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Partial Differential Equations and Mathematica




ISBN 9781584883142
Published November 12, 2002 by Chapman and Hall/CRC
440 Pages 74 B/W Illustrations

 
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Book Description

Early training in the elementary techniques of partial differential equations is invaluable to students in engineering and the sciences as well as mathematics. However, to be effective, an undergraduate introduction must be carefully designed to be challenging, yet still reasonable in its demands. Judging from the first edition's popularity, instructors and students agree that despite the subject's complexity, it can be made fairly easy to understand.

Revised and updated to reflect the latest version of Mathematica, Partial Differential Equations and Boundary Value Problems with Mathematica, Second Edition meets the needs of mathematics, science, and engineering students even better. While retaining systematic coverage of theory and applications, the authors have made extensive changes that improve the text's accessibility, thoroughness, and practicality.

New in this edition:

  • Upgraded and expanded Mathematica sections that include more exercises
  • An entire chapter on boundary value problems
  • More on inverse operators, Legendre functions, and Bessel functions
  • Simplified treatment of Green's functions that make it more accessible to undergraduates
  • A section on the numerical computation of Green's functions
  • Mathemcatica codes for solving most of the problems discussed
  • Boundary value problems from continuum mechanics, particularly on boundary layers and fluctuating flows
  • Wave propagation and dispersion

    With its emphasis firmly on solution methods, this book is ideal for any mathematics curricula. It succeeds not only in preparing readers to meet the challenge of PDEs, but also in imparting the inherent beauty and applicability of the subject.
  • Table of Contents

    INTRODUCTION TO MATHEMATICA
    Introduction
    Conventions
    Getting Started
    File Manipulation
    Differential Equations
    To the Instructor
    To the Student
    MathSource
    INTRODUCTION
    Notation and Definitions
    Initial and Boundary Conditions
    Classification of Second Order Equations
    Some Known Equations
    Superposition Principle
    METHOD OF CHARACTERISTICS
    First Order Equations
    Linear Equations with Constant Coefficients
    Linear Equations with Variable Coefficients
    First Order Quasi-Linear Equations
    First Order Nonlinear Equations
    Geometrical Considerations
    Some Theorems on Characteristics
    Second Order Equations
    Linear and Quasi-linear Equations
    LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
    Inverse Operators
    Homogeneous Equations
    Nonhomogeneous Equations
    ORTHOGONAL EXPANSIONS
    Orthogonality
    Orthogonal Polynomials
    Series of Orthogonal Functions
    Trigonometric Fourier Series
    Eigenfunction Expansions
    Bessel Functions
    SEPARATION OF VARIABLES
    Introduction
    Hyperbolic Equations
    Parabolic Equations
    Elliptic Equations
    Cylindrical Coordinates
    Spherical Coordinates
    Nonhomogeneous Problems
    INTEGRAL TRANSFORMS
    Laplace Transforms
    Notation
    Basic Laplace Transforms
    Inversion Theorem
    Fourier Transforms
    Fourier Integral Theorems
    Properties of Fourier Transforms
    Fourier Sine and Cosine Transforms
    Finite Fourier Transforms
    GREEN'S FUNCTIONS
    Generalized Functions
    Green's Functions
    Elliptic Equations
    Parabolic Equations
    Hyperbolic Equations
    Applications of Green's Functions
    Computation of Green's Functions
    BOUNDARY VALUE PROBLEMS
    Initial and Boundary Conditions
    Implicit Conditions
    Periodic Conditions
    Wave Propagation and Dispersion
    Boundary Layer Flows
    Miscellaneous Problems
    WEIGHTED RESIDUAL METHODS
    Line Integrals
    Variational Notation
    Multiple Integrals
    Weak Variational Formulation
    Gauss-Jacobi Quadrature
    Rayleigh-Ritz Method
    Choice of Test Functions
    Transient Problems
    Other Methods
    PERTURBATION METHODS
    Perturbation Problem
    Taylor Series Expansions
    Successive Approximations
    Boundary Perturbations
    Fluctuating Flows
    FINITE DIFFERENCE METHODS
    Finite Difference Schemes
    First Order Equations
    Second Order Equations

    Appendix A: Green's Identities
    Appendix B: Orthogonal Polynomials
    Appendix C: Tables of Transform Pairs
    Appendix D: Glossary of Mathematica Functions
    Appendix E: Mathematica Packages and Notebooks
    Bibliography
    Index

    Each chapter also contains sections of Mathematica Projects and Exercises

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