An Introduction to Partial Differential Equations with MATLAB (Chapman & Hall/CRC Applied Mathematics & Nonlinear Science 27 2nd Revised edition)
By: Matthew P. Coleman (author)Hardback
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An Introduction to Partial Differential Equations with MATLAB(R), Second Edition illustrates the usefulness of PDEs through numerous applications and helps students appreciate the beauty of the underlying mathematics. Updated throughout, this second edition of a bestseller shows students how PDEs can model diverse problems, including the flow of heat, the propagation of sound waves, the spread of algae along the ocean's surface, the fluctuation in the price of a stock option, and the quantum mechanical behavior of a hydrogen atom. Suitable for a two-semester introduction to PDEs and Fourier series for mathematics, physics, and engineering students, the text teaches the equations based on method of solution. It provides both physical and mathematical motivation as much as possible. The author treats problems in one spatial dimension before dealing with those in higher dimensions. He covers PDEs on bounded domains and then on unbounded domains, introducing students to Fourier series early on in the text. Each chapter's prelude explains what and why material is to be covered and considers the material in a historical setting.
The text also contains many exercises, including standard ones and graphical problems using MATLAB. While the book can be used without MATLAB, instructors and students are encouraged to take advantage of MATLAB's excellent graphics capabilities. The MATLAB code used to generate the tables and figures is available in an appendix and on the author's website.
Introduction What are Partial Differential Equations? PDEs We Can Already Solve Initial and Boundary Conditions Linear PDEs-Definitions Linear PDEs-The Principle of Superposition Separation of Variables for Linear, Homogeneous PDEs Eigenvalue Problems The Big Three PDEs Second-Order, Linear, Homogeneous PDEs with Constant Coefficients The Heat Equation and Diffusion The Wave Equation and the Vibrating String Initial and Boundary Conditions for the Heat and Wave Equations Laplace's Equation-The Potential Equation Using Separation of Variables to Solve the Big Three PDEs Fourier Series Introduction Properties of Sine and Cosine The Fourier Series The Fourier Series, Continued The Fourier Series-Proof of Pointwise Convergence Fourier Sine and Cosine Series Completeness Solving the Big Three PDEs Solving the Homogeneous Heat Equation for a Finite Rod Solving the Homogeneous Wave Equation for a Finite String Solving the Homogeneous Laplace's Equation on a Rectangular Domain Nonhomogeneous Problems Characteristics First-Order PDEs with Constant Coefficients First-Order PDEs with Variable Coefficients The Infinite String Characteristics for Semi-Infinite and Finite String Problems General Second-Order Linear PDEs and Characteristics Integral Transforms The Laplace Transform for PDEs Fourier Sine and Cosine Transforms The Fourier Transform The Infinite and Semi-Infinite Heat Equations Distributions, the Dirac Delta Function and Generalized Fourier Transforms Proof of the Fourier Integral Formula Bessel Functions and Orthogonal Polynomials The Special Functions and Their Differential Equations Ordinary Points and Power Series Solutions; Chebyshev, Hermite and Legendre Polynomials The Method of Frobenius; Laguerre Polynomials Interlude: The Gamma Function Bessel Functions Recap: A List of Properties of Bessel Functions and Orthogonal Polynomials Sturm-Liouville Theory and Generalized Fourier Series Sturm-Liouville Problems Regular and Periodic Sturm-Liouville Problems Singular Sturm-Liouville Problems; Self-Adjoint Problems The Mean-Square or L2 Norm and Convergence in the Mean Generalized Fourier Series; Parseval's Equality and Completeness PDEs in Higher Dimensions PDEs in Higher Dimensions: Examples and Derivations The Heat and Wave Equations on a Rectangle; Multiple Fourier Series Laplace's Equation in Polar Coordinates: Poisson's Integral Formula The Wave and Heat Equations in Polar Coordinates Problems in Spherical Coordinates The Infinite Wave Equation and Multiple Fourier Transforms Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator; Green's Identities for the Laplacian Nonhomogeneous Problems and Green's Functions Green's Functions for ODEs Green's Function and the Dirac Delta Function Green's Functions for Elliptic PDEs (I): Poisson's Equation in Two Dimensions Green's Functions for Elliptic PDEs (II): Poisson's Equation in Three Dimensions; the Helmholtz Equation Green's Functions for Equations of Evolution Numerical Methods Finite Difference Approximations for ODEs Finite Difference Approximations for PDEs Spectral Methods and the Finite Element Method Appendix A: Uniform Convergence; Differentiation and Integration of Fourier Series Appendix B: Other Important Theorems Appendix C: Existence and Uniqueness Theorems Appendix D: A Menagerie of PDEs Appendix E: MATLAB Code for Figures and Exercises Appendix F: Answers to Selected Exercises References Index
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- ID: 9781439898468
2nd Revised edition
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