Numerical Methods and Programming

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    Information about the content Lesson

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    Course Description:

    This course aims to introduce students to the fundamental principles of numerical methods and computer programming. It will cover the basic concepts of numerical analysis, approximation methods, and algorithms commonly used in solving numerical problems. Students will also have the opportunity to develop their programming skills through practical exercises.

    Course Objectives:

    • Understand the fundamental concepts of numerical analysis.
    • Be able to apply different numerical methods to solve mathematical problems.
    • Acquire programming skills using a language suitable for numerical analysis.
    • Know how to evaluate the efficiency and accuracy of the numerical methods used.

    Course Content:

    1. Basic syntax of a programming language

    2. Numerical Integration

       • Trapezoidal Method
       • Simpson's Method

    3. Solving Nonlinear Equations

       • Bisection Method
       • Newton-Raphson Method

    4. Solving Ordinary Differential Equations

       • Euler's Method
       • Second-Order Runge-Kutta Method
       • Fourth-Order Runge-Kutta Method

    5. Solving Linear Systems

       • Jacobi Method
       • Gauss-Seidel Method

    Evaluation:

    • Practical Exam: 50%
    • Final Exam: 50%

    
    

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    Glossary of Numerical Methods and Programming

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    Numerical Method: A method that allows solving mathematical problems using algorithms instead of analytical methods.

    Algorithm: A sequence of instructions that solves a given problem.

    Iteration: A set of instructions and operations that are repeated a certain number of times.

    Variable: A symbol used to represent a value that changes during the execution of a program.

    Function: A collection of instructions grouped under a specific name, designed to perform a specific task.

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    Books/References File

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    This file contains a book related to this module. Furthermore, the following links corresponds to some downloaded pdf file.
    

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• Introduction
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    Introduction File

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    Engineers are constantly confronted with concrete problems in their respective fields. The majority of these problems can be formulated as mathematical problems (such as solving an equation, calculating an integral, and so on), and the majority of these problems are not resolvable by traditional analytical methods, or we are convinced that it will take a long time to solve them analytically if this is not possible, that is why we then resort to numerical methods.

    For example, the integrals
    $\int_{-2}^3 e^{-x^2}\,dx$
    and
    $\int_{-\pi}^\pi \sin(x^2)\,dx$
    cannot be calculated using classical methods such as integration by parts, substitution, etc. Using a numerical method, this type of integration can be approximated numerically with a given accuracy.

    Over the past few decades, numerous algorithms based on theoretical studies have been developed to solve various mathematical problems. This course is designed for second-year students of Material Science. It constitutes an introduction to numerical calculation and consequently to the different techniques that these students will frequently have to use, without delving into the theoretical considerations and foundations of the methods presented. Indeed, we have described the most well-known and widely used numerical methods for solving various problems encountered during their studies, focusing on the practical aspects in the presentation of these methods.

    In this booklet, each section is followed by detailed examples, and at the end of each chapter, students are encouraged to work on additional exercises. The course is structured into five main chapters: The first chapter is dedicated to numerical integration. The second chapter covers two methods of polynomial interpolation. The third chapter focuses on the numerical solution of nonlinear equations. The fourth chapter describes numerical techniques for solving ordinary differential equations. Finally, the last chapter is dedicated to methods for solving systems of linear equations.

    
    

• Chapter 1: Numerical Integration

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  Often, the explicit computation of the integral of a continuous function $f$ over an interval $[a, b]$ of $\mathbb{R}$ can be very expensive or cannot be solved analytically. To overcome these difficulties, we use numerical methods to calculate an approximation of an integral

  $$\int_a^b f(x)\,dx$$within a given precision.
  

  In this chapter, we present two common integration techniques:  Trapezoidal method and Simpson method. Using these algorithms, we can even approximate the integral of functions known only through discrete data points.

  
  

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    Chapitre 1 : Numerical Integration File

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    Explanatory video: Trapezoidal method Page

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    Practical Work 1: Matlab (Sylab) code of the trapezoidal method for Numerical Integration File

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    Here is a MATLAB (Scilab) code of the trapezoidal rule for numerical integration.

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    Explanatory video: Simpson's method Page

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    Practical Work 2: Matlab (Sylab) code of Simson's method for Numerical Integration File

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    Here is a MATLAB (Scilab) code of the trapezoidal Simpson's rule for numerical integration.

    
    

    
    

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    Solved Exercises File

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    This file contains some solved exercises.

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    Forum on Chapter 1

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    Dear students, if you have any questions about Chapter 1, please feel free to contact me.

    
    Best regards.

    
    

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    Test 1: Test about Numerical Integration Quiz

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    This file contains some test questions. The students have to answer them before 10th June 2025.

    
    

• Chapter 2: Numerical Solution of Nonlinear Equations

  In mathematics and applied sciences, nonlinear equations play a crucial role in modeling complex phenomena across various disciplines. Unlike linear equations, which can be solved analytically using algebraic methods, finding exact solutions for nonlinear equations is often difficult or even impossible.
  

  For example, consider the equation:

  $$\cos (x^3)\sin (2x^2-3)+0.5=0.$$
  

  It is evident that solving this equation analytically would be extremely time-consuming, if not impossible. These types of equations, known as nonlinear (transcendental) equations, can instead be solved numerically using methods that allow us to compute approximate roots with a specified level of precision.

  In this chapter, we will explore two numerical methods for solving nonlinear univariate equations of the form $f(x) = 0$.

  
  

  
  

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    Numerical Solution of Nonlinear Equations File

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    This video explains the principle of the Bisection (Dichotomy) method. Page

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    This video explains the principle of the Bisection (Dichotomy) method.

    
    

    
    

    
    

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    Practical Work 3: Matlab (Sylab) code of Bissection method for Numerical Integration File

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    Example of MATLAB code for implementing the bisection method for numerical solution of a nonlinear equation.

    
    

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    Explainer Video: Newton-Raphson Method Page

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    This video explains the principle of the Newton-Raphson method.

    
    

    
    

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    Practical Work 4: Matlab (Sylab) code of Newton-Raphson method for Numerical Integration File

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    Example of MATLAB code for implementing the  Newton- Raphson method for numerical solution of a nonlinear equation.

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    Solved Exercises File

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    This file contains some solved exercises.

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    Forum on Chapitre 02

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    Dear students, if you have any questions about Chapter 2, please feel free to contact me.

    Best regards.

    
    
    

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    Test 2: Test about Numerical Solution of Nonlinear Equations Quiz

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    This file contains some test questions. The students have to answer them before 10-th June 2025.
    

    
    

• Chapter 3: Numerical Solution of Ordinary Differential Equations of First Order

  Many challenges in science and engineering can be reduced to the task of solving differential equations while satisfying certain predefined conditions. Traditional analytical techniques, which are assumed to be understood by the reader, are suitable for solving only a subset of these equations. However, the differential equations that govern the behavior of physical systems often lack closed-form solutions. Therefore, it is crucial to use numerical methods to solve these problems.
  

  In this chapter, we will study three methods for the numerical solution of a first-order ordinary differential equation: the Euler method, the second-order Runge-Kutta method (Heun), and the fourth-order Runge-Kutta method.

  
  

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    Numerical Solution of Ordinary Differential Equations of First Order File

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    This video explains the principle of Euler method Page

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    This video explains the principle of the Euler method.

    
    

    
    

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    Practical Work 5: MATLAB (Sylab) code of Euler method File

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    This video explains the principle of the Second- Order- Runge Kutta method (RK2) Page

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    Practical Work 6: MATLAB (Sylab) code of the Second-Order Runge-Kutta (RK2) Method File

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    This video explains the principle of the Fourth- Order- Runge Kutta method (RK4) Page

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    Practical Work 7: Matlab (Sylab) code of Fourth-Order Runge-Kutta (RK4) method File

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    Solved Exercises File

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    Forum on Chapitre 03

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    Dear students, if you have any questions about Chapter 3, please feel free to contact me.

    Best regards.

    
    

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    Test 3: Test about Numerical Solution of Ordinary Differential Equations of First Order Quiz

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    This file contains some test questions. The students have to answer them before 10-th June 2025.

    
    

• Chapter 4: Numerical Solution of Systems of Linear Equations

  In practice, engineers often encounter problems whose solution requires solving a system of linear equations that models the problem under consideration. For example, determining currents and voltages in electrical networks requires solving a system of linear equations. That is, we seek the vector $X \in \mathbb{R}^n$, where $X = (x_1, x_2, \dots, x_n)$, which is the solution of the following linear system:

  $$AX = b \Longleftrightarrow \left\lbrace \begin{array}{lllll} a_{11} x_1 + a_{12} x_2 + \dots + a_{1n} x_n = b_1\\ a_{21} x_1 + a_{22} x_2 + \dots+ a_{2n} x_n = b_2\\ \vdots\\ a_{n1} x_1 + a_{n2} x_2 + \dots+ a_{nn} x_n = b_n\\ \end{array} \right.$$

  This system admits a unique solution when the determinant of $A$ is nonzero, which we will assume below. Solving this system using direct methods becomes impractical when $n$ is relatively high. Therefore, it is preferable to use numerical methods that construct a sequence converging to the solution of the system.

  In this chapter, we present two numerical methods that provide approximate solutions to systems of linear equations using a linear function $f$ such that

  $$X^{k+1} = f(X^k), \quad k \in \mathbb{N}.$$

  These methods are easy to implement, require minimal memory, and produce results with the desired accuracy.
  

  
  

  
  

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    Numerical Solution of Systems of Linear Equations File

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    This video explains the principle of Jaccobi method Page

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    MATLAB (Scilab) code of the Jacobi method for solving systems of linear equations File

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    Here is a MATLAB (Scilab) code of the Jacobi method for solving systems of linear equations

    
    

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    This video explains the principle of Gauss-Seidel method Page

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    Practical Work 9: Matlab (Sylab) code of Gauss-Seidel method for solving systems of linear equations File

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    Here is a MATLAB (Scilab) code of the Gauss-Seidel for solving systems of linear equations.

    
    

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    Solved Exercices File

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    This file contains some solved exercises.

    
    

• Exam 2025
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    Exam 2025 File

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    Solution of the exam File

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