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An Introduction to Partial Differential Equations with MATLAB, 2nd Rev. Ed. - ISBN 9781439898468

An Introduction to Partial Differential Equations with MATLAB, 2nd Rev. Ed.

ISBN 9781439898468

Autor: Matthew P. Coleman

Wydawca: CRC Press

Dostępność: Wysyłka w ciągu 2-3 dni

Cena: 240,45 zł


ISBN13:      

9781439898468

ISBN10:      

1439898464

Autor:      

Matthew P. Coleman

Oprawa:      

Hardback

Rok Wydania:      

2013-08-16

Numer Wydania:      

2

Ilość stron:      

683

Features

 

Gives a thorough yet accessible treatment of PDEs and their applications Requires basic knowledge of multivariable calculus and ODEs Emphasizes early on the solution of separable linear equations based on the application of Fourier series Presents an introduction and historical overview at the beginning of each chapter Includes numerous problem-solving exercises, proofs, extended challenges, and MATLAB exercises Provides MATLAB code for tables and figures on the 

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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