to calculate the centripetal force.

A Ferris wheel has a radius of 20 meters and completes one revolution in 40 seconds. Calculate the linear speed of a passenger at the top and bottom of the wheel. Solution: Given: Radius of the Ferris wheel (r) = 20 meters, Time for one revolution (T) = 40 seconds Linear speed at the top can be calculated using the equation: v_top = 2 * π * r / T v_top = 2 * 3.14 * 20 m / 40 s ≈ 3.14 m/s (rounded to two decimal places) Linear speed at the bottom is the same as at the top because the distance traveled in one revolution is the same.

A car moves around a horizontal circular track with a radius of 100 meters. If the car completes one revolution in 60 seconds, calculate the magnitude of the centripetal acceleration and the net force acting on the car if its mass is 1500 kg. Solution: Given: Radius of the circular track (r) = 100 meters, Time for one revolution (T) = 60 seconds, Mass of the car (m) = 1500 kg Centripetal acceleration (a) can be calculated using the equation: a = 4 * π² * r / T² a = 4 * 3.14² * 100 m / (60 s)² ≈ 2.62 m/s² (rounded to two decimal places) Net force (F) acting on the car can be calculated using the equation: F = m * a F = 1500 kg * 2.62 m/s² ≈ 3930 N (rounded to two decimal places)

A small object is tied to a string and whirled around in a horizontal circle with a radius of 0.5 meters. If the speed of the object is 4 m/s, calculate the period of its motion and the centripetal acceleration. Solution : Given: Radius of the circular path (r) = 0.5 meters, Speed of the object (v) = 4 m/s Period (T) of motion can be calculated using the equation: T = 2 * π * r / v T = 2 * 3.14 * 0.5 m / 4 m/s ≈ 0.785 seconds (rounded to three decimal places) Centripetal acceleration (a) can be calculated using the equation: a = v² / r a = (4 m/s)² / 0.5 m ≈ 32 m/s²

By solving these kinematics and circular motion numericals, students can gain a strong grasp of the fundamental principles of motion and further enhance their problem-solving skills in physics. Regular practice and a solid understanding of these concepts will enable them to tackle more complex physics problems with confidence.

3. Laws of Motion Numericals

3.1 newton’s first law applications.

Newton’s First Law of Motion states that an object at rest will remain at rest, and an object in motion will continue moving with a constant velocity unless acted upon by an external force. This law is also known as the law of inertia. Let’s explore some numerical problems that apply Newton’s First Law.

A book is placed on a table and remains at rest. Explain the forces acting on the book and how Newton’s First Law is demonstrated in this scenario. Solution: In this scenario, the forces acting on the book are gravity (acting downwards) and the normal force (acting upwards) exerted by the table. According to Newton’s First Law, the book remains at rest because the forces acting on it are balanced. The gravitational force pulling the book downwards is balanced by the normal force exerted by the table in the opposite direction. As a result, there is no net force acting on the book, and it stays at rest.

A hockey puck slides on a frictionless ice surface and keeps moving with a constant velocity. Explain how Newton’s First Law is demonstrated in this situation. Solution: In this situation, the only force acting on the hockey puck is its initial forward push. Once the puck starts moving, there is no significant external force acting on it to change its velocity or direction. Newton’s First Law explains that the puck will continue moving with the same velocity (constant speed and direction) as there is no net external force acting on it. The absence of friction on the ice surface allows the puck to maintain its motion without slowing down or changing direction.

A car comes to a stop after hitting the brakes. Explain how Newton’s First Law is involved in this scenario. Solution: When the driver hits the brakes of the car, frictional forces between the tires and the road come into play. The frictional force opposes the forward motion of the car and acts in the opposite direction. As a result, there is a net external force acting on the car in the direction opposite to its initial motion. According to Newton’s First Law, the car will tend to maintain its initial forward motion due to its inertia, but the net external force from braking overcomes this inertia, causing the car to slow down and eventually come to a stop.

3.2 Newton’s Second Law Problems

Newton’s Second Law of Motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The law is mathematically expressed as F = ma, where F is the net force, m is the mass of the object, and a is the acceleration. Let’s solve some numerical problems involving Newton’s Second Law.

A force of 40 N is applied to an object with a mass of 5 kg. Calculate the acceleration of the object. Solution: Given: Force (F) = 40 N, Mass (m) = 5 kg Using Newton’s Second Law: a = F / m a = 40 N / 5 kg = 8 m/s²

A 20 kg box is pushed with a force of 100 N. If the coefficient of friction between the box and the floor is 0.4, calculate the acceleration of the box. Solution: Given: Force applied (F) = 100 N, Mass of the box (m) = 20 kg, Coefficient of friction (μ) = 0.4 Frictional force (F_friction) = μ * (normal force) Normal force (F_normal) = m * g (where g is the acceleration due to gravity, approximately 9.8 m/s²) F_friction = 0.4 * (20 kg * 9.8 m/s²) = 78.4 N Net force (F_net) = F – F_friction = 100 N – 78.4 N = 21.6 N Using Newton’s Second Law: a = F_net / m a = 21.6 N / 20 kg ≈ 1.08 m/s² (rounded to two decimal places)

An object is pushed with a force of 30 N and accelerates at 5 m/s². Calculate the mass of the object. Solution: Given: Force (F) = 30 N, Acceleration (a) = 5 m/s² Using Newton’s Second Law: m = F / a m = 30 N / 5 m/s² = 6 kg

3.3 Newton’s Third Law Numericals

Newton’s Third Law of Motion states that for every action, there is an equal and opposite reaction. In other words, if object A exerts a force on object B, then object B exerts an equal and opposite force on object A. Let’s solve some numerical problems based on Newton’s Third Law.

A person pushes a wall with a force of 50 N. Explain the reaction force according to Newton’s Third Law.

According to Newton’s Third Law, the wall exerts an equal and opposite force of 50 N on the person. When the person pushes the wall, the wall pushes back on the person with the same magnitude of force but in the opposite direction.

A rocket is launched into space by expelling gases at a high velocity. Explain the forces involved and their directions based on Newton’s Third Law.

The rocket propulsion is based on Newton’s Third Law. The rocket engine expels high-velocity gases downward (action). As a reaction to the downward expulsion of gases, the rocket experiences an equal and opposite force pushing it upward. This upward force propels the rocket into space.

A ball is dropped on the floor, and it bounces back upward. Explain the forces involved and their directions according to Newton’s Third Law.

When the ball hits the floor, it exerts a downward force on the floor (action). According to Newton’s Third Law, the floor exerts an equal and opposite force on the ball (reaction). This upward reaction force causes the ball to bounce back upward.

By practicing these numerical problems related to Newton’s Laws of Motion, students can better comprehend the practical applications of these laws and improve their problem-solving abilities in physics . Understanding these foundational principles is crucial for building a strong foundation in mechanics and other branches of physics.

4. Work, Energy, and Power Numericals

4.1 work-energy theorem applications.

The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy. Work is done on an object when a force is applied to it and it moves in the direction of the force. Let’s explore some numerical problems that apply the Work-Energy Theorem.

A force of 20 N is applied to push a box a distance of 5 meters along a horizontal surface. Calculate the work done on the box. Solution: Given: Force (F) = 20 N, Displacement (s) = 5 meters Work done (W) can be calculated using the equation: W = F * s W = 20 N * 5 m = 100 Joules

A spring with a spring constant of 200 N/m is compressed by 0.1 meters. Calculate the work done to compress the spring. Solution: Given: Spring constant (k) = 200 N/m, Compression (x) = 0.1 meters Work done (W) to compress the spring can be calculated using the equation: W = (1/2) * k * x² W = (0.5) * 200 N/m * (0.1 m)² = 1 Joule

A ball is thrown vertically upward with an initial velocity of 15 m/s. Calculate the work done by gravity on the ball during its ascent until it reaches its highest point. Solution: At the highest point of its trajectory, the ball’s vertical velocity becomes 0 m/s. Therefore, its kinetic energy at the highest point is 0. Work done by gravity (W) can be calculated using the equation: W = ΔKE (change in kinetic energy) Since the kinetic energy at the highest point is 0, the work done by gravity is also 0.

4.2 Conservation of Mechanical Energy Problems

Conservation of Mechanical Energy states that the total mechanical energy of an isolated system remains constant if only conservative forces, like gravity, are acting on it. In these numerical problems, we’ll explore scenarios where mechanical energy is conserved.

A roller coaster car with a mass of 500 kg starts from the top of a hill with an initial velocity of 10 m/s. Calculate its total mechanical energy at the top of the hill and when it reaches the bottom, assuming no frictional forces are acting. Solution: Given: Mass (m) = 500 kg, Initial velocity (v_initial) = 10 m/s, Height of the hill (h) = ? At the top of the hill, the roller coaster’s kinetic energy is 0 since its velocity is 0. Therefore, the total mechanical energy is equal to its potential energy: Potential energy (PE) at the top = m * g * h At the bottom of the hill, the roller coaster’s height is 0, and its potential energy is 0. Therefore, the total mechanical energy is equal to its kinetic energy: Kinetic energy (KE) at the bottom = (1/2) * m * v_final² Since mechanical energy is conserved, the total mechanical energy at the top is equal to the total mechanical energy at the bottom: m * g * h = (1/2) * m * v_final² Solving for v_final: v_final = √(2 * g * h) Substitute the given values to find v_final.

A pendulum swings back and forth between two points. At its highest point (Point A), it has a potential energy of 80 J, and at its lowest point (Point B), it has a potential energy of 20 J. Assuming no energy losses due to friction, calculate its speed at Point B. Solution: At Point A, the kinetic energy is 0, so the total mechanical energy is equal to its potential energy: Total mechanical energy at A = Potential energy (PE) at A = 80 J At Point B, the potential energy is 20 J, and the kinetic energy is: Total mechanical energy at B = Potential energy (PE) at B + Kinetic energy (KE) at B Total mechanical energy at B = 20 J + KE at B Since mechanical energy is conserved, the total mechanical energy at A is equal to the total mechanical energy at B: 80 J = 20 J + KE at B Solving for KE at B: KE at B = 80 J – 20 J = 60 J The kinetic energy at Point B is 60 J. Use the formula for kinetic energy (KE = (1/2) * m * v²) to find the speed (v) at Point B.

A skier starts from rest at the top of a hill and slides down. At the bottom of the hill, the skier has a speed of 15 m/s. Calculate the loss in mechanical energy due to non-conservative forces (like friction). Solution: Given: Initial velocity (v_initial) = 0 m/s, Final velocity (v_final) = 15 m/s The change in kinetic energy (ΔKE) can be calculated using the equation: ΔKE = (1/2) * m * (v_final² – v_initial²) Since the skier starts from rest (v_initial = 0 m/s), the change in kinetic energy simplifies to: ΔKE = (1/2) * m * v_final² Calculate the change in kinetic energy, and this value represents the loss in mechanical energy due to non-conservative forces like friction.

4.3 Power Calculations

Power is the rate at which work is done or the rate at which energy is transferred or converted. It is the time derivative of work or energy. Let’s explore some numerical problems involving power calculations.

A motor lifts a 500 kg crate vertically at a constant speed of 2 m/s. Calculate the power output of the motor. Solution: Given: Mass (m) = 500 kg, Vertical speed (v) = 2 m/s Work done (W) by the motor to lift the crate can be calculated using the equation: W = m * g * h, where h is the height lifted. Since the crate moves at a constant speed, there is no change in height (h) during the lift. Therefore, the work done (W) is 0, as no energy is transferred in the vertical direction. Power (P) can be calculated using the equation: P = W / t, where t is the time taken to lift the crate. Since W = 0, the power output of the motor is also 0.

A machine does 2000 Joules of work in 5 seconds. Calculate the power output of the machine. Solution: Given: Work done (W) = 2000 Joules, Time (t) = 5 seconds Power (P) can be calculated using the equation: P = W / t P = 2000 Joules / 5 seconds = 400 Watts

A pump lifts 100 liters of water from a well to the ground level in 2 minutes. The height of the well is 5 meters. Calculate the power output of the pump. Solution: Given: Volume of water (V) = 100 liters = 100,000 cm³ = 100,000 mL, Height (h) = 5 meters, Time (t) = 2 minutes = 120 seconds Work done (W) by the pump to lift the water can be calculated using the equation: W = m * g * h, where m is the mass of the water. Since the density of water is approximately 1 g/cm³, the mass of 100 liters (100,000 mL) of water is 100,000 grams (100 kg). W = 100 kg * 9.8 m/s² * 5 meters = 4,900 Joules Power (P) can be calculated using the equation: P = W / t P = 4,900 Joules / 120 seconds ≈ 40.83 Watts (rounded to two decimal places)

By solving these work, energy, and power numericals, students can better understand the concepts of energy transformation, conservation, and power calculations in various physical scenarios. These concepts play a fundamental role in understanding how energy works in real-world situations and are essential for solving more complex problems in physics .

FAQ’s

1. what is the purpose of solving physics numericals in class 11.

Solving physics numericals in Class 11 serves multiple purposes. It helps students develop problem-solving skills, gain a deeper understanding of theoretical concepts, and apply mathematical techniques to real-world scenarios. Numericals also reinforce the application of fundamental principles, preparing students for more complex topics in higher classes and future scientific endeavors.

2. How do I approach solving physics numericals effectively?

To effectively solve physics numericals, follow these steps:

a) Carefully read the problem statement and identify the given data and required quantities.
b) Choose the appropriate formula or equations that relate to the given scenario.
c) Substitute the given values into the equations and solve for the unknown quantities.
d) Pay attention to units and dimensions to ensure correct results.
e) Check your answers and reevaluate if needed.

3. I find physics numericals challenging. How can I improve my problem-solving skills?

Improving problem-solving skills in physics requires practice and a systematic approach. Start with simple numericals and gradually progress to more complex ones. Seek help from your teachers or peers if you encounter difficulties. Analyze your mistakes and learn from them. Engage in group discussions, participate in online forums, and explore additional resources like textbooks and video tutorials to reinforce your understanding.

4. Can I rely solely on theory and skip practicing physics numericals?

While understanding theoretical concepts is essential, practicing physics numericals is equally crucial. Numericals provide a practical application of theory, helping you grasp the concepts more effectively. They also enhance problem-solving abilities, critical thinking, and mathematical skills, which are valuable in various fields of study and everyday life.

5. Are there any specific strategies to excel in physics numericals during exams?

Yes, there are strategies to excel in physics numericals during exams:

a) Familiarize yourself with common numerical patterns and formulas beforehand.
b) Practice solving a variety of numericals to gain confidence and speed.
c) Create a formula sheet with key equations for quick reference during exams.
d) Read each question carefully, paying attention to units and significant figures.
e) If you get stuck, move on to the next question and come back later with a fresh perspective.
f) Review and revise your solutions to avoid simple mistakes and maximize your score.

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An algorithm is a process or set of rules which must be followed to complete a particular task. This is basically the step-by-step procedure to complete any task. All the tasks are followed a particular algorithm, from making a cup of tea to make high scalable software. This is the way to divide a task into several parts. If we draw an algorithm to complete a task then the task will be easier to complete.

The algorithm is used for,

To develop a framework for instructing computers.
Introduced notation of basic functions to perform basic tasks.
For defining and describing a big problem in small parts, so that it is very easy to execute.

Characteristics of Algorithm

An algorithm should be defined clearly.
An algorithm should produce at least one output.
An algorithm should have zero or more inputs.
An algorithm should be executed and finished in finite number of steps.
An algorithm should be basic and easy to perform.
Each step started with a specific indentation like, “Step-1”,
There must be “Start” as the first step and “End” as the last step of the algorithm.

Let’s take an example to make a cup of tea,

Step 1: Start

Step 2: Take some water in a bowl.

Step 3: Put the water on a gas burner .

Step 4: Turn on the gas burner

Step 5: Wait for some time until the water is boiled.

Step 6: Add some tea leaves to the water according to the requirement.

Step 7: Then again wait for some time until the water is getting colorful as tea.

Step 8: Then add some sugar according to taste.

Step 9: Again wait for some time until the sugar is melted.

Step 10: Turn off the gas burner and serve the tea in cups with biscuits.

Step 11: End

Here is an algorithm for making a cup of tea. This is the same for computer science problems.

There are some basics steps to make an algorithm:

Start – Start the algorithm
Input – Take the input for values in which the algorithm will execute.
Conditions – Perform some conditions on the inputs to get the desired output.
Output – Printing the outputs.
End – End the execution.

Let’s take some examples of algorithms for computer science problems.

Example 1. Swap two numbers with a third variable

Step 1: Start Step 2: Take 2 numbers as input. Step 3: Declare another variable as “temp”. Step 4: Store the first variable to “temp”. Step 5: Store the second variable to the First variable. Step 6: Store the “temp” variable to the 2nd variable. Step 7: Print the First and second variables. Step 8: End

Example 2. Find the area of a rectangle

Step 1: Start Step 2: Take the Height and Width of the rectangle as input. Step 3: Declare a variable as “area” Step 4: Multiply Height and Width Step 5: Store the multiplication to “Area”, (its look like area = Height x Width) Step 6: Print “area”; Step 7: End

Example 3. Find the greatest between 3 numbers.

Step 1: Start Step 2: Take 3 numbers as input, say A, B, and C. Step 3: Check if(A>B and A>C) Step 4: Then A is greater Step 5: Print A Step 6 : Else Step 7: Check if(B>A and B>C) Step 8: Then B is greater Step 9: Print B Step 10: Else C is greater Step 11 : Print C Step 12: End

Advantages of Algorithm

An algorithm uses a definite procedure.
It is easy to understand because it is a step-by-step definition.
The algorithm is easy to debug if there is any error happens.
It is not dependent on any programming language
It is easier for a programmer to convert it into an actual program because the algorithm divides a problem into smaller parts.

Disadvantages of Algorithms

An algorithm is Time-consuming, there is specific time complexity for different algorithms.
Large tasks are difficult to solve in Algorithms because the time complexity may be higher, so programmers have to find a good efficient way to solve that task.
Looping and branching are difficult to define in algorithms.

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What is Problem Solving Algorithm?, Steps, Representation

Table of Contents

1 What is Problem Solving Algorithm?
2 Definition of Problem Solving Algorithm
3.1 Analysing the Problem
3.2 Developing an Algorithm
3.4 Testing and Debugging
4.1 Flowchart
4.2 Pseudo code

What is Problem Solving Algorithm?

Computers are used for solving various day-to-day problems and thus problem solving is an essential skill that a computer science student should know. It is pertinent to mention that computers themselves cannot solve a problem. Precise step-by-step instructions should be given by us to solve the problem.

Thus, the success of a computer in solving a problem depends on how correctly and precisely we define the problem, design a solution (algorithm) and implement the solution (program) using a programming language.

Thus, problem solving is the process of identifying a problem, developing an algorithm for the identified problem and finally implementing the algorithm to develop a computer program.

Definition of Problem Solving Algorithm

These are some simple definition of problem solving algorithm which given below:

Steps for Problem Solving

When problems are straightforward and easy, we can easily find the solution. But a complex problem requires a methodical approach to find the right solution. In other words, we have to apply problem solving techniques.

Problem solving begins with the precise identification of the problem and ends with a complete working solution in terms of a program or software. Key steps required for solving a problem using a computer.

For Example: Suppose while driving, a vehicle starts making a strange noise. We might not know how to solve the problem right away. First, we need to identify from where the noise is coming? In case the problem cannot be solved by us, then we need to take the vehicle to a mechanic.

The mechanic will analyse the problem to identify the source of the noise, make a plan about the work to be done and finally repair the vehicle in order to remove the noise. From the example, it is explicit that, finding the solution to a problem might consist of multiple steps.

Following are Steps for Problem Solving :

Analysing the Problem

Developing an algorithm, testing and debugging.

It is important to clearly understand a problem before we begin to find the solution for it. If we are not clear as to what is to be solved, we may end up developing a program which may not solve our purpose.

Thus, we need to read and analyse the problem statement carefully in order to list the principal components of the problem and decide the core functionalities that our solution should have. By analysing a problem, we would be able to figure out what are the inputs that our program should accept and the outputs that it should produce.

It is essential to device a solution before writing a program code for a given problem. The solution is represented in natural language and is called an algorithm. We can imagine an algorithm like a very well-written recipe for a dish, with clearly defined steps that, if followed, one will end up preparing the dish.

We start with a tentative solution plan and keep on refining the algorithm until the algorithm is able to capture all the aspects of the desired solution. For a given problem, more than one algorithm is possible and we have to select the most suitable solution.

After finalising the algorithm, we need to convert the algorithm into the format which can be understood by the computer to generate the desired solution. Different high level programming languages can be used for writing a program. It is equally important to record the details of the coding procedures followed and document the solution. This is helpful when revisiting the programs at a later stage.

The program created should be tested on various parameters. The program should meet the requirements of the user. It must respond within the expected time. It should generate correct output for all possible inputs. In the presence of syntactical errors, no output will be obtained. In case the output generated is incorrect, then the program should be checked for logical errors, if any.

Software industry follows standardised testing methods like unit or component testing, integration testing, system testing, and acceptance testing while developing complex applications. This is to ensure that the software meets all the business and technical requirements and works as expected.

The errors or defects found in the testing phases are debugged or rectified and the program is again tested. This continues till all the errors are removed from the program. Once the software application has been developed, tested and delivered to the user, still problems in terms of functioning can come up and need to be resolved from time to time.

The maintenance of the solution, thus, involves fixing the problems faced by the user, answering the queries of the user and even serving the request for addition or modification of features.

Representation of Algorithms

Using their algorithmic thinking skills, the software designers or programmers analyse the problem and identify the logical steps that need to be followed to reach a solution. Once the steps are identified, the need is to write down these steps along with the required input and desired output.

There are two common methods of representing an algorithm —flowchart and pseudocode. Either of the methods can be used to represent an algorithm while keeping in mind the following:

It showcases the logic of the problem solution, excluding any implementational details.
It clearly reveals the flow of control during execution of the program.

A flowchart is a visual representation of an algorithm . A flowchart is a diagram made up of boxes, diamonds and other shapes, connected by arrows. Each shape represents a step of the solution process and the arrow represents the order or link among the steps.

A flow chart is a step by step diagrammatic representation of the logic paths to solve a given problem. Or A flowchart is visual or graphical representation of an algorithm .

The flowcharts are pictorial representation of the methods to b used to solve a given problem and help a great deal to analyze the problem and plan its solution in a systematic and orderly manner. A flowchart when translated in to a proper computer language, results in a complete program.

Advantages of Flowcharts:

The flowchart shows the logic of a problem displayed in pictorial fashion which felicitates easier checking of an algorithm
The Flowchart is good means of communication to other users. It is also a compact means of recording an algorithm solution to a problem.
The flowchart allows the problem solver to break the problem into parts. These parts can be connected to make master chart.
The flowchart is a permanent record of the solution which can be consulted at a later time.

Differences between Algorithm and Flowchart


1	A method of representing the step-by-step logical procedure for solving a problem.	Flowchart is diagrammatic representation of an algorithm. It is constructed using different types of boxes and symbols.
2	It contains step-by-step English descriptions, each step representing a particular operation leading to solution of problem.	The flowchart employs a series of blocks and arrows, each of which represents a particular step in an algorithm.
3	These are particularly useful for small problems.	These are useful for detailed representations of complicated programs.
4	For complex programs, algorithms prove to be Inadequate.	For complex programs, Flowcharts prove to be adequate.

Pseudo code

The Pseudo code is neither an algorithm nor a program. It is an abstract form of a program. It consists of English like statements which perform the specific operations. It is defined for an algorithm. It does not use any graphical representation.

In pseudo code , the program is represented in terms of words and phrases, but the syntax of program is not strictly followed.

Advantages of Pseudocode

Before writing codes in a high level language, a pseudocode of a program helps in representing the basic functionality of the intended program.
By writing the code first in a human readable language, the programmer safeguards against leaving out any important step. Besides, for non-programmers, actual programs are difficult to read and understand.
But pseudocode helps them to review the steps to confirm that the proposed implementation is going to achieve the desire output.

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Solving Complex Problems with Numerical Methods: Case Studies

In the ever-evolving landscape of science and engineering, solving complex problems often requires a sophisticated approach that transcends conventional analytical solutions. At the forefront of this innovative approach is the realm of numerical methods, a dynamic branch of applied mathematics that assumes a pivotal role in addressing challenges deemed too intricate or time-consuming for traditional methodologies. Within the confines of this comprehensive blog, we embark on an exploration of the multifaceted world of numerical methods, unraveling the intricate algorithms and computational techniques that underpin their efficacy. Delving deeper, we scrutinize real-world case studies that stand as testaments to the instrumental role these numerical techniques play in unraveling and solving complex problems across diverse domains. From the intricacies of fluid dynamics in aircraft design to the structural analysis inherent in civil engineering, numerical methods emerge as indispensable tools, facilitating simulations and computations that pave the way for enhanced problem-solving capabilities. This examination not only sheds light on the versatility of numerical methods but also underscores their practical significance in pushing the boundaries of what is achievable in the fields of science and engineering. If you are seeking assistance with your numerical methods assignment , this blog will provide valuable insights and guidance to help you master these essential techniques.

Solving Complex Problems with Numerical Methods-Case Studies

As we navigate through these case studies, it becomes evident that numerical methods are not mere computational tools; they are enablers of innovation, providing engineers and scientists with the means to analyze, optimize, and design intricate systems with unprecedented precision. The synergy between theoretical understanding and computational prowess becomes apparent as numerical simulations facilitate a deeper comprehension of complex phenomena, guiding decision-making processes and design iterations. Within the dynamic landscape of numerical methods, the amalgamation of Finite Element Analysis (FEA), Computational Fluid Dynamics (CFD), linear programming, genetic algorithms, and Fourier Transforms showcases the diversity of techniques employed to surmount specific challenges. Yet, amidst the triumphs lie challenges and limitations that demand meticulous attention—numerical stability, convergence, and dimensionality complexities necessitate a judicious approach to ensure the reliability and accuracy of results. As we reflect on the present and gaze into the future, the trajectory of numerical methods seems poised for continued evolution. The integration of high-performance computing, parallel processing, and machine learning promises to push the boundaries further, enabling the resolution of increasingly intricate and large-scale problems. Moreover, emerging technologies like meshless methods and quantum computing tantalize with the prospect of transforming the landscape of numerical solutions. In essence, this exploration serves as a testament to the enduring significance of numerical methods in shaping the trajectory of problem-solving in the realms of science and technology, heralding a future where innovation and numerical ingenuity converge to unravel the mysteries of the most complex challenges.

Understanding Numerical Methods:

Understanding numerical methods is paramount in addressing complex problems that transcend the realm of analytical solutions. Numerical methods, a branch of applied mathematics, provide a systematic approach to solving mathematical problems through computational techniques and algorithms. In scenarios where equations lack closed-form solutions, numerical methods offer a practical means of obtaining approximate answers. This section delves into the foundational aspects of numerical methods, exploring their role in tackling intricate problems across diverse fields. These methods become particularly crucial when dealing with real-world phenomena characterized by complex equations, such as fluid dynamics, structural analysis, optimization, and data fitting. By breaking down these intricate problems into manageable components, numerical methods empower scientists and engineers to simulate and analyze systems that would be otherwise intractable. The discussion encompasses the significance of numerical simulations in fields like aircraft design, where Computational Fluid Dynamics (CFD) revolutionizes aerodynamic analyses, and civil engineering, where the Finite Element Method (FEM) aids in structural optimization. Through a deeper understanding of numerical methods, professionals can harness these tools to navigate the complexities of their respective fields, pushing the boundaries of problem-solving capabilities and contributing to advancements in science and engineering.

Case Study 1: Fluid Dynamics Simulation for Aircraft Design:

Delves into the realm of Fluid Dynamics Simulation, specifically focusing on its pivotal role in aircraft design. The aerodynamics of aircraft, governed by the complex Navier-Stokes equations, pose intricate challenges that lack analytical solutions, particularly when dealing with intricate geometries and turbulent flows. Numerical methods, such as Finite Element Analysis (FEA) and Computational Fluid Dynamics (CFD), have become indispensable tools in this domain. Engineers leverage these methods to simulate the airflow around an aircraft by inputting its geometry, providing essential data on lift, drag, and stability. This approach revolutionizes the traditional aircraft design process, replacing time-consuming and costly wind tunnel testing with efficient computational simulations. The ability to iterate designs rapidly enhances the optimization of aerodynamic performance and fuel efficiency, crucial factors in the aerospace industry. Fluid dynamics simulations not only aid in understanding the complex interactions between air and aircraft surfaces but also contribute to the development of innovative designs that push the boundaries of aerodynamic efficiency. As a result, numerical methods in Fluid Dynamics Simulation have become integral to the continuous evolution of aircraft design, showcasing the transformative power of computational techniques in addressing intricate challenges in engineering and pushing the boundaries of what is achievable in the aerospace sector.

Case Study 2: Structural Analysis in Civil Engineering:

In the realm of civil engineering, where the safety and reliability of structures are paramount, numerical methods have proven instrumental in tackling the intricacies of structural analysis. The complexity of real-world structures often leads to mathematical models that defy analytical solutions. Enter the Finite Element Method (FEM), a numerical technique that has revolutionized the field. Engineers employ FEM to break down complex structures into smaller, manageable elements, allowing them to simulate and understand the behavior of the entire structure under various loads and environmental conditions. This approach provides crucial insights into factors such as stress distribution, deformation, and potential failure points. By leveraging numerical simulations, civil engineers can optimize designs for factors like safety, cost-effectiveness, and structural efficiency. From skyscrapers to bridges, FEM enables engineers to explore diverse scenarios and refine designs iteratively. The ability to predict how a structure will respond to different forces and conditions is invaluable, reducing the need for costly physical prototypes and ensuring that structures meet stringent safety standards. As urbanization accelerates and infrastructure demands intensify, the role of numerical methods in structural analysis becomes increasingly vital, offering a sophisticated means to design and assess structures with a level of detail and accuracy that analytical methods alone cannot provide. Through numerical simulations, civil engineers are not merely constructing buildings and bridges; they are crafting structures with a profound understanding of their dynamic behavior and resilience in the face of real-world challenges.

Case Study 3: Optimization in Operations Research:

Case Study 3 focuses on the critical role of numerical methods in optimization problems within the realm of operations research. In the complex landscape of supply chain management, where numerous variables influence decision-making, numerical methods offer indispensable solutions. Algorithms like the Simplex method for linear programming and genetic algorithms for nonlinear optimization prove instrumental in streamlining logistics, minimizing costs, and enhancing overall operational efficiency. Consider a scenario where a company aims to optimize its supply chain by balancing transportation costs, inventory levels, and production rates. Numerical optimization techniques allow for the identification of optimal solutions that minimize expenses while maximizing efficiency. The Simplex method, a widely used linear programming algorithm, iteratively refines the solution space until an optimal solution is reached. On the other hand, genetic algorithms, inspired by natural selection, provide robust solutions in nonlinear optimization scenarios by mimicking evolutionary processes. These numerical methods empower businesses to make data-driven decisions, leading to improved resource allocation, reduced wastage, and heightened competitiveness. As industries become increasingly complex and interconnected, the ability to leverage numerical optimization techniques becomes paramount in navigating the intricate web of logistical challenges, ensuring sustainable and efficient operations in the dynamic landscape of operations research.

Case Study 4: Image and Signal Processing:

Image and signal processing constitute pivotal domains in the application of numerical methods, where computational techniques are harnessed to extract meaningful information from visual or auditory data. In image processing, algorithms such as convolution, edge detection, and image segmentation play a crucial role in tasks ranging from facial recognition to medical imaging. Numerical methods, like Fourier Transforms, are indispensable in signal processing, allowing the analysis and manipulation of signals in the frequency domain. This proves particularly beneficial in fields such as telecommunications, where noise reduction, compression, and modulation are essential for reliable data transmission. Moreover, the fusion of numerical methods with machine learning techniques has revolutionized image and signal processing, enabling the development of sophisticated algorithms for pattern recognition and feature extraction. The application of these methods extends to medical diagnostics, where accurate processing of images and signals is critical for identifying anomalies or diseases. As technology advances, the integration of numerical methods with artificial intelligence continues to push the boundaries of what can be achieved in image and signal processing. The quest for faster and more accurate algorithms, coupled with the increasing availability of computational resources, promises a future where these numerical techniques will continue to play a transformative role in extracting valuable information from diverse data sources, ultimately contributing to advancements in fields as varied as healthcare, communications, and multimedia.

Challenges and Limitations:

In the realm of numerical methods, despite their remarkable utility, challenges and limitations persist. One of the primary concerns is the issue of numerical stability, where small errors in calculations can accumulate and lead to significant discrepancies in results. Convergence, the tendency of iterative numerical algorithms to approach a solution, is another challenge, as some methods may struggle to converge within reasonable time frames or fail to converge altogether. The curse of dimensionality poses a substantial limitation, especially in optimization problems with a large number of variables, making computations exponentially more complex. Additionally, the choice of algorithms and parameters can significantly impact the accuracy and efficiency of numerical solutions, necessitating a careful balance between computational resources and precision. Validation and verification become critical, requiring a thorough understanding of both the numerical methods employed and the underlying physics or mathematics of the problem. Ensuring the reliability of numerical models is a continual challenge, demanding diligence in the face of potential inaccuracies. Despite these challenges, the field of numerical methods continues to evolve, driven by the pursuit of more robust algorithms and innovative approaches. As researchers strive to overcome these limitations, the future holds promise for enhanced accuracy and efficiency in solving increasingly intricate and large-scale problems through numerical techniques.

Future Trends and Innovations:

Future trends and innovations in numerical methods promise to reshape the landscape of problem-solving across diverse domains. High-performance computing, marked by exponential increases in processing power, enables simulations of unprecedented complexity. The integration of parallel processing further accelerates computations, paving the way for real-time simulations and analyses. Machine learning algorithms, with their ability to recognize patterns and learn from data, are becoming integral to numerical methods, offering a paradigm shift in optimization and decision-making processes. As technology evolves, researchers are exploring unconventional numerical techniques, including meshless methods that eliminate the need for a predefined mesh, providing more flexibility in handling complex geometries. Quantum computing, with its capacity for parallelism and superposition, holds the potential to revolutionize numerical simulations by solving certain types of problems exponentially faster than classical computers. These advancements not only enhance the accuracy and efficiency of numerical solutions but also enable the exploration of larger problem spaces previously deemed impractical. Collaborative efforts between mathematicians, scientists, and engineers are crucial for pushing the boundaries of numerical methods. As we look ahead, the synergy between numerical methods and emerging technologies promises a future where solving increasingly complex problems becomes not only achievable but also routine, ushering in a new era of innovation and discovery.

Conclusion:

In conclusion, the realm of numerical methods emerges as a transformative force in solving intricate problems across diverse fields. From simulating fluid dynamics for aircraft design to optimizing logistics in operations research, these methods have proven indispensable in addressing challenges that defy analytical solutions. The case studies presented underscore the versatility and efficacy of numerical techniques, showcasing their ability to unravel complex phenomena and facilitate informed decision-making. Despite their undeniable advantages, it is crucial to acknowledge the challenges inherent in numerical methods, such as numerical stability and convergence issues, emphasizing the need for meticulous validation and verification processes. Looking ahead, the integration of high-performance computing, parallel processing, and machine learning signals a promising future for numerical methods, pushing the boundaries of problem-solving capabilities. As technology advances, the collaboration between mathematicians, scientists, and engineers will continue to drive innovation, potentially unlocking new frontiers in numerical simulations. In essence, numerical methods stand as pillars of progress, offering a robust framework for tackling the intricate challenges that define the ever-evolving landscape of science and engineering. Through ongoing exploration, refinement, and application, the significance of numerical methods in shaping the future of problem-solving remains resolute, promising continued advancements and breakthroughs on the frontier of scientific inquiry and technological innovation.

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What does it mean to solve something numerically?

What does it mean to solve something numerically? Why it is important to solve something numerically?

numerical-methods
discrete-mathematics
numerical-integration

2 I'm voting to close this question as off-topic because it's about Mathematics and not directly about programming or coding. – Pang Commented Dec 21, 2017 at 6:40

2 Answers 2

Solving something numerically means you use approximations that make a problem easier to solve.

Generally it refers to the difficulty of solving problems mathematically that give you the exact answer, or trying to get approximate answers using techniques that involve numeric approximations, that allow you to get close to the solution sooner or more easily.

For instance, since computers can only store numbers with bits and bytes, many solutions you obtain are only approximations. You cannot store the number known as Pi, for instance. But you can use an approximation of Pi with a large number of bits.

Another example is that certain classes or problems, like partial differential equations, are very difficult to solve. However, you can use methods that are known to give you something that is close to the answer, a numerical approximation. Typically such methods use approximations and computer power. So you never get the exact answer but you get something close to it.

There is a entire field of science known as numerical analysis based on this topic, so my answer cannot possibly be complete, but I'm giving you a very basic take on it.

The tags you used are all related. Numerical integration is what I just described for the problem of integration. Discrete mathematics can be used to approximate infinite or continuous values to get solutions numerically. Numerical methods are a collection of such methods to solve problems numerically.

It simply means to evaluate the numerical value of the solution, say return 0.250 rather than 1/4 or arctan(1)/π (called a symbolic value). Numbers instead of formulas.

For applications, you need the numerical value (you need to know the area in m² to purchase fabric, you don't care about the formula that gave it). But for insight into a problem, the formulas are more useful as they give you a feeling on trends when you change parameters in the problem setting.

I will not expand on the fact that numerical and symbolic approaches use very different methods and the problems that they can address only partly overlap. This is a large topic.

Computers are pretty good at handling numerical computations in a large scale, due to their 'brute force' arithmetic capabilities. Humans are better at... thinking.

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Numerical methods to solve mathematical problems

Numerical methods are employed in solving mathematical problems. With the increase in the difficulty of the equation, it becomes hectic to solve the equations by analytical methods. Learn through this article how numerical methods prove to be the best option to solve mathematical equations.

What are numerical methods?

Why should we use numerical techniques, types of numerical methods, accuracy of numerical methods.

A nice article explaining the methodology used in numerical analysis. We are generally habitual of solving problems by direct methods but sometimes it is not possible to do so directly and then we take help of calculators and computers for the same. Today the methods of the numerical analysis are used in computer algorithms for calculating the solution of a problem with a good degree of certainty. Let us understand this by solving a simple problem to find value of x in the equation x^3 + 4 = 31 If we use the direct method then we would say that - x^3 = 31 - 4 so x^3 = 27 and hence x = 3 It was so easy to find x in the above case. Sometimes problem might be a complex one and it would not be possible to solve it in a direct way. For example if x^3 = 11 (which also means that x^3 - 11 = 0) then it is not easy to find x directly. In such cases the numerical analysis helps us and one of the most common method in numerical analysis goes like - let f(x) = x^3 - 11 Now we can see that x is more than 2 but less than 3 as 2^3 = 8 which is less than 11 and 3^3 = 27 which is more than 11. Let us iterate the value of x by increasing it from 2 and decreasing it from 3 and get some values of f(x) at those points. The value of f(x) will slowly converge to zero and we would get a solution for x somewhere in between 2 and 3. It actually comes approximately as x = 2.2235 The interesting point is that more number of iterations we do we will get a more accurate value of x somewhere between 2.223 to 2.224. But in practice we cannot go for an infinite number of iterations and we have to accept a value of x with some minimal error. That is what many computer programmes do and we get an approximate solution as x = 2.2235 which is a number between 2.223 and 2.224 Numerical analysis is definitely a great tool for solving complex mathematical problems.

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DOI: 10.3390/biomimetics9060361
Corpus ID: 270551422

An Improved Multi-Strategy Crayfish Optimization Algorithm for Solving Numerical Optimization Problems

Ruitong Wang , Shuishan Zhang , Guangyu Zou
Published in Biomimetics 14 June 2024
Computer Science

42 References

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Solving a system of complex matrix equations using a gradient-based method and its application in image restoration

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Akbar Shirilord 1 &
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This study presents some new iterative algorithms based on the gradient method to solve general constrained systems of conjugate transpose matrix equations for both real and complex matrices. In addition, we analyze the convergence properties of these methods and provide numerical techniques to determine the solutions. Then we prove that the optimal parameters of the new algorithm satisfy a constrained optimization problem. The effectiveness of the proposed iterative methods is demonstrated through various numerical examples employed in this study and compared the results by some existing algorithms.

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The proof of Theorem 3.8 : Initially, we define the error matrix as follows:

Employing the procedure outlined in Algorithm 4 will yield the subsequent outcome

By considering the definition of the Frobenius norm, we obtain:

It is readily apparent that

Merging these relationships with equation ( 6.2 ) yields:

Now from the definition of Frobenius inner product and the fact that for two matrices $ L_1 $ and $ L_2 $ , $ \left\langle L_1, L_2 \right\rangle _{F}+ \left\langle L_2, L_1 \right\rangle _{F}$ is real and $ -\left\langle L_1, L_2 \right\rangle _{F}- \left\langle L_2, L_1 \right\rangle _{F} \le \Vert L_{1}\Vert _F^2+\Vert L_2\Vert _F^2$ we obtain

Then by using Lemmas 2.2 and 2.3 , one can ascertain that:

Given these inequalities and ( 6.7 ), it can be concluded that:

where $ \theta _{1} $ and $\theta _{2} $ are defined in ( 3.39 ) and ( 3.40 ), respectively. Now there are $ \epsilon _{\tau }>0, \tau =1,..., \gamma , $ such that

where $ \epsilon =\max _{1\le \tau \le \gamma } \{\epsilon _{\tau }\} $ . Hence we conclude

Now ( 6.11 ) yields:

If the parameters $ \omega _{1} $ and $ \omega _{2} $ are selected according to ( 3.38 ), then it follows that:

Consequences drawn from the convergence theorem of series indicate that:

Finally given that the matrix equation ( 3.31 ) has a unique solution, it can be concluded that $ \lim \limits _{\varsigma \longrightarrow \infty } L(\varsigma )=L^{\star }$ .

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Shirilord, A., Dehghan, M. Solving a system of complex matrix equations using a gradient-based method and its application in image restoration. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01856-2

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Iterative methods
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Interpolation and regression models for the chemical engineer : solving numerical problems

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Preface INTERPOLATION Introduction Classes for Function Interpolation Polynomial Interpolation Roots-Product Form Standard Form Lagrange Method Newton Method Neville Algorithm Hermite Polynomial Interpolation Interpolation with Rational Functions Inverse Interpolation Successive Polynomial Interpolation Two-Dimensional Curves Orthogonal Polynomials FUNDAMENTALS OF STATISTICS Introduction Fundamentals Estimation of Expected Value Estimation of Variance Estimation of Standard Deviation Outlier Detection Relevant Probability Distributions Correct Meaning of Statistical Tests and Confidence Regions Nonparametric Statistics Conditional Probability LINEAR REGRESSIONS Introduction Least Sum of Squares Methods Some Caveat Class for Linear Regressions Generalized Toolkit for Linear Problems Data Modification Data Deletion Preliminary Analysis Multicollinearity Best Model Selection Principal Components ROBUST LINEAR REGRESSIONS Introduction Some Caveat Outliers and Gross Errors Studentized Residuals M-Estimators Influential Observations Y-Outliers, X-Outliers, and F-Outliers Secluded Observations Robust Indices Normality Condition Heteroscedasticity Condition LINEAR REGRESSION CASE STUDIES Introduction Ferrari F1's Test Best Model Formulation Outliers Best Model Selection Principal Components NONLINEAR REGRESSIONS Nonlinear Regression Problems Some Caveat Parameter Evaluation BzzNonLinearRegression Class Nonalgebraic Constraints Algorithms for Outlier Detection Correlations Among Model Parameters Preventative Model Analysis Model Discrimination Model Collection and Model Selection MONLINEAR REGRESSION CASE STUDIES Introduction One Dependent Variable with Constant Variance Multicubic Piecewise Models One Dependent Variable and Nonconstant Variance More Dependent Variable and Constant Variance More Dependent Variable and Nonconstant Variance Model Consisting of Ordinary Differential Equations Model Consisting of Differential Algebraic Equations Analysis of Alternative Models Independent Variables Subject to Experimental Error Variables with Missing Experiments Outliers Independent Variables Subject to Experimental Error and Model with Outliers REASONABLE DESIGN OF EXPERIMENTS Introduction Preliminary Experiments Using Models to Suggest New Experiments New Experiments to Improve the Parameter Estimation Model Selection: The Bayesian Approach New Experiments for Model Discrimination Criterion Used in BzzNonLinearRegression Class to Generate New Experiments APPENDIX A: Mixed-Language: Fortan and C++ APPENDIX B: Basic Requirements for Using the BzzMath Library APPENDIX C: Copyrights.
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Interface between ansys and matlab for solving elastic problems with non-conforming meshes

Světlík, Tadeáš
Varga, Radek
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In our research, we are focusing on the development of iterative algorithms for solving contact problems emerging in the analysis of steel structures bolt connections. Despite the project is focusing mainly on the development of efficient numerical algorithms, several supplementary problems have to be solved. In our research, we decided to utilize widely-used popular commercial software Ansys to provide the data for our algorithms implemented in Matlab as well as for the comparison of results. This paper presents experiences with our newly developed data interface.

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Physics Numericals for Class 11: Mastering Concepts with Practical Problems

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Numerical 2:

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Numerical 5:

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