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Introduction to Numerical Methods for Time Dependent Differential Equations

Book Description

Introduces both the fundamentals of time dependent differential equations and their numerical solutions

Introduction to Numerical Methods for Time Dependent Differential Equations delves into the underlying mathematical theory needed to solve time dependent differential equations numerically. Written as a self-contained introduction, the book is divided into two parts to emphasize both ordinary differential equations (ODEs) and partial differential equations (PDEs).

Beginning with ODEs and their approximations, the authors provide a crucial presentation of fundamental notions, such as the theory of scalar equations, finite difference approximations, and the Explicit Euler method. Next, a discussion on higher order approximations, implicit methods, multistep methods, Fourier interpolation, PDEs in one space dimension as well as their related systems is provided.

Introduction to Numerical Methods for Time Dependent Differential Equations features:

  • A step-by-step discussion of the procedures needed to prove the stability of difference approximations

  • Multiple exercises throughout with select answers, providing readers with a practical guide to understanding the approximations of differential equations

  • A simplified approach in a one space dimension

  • Analytical theory for difference approximations that is particularly useful to clarify procedures

  • Introduction to Numerical Methods for Time Dependent Differential Equations is an excellent textbook for upper-undergraduate courses in applied mathematics, engineering, and physics as well as a useful reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs or predict and investigate phenomena from many disciplines.

    Table of Contents

    1. Cover
    2. Half Title page
    3. Title page
    4. Copyright page
    5. Dedication
    6. Preface
    7. Acknowledgements
    8. Part I: Ordinary Differential Equations and Their Approximations
      1. Chapter 1: First-Order Scalar Equations
        1. 1.1 Constant coefficient linear equations
        2. 1.2 Variable coefficient linear equations
        3. 1.3 Perturbations and the concept of stability
        4. 1.4 Nonlinear equations: the possibility of blow-up
        5. 1.5 Principle of linearization
      2. Chapter 2: Method of Euler
        1. 2.1 Explicit Euler method
        2. 2.2 Stability of the explicit Euler method
        3. 2.3 Accuracy and truncation error
        4. 2.4 Discrete Duhamel’s principle and global error
        5. 2.5 General one-step methods
        6. 2.6 How to test the correctness of a program
        7. 2.7 Extrapolation
      3. Chapter 3: Higher-Order Methods
        1. 3.1 Second-order Taylor method
        2. 3.2 Improved Euler’s method
        3. 3.3 Accuracy of the solution computed
        4. 3.4 Runge-Kutta methods
        5. 3.5 Regions of stability
        6. 3.6 Accuracy and truncation error
        7. 3.7 Difference approximations for unstable problems
      4. Chapter 4: Implicit Euler Method
        1. 4.1 Stiff equations
        2. 4.2 Implicit Euler method
        3. 4.3 Simple variable-step-size strategy
      5. Chapter 5: Two-Step and Multistep Methods
        1. 5.1 Multistep methods
        2. 5.2 Leapfrog method
        3. 5.3 Adams methods
        4. 5.4 Stability of multistep methods
      6. Chapter 6: Systems of Differential Equations
    9. Part II: Partial Differential Equations and Their Approximations
      1. Chapter 7: Fourier Series and Interpolation
        1. 7.1 Fourier expansion
        2. 7.2 L2-norm and scalar product
        3. 7.3 Fourier interpolation
      2. Chapter 8: 1-Periodic Solutions of time Dependent Partial Differential Equations with Constant Coefficients
        1. 8.1 Examples of equations with simple wave solutions
        2. 8.2 Discussion of well posed problems for time dependent partial differential equations with constant coefficients and with 1-periodic boundary conditions
      3. Chapter 9: Approximations of 1-Periodic Solutions of Partial Differential Equations
        1. 9.1 Approximations of space derivatives
        2. 9.2 Differentiation of Periodic Functions
        3. 9.3 Method of lines
        4. 9.4 Time Discretizations and Stability Analysis
      4. Chapter 10: Linear Initial Boundary Value Problems
        1. 10.1 Well-Posed Initial Boundary Value Problems
        2. 10.2 Method of lines
      5. Chapter 11: Nonlinear Problems
        1. 11.1 Initial value problems for ordinary differential equations
        2. 11.2 Existence theorems for nonlinear partial differential equations
        3. 11.3 Nonlinear example: Burgers’ equation
      6. Appendix A: Auxiliary Material
        1. A.1 Some useful Taylor series
        2. A.2 “O” notation
        3. A.3 Solution expansion
      7. Appendix B: Solutions to Exercises
      8. References
      9. Index