Introduction to Partial Differential Equations With Matlab

by Cooper, Jeffery M.; Benedetto, J. J.

• ISBN13: 9780817639679
• ISBN10: 0817639675
• Format: Hardcover
• Copyright: 1998-01-01
• Publisher: BIRKHAUSER

Summary

The subject of partial differential equations has an unchanging core of material but is constantly expanding and evolving. Introduction to Partial Differential Equations with MATLAB is a careful integration of traditional core topics with modern topics, taking full advantage of the computational power of MATLAB to enhance the learning experience. This advanced text/reference is an introduction to partial differential equations covering the traditional topics within a modern context. To provide an up-to-date treatment, techniques of numerical computation have been included with carefully selected nonlinear topics, including nonlinear first order equations. Each equation studied is placed in the appropriate physical context. The analytical aspects of solutions are discussed in an integrated fashion with extensive examples and exercises, both analytical and computational. The book is excellent for classroom use and can be used for self-study purposes. Topic and Features: a?? Nonlinear equations including nonlinear conservation laws; a?? Dispersive wave equations and the Schrodinger equation; a?? Numerical methods for each core equation including finite difference methods, finite element methods, and the fast Fourier transform; a?? Extensive use of MATLAB programs in exercise sets. MATLAB m files for numerical and graphics programs available by ftp from this web site. This text/reference is an excellent resources designed to introduce advanced students in mathematics, engineering and sciences to partial differential equations. It is also suitable as a self-study resource for professionals and practitioners.

Table of Contents

Preface	xiii
1 Preliminaries	1	(18)
1.1 Elements of analysis	1	(13)
1.1.1 Sets and their boundaries	1	(2)
1.1.2 Integration and differentiation	3	(2)
1.1.3 Sequences and series of functions	5	(6)
1.1.4 Functions of several variables	11	(3)
1.2 Vector spaces and linear operators	14	(3)
1.3 Review of facts about ordinary differential equations	17	(2)
2 First-Order Equations	19	(54)
2.1 Generalities	19	(2)
2.2 First-order linear PDE's	21	(9)
2.2.1 Constant coefficients	22	(3)
2.2.2 Spatially dependent velocity of propagation	25	(5)
2.3 Nonlinear conservation laws	30	(9)
2.4 Linearization	39	(2)
2.5 Weak solutions	41	(12)
2.5.1 The notion of a weak solution	41	(2)
2.5.2 Weak solutions of u(1) + F(u)(x) = 0	43	(2)
2.5.3 The Riemann problem	45	(2)
2.5.4 Formation of shock waves	47	(1)
2.5.5 Nonuniqueness and stability of weak solutions	48	(5)
2.6 Numerical methods	53	(11)
2.6.1 Difference quotients	53	(2)
2.6.2 A finite difference scheme	55	(2)
2.6.3 An upwind scheme and the CFL condition	57	(3)
2.6.4 A scheme for the nonlinear conservation law	60	(4)
2.7 A conservation law for cell dynamics	64	(6)
2.7.1 A nonreproducing model	64	(3)
2.7.2 The mitosis boundary condition	67	(3)
2.8 Projects	70	(3)
3 Diffusion	73	(38)
3.1 The diffusion equation	73	(4)
3.2 The maximum principle	77	(4)
3.3 The heat equation without boundaries	81	(14)
3.3.1 The fundamental solution	81	(4)
3.3.2 Solution of the initial-value problem	85	(4)
3.3.3 Sources and the principle of Duhamel	89	(6)
3.4 Boundary value problems on the half-line	95	(6)
3.5 Diffusion and nonlinear wave motion	101	(4)
3.6 Numerical methods for the heat equation	105	(5)
3.7 Projects	110	(1)
4 Boundary Value Problems for the Heat Equation	111	(46)
4.1 Separation of variables	111	(5)
4.2 Convergence of the eigenfunction expansions	116	(14)
4.3 Symmetric boundary conditions	130	(11)
4.4 Inhomogeneous problems and asymptotic behavior	141	(12)
4.5 Projects	153	(4)
5 Waves Again	157	(62)
5.1 Acoustics	157	(3)
5.1.1 The equations of gas dynamics	157	(2)
5.1.2 The linearized equations	159	(1)
5.2 The vibrating string	160	(5)
5.2.1 The nonlinear model	160	(3)
5.2.2 The linearized equation	163	(2)
5.3 The wave equation without boundaries	165	(16)
5.3.1 The initial-value problem and d'Alembert's formula	165	(5)
5.3.2 Domains of influence and dependence	170	(1)
5.3.3 Conservation of energy on the line	170	(4)
5.3.4 An inhomogeneous problem	174	(7)
5.4 Boundary value problems on the half-line	181	(11)
5.4.1 d'Alembert's formula extended	181	(5)
5.4.2 A transmission problem	186	(1)
5.4.3 Inhomogeneous problems	187	(5)
5.5 Boundary value problems on a finite interval	192	(16)
5.5.1 A geometric construction	192	(1)
5.5.2 Modes of vibration	193	(3)
5.5.3 Conservation of energy for the finite interval	196	(2)
5.5.4 Other boundary conditions	198	(1)
5.5.5 Inhomogeneous equations	199	(2)
5.5.6 Boundary forcing and resonance	201	(7)
5.6 Numerical methods	208	(3)
5.7 A nonlinear wave equation	211	(6)
5.8 Projects	217	(2)
6 Fourier Series and Fourier Transform	219	(40)
6.1 Fourier series	219	(4)
6.2 Convergence of Fourier series	223	(8)
6.3 The Fourier transform	231	(5)
6.4 The heat equation again	236	(2)
6.5 The discrete Fourier transform	238	(12)
6.5.1 The DFT and Fourier series	238	(6)
6.5.2 The DFT and the Fourier transform	244	(6)
6.6 The fast Fourier transform (FFT)	250	(7)
6.7 Projects	257	(2)
7 Dispersive Waves and the Schrodinger Equation	259	(38)
7.1 Oscillatory integrals and the method of stationary phase	259	(4)
7.2 Dispersive equation	263	(11)
7.2.1 The wave equation	263	(1)
7.2.2 Dispersion relations	264	(3)
7.2.3 Group velocity and phase velocity	267	(7)
7.3 Quantum mechanics and the uncertainty principle	274	(4)
7.4 The Schrodinger equation	278	(9)
7.4.1 The dispersion relation of the Schrodinger equation	278	(3)
7.4.2 The correspondence principle	281	(1)
7.4.3 The initial-value problem for the free Schrodinger equation	282	(5)
7.5 The spectrum of the Schrodinger operator	287	(9)
7.5.1 Continuous spectrum	287	(3)
7.5.2 Bound states of the square well potential	290	(6)
7.6 Projects	296	(1)
8 The Heat and Wave Equations in Higher Dimensions	297	(70)
8.1 Diffusion in higher dimensions	297	(7)
8.1.1 Derivation of the heat equation	297	(1)
8.1.2 The fundamental solution of the heat equation	298	(6)
8.2 Boundary value problems for the heat equation	304	(6)
8.3 Eigenfunctions for the rectangle	310	(5)
8.4 Eigenfunctions for the disk	315	(7)
8.5 Asymptotics and steady-state solutions	322	(9)
8.5.1 Approach to the steady state	322	(3)
8.5.2 Compatibility of source and boundary flux	325	(6)
8.6 The wave equation	331	(8)
8.6.1 The initial-value problem	331	(4)
8.6.2 The method of descent	335	(4)
8.7 Energy	339	(4)
8.8 Sources	343	(4)
8.9 Boundary value problems for the wave equation	347	(9)
8.9.1 Eigenfunction expansions	347	(2)
8.9.2 Nodal curves	349	(1)
8.9.3 Conservation of energy	349	(2)
8.9.4 Inhomogeneous problems	351	(5)
8.10 The Maxwell equations	356	(9)
8.10.1 The electric and magnetic fields	356	(3)
8.10.2 The initial-value problem	359	(1)
8.10.3 Plane waves	359	(1)
8.10.4 Electrostatics	360	(1)
8.10.5 Conservation of energy	361	(4)
8.11 Projects	365	(2)
9 Equilibrium	367	(58)
9.1 Harmonic functions	367	(10)
9.1.1 Examples	367	(1)
9.1.2 The mean value property	368	(4)
9.1.3 The maximum principle	372	(5)
9.2 The Dirichlet problem	377	(12)
9.2.1 Fourier series solution in the disk	377	(7)
9.2.2 Liouville's theorem	384	(5)
9.3 The Dirichlet problem in a rectangle	389	(5)
9.4 The Poisson equation	394	(20)
9.4.1 The Poisson equation without boundaries	394	(5)
9.4.2 The Green's function	399	(15)
9.5 Variational methods and weak solutions	414	(9)
9.5.1 Problems in variational form	414	(3)
9.5.2 The Rayleigh-Ritz procedure	417	(6)
9.6 Projects	423	(2)
10 Numerical Methods for Higher Dimensions	425	(30)
10.1 Finite differences	425	(8)
10.2 Finite elements	433	(9)
10.3 Galerkin methods	442	(8)
10.4 A reaction-diffusion equation	450	(5)
11 Epilogue: Classification	455	(4)
Appendices	459	(74)
A Recipes and Formulas	459	(18)
A.1 Separation of variables in space-time problems	459	(5)
A.2 Separation of variables in steady-state problems	464	(7)
A.3 Fundamental solutions	471	(3)
A.4 The Laplace operator in polar and spherical coordinates	474	(3)
B Elements of MATLAB	477	(20)
B.1 Forming vectors and matrices	477	(3)
B.2 Operations on matrices	480	(1)
B.3 Array operations	481	(1)
B.4 Solution of linear systems	482	(1)
B.5 MATLAB functions and mfiles	483	(2)
B.6 Script mfiles and programs	485	(1)
B.7 Vectorizing computations	486	(2)
B.8 Function functions	488	(2)
B.9 Plotting 2-D graphs	490	(2)
B.10 Plotting 3-D graphs	492	(4)
B.11 Movies	496	(1)
C References	497	(4)
D Solutions to Selected Problems	501	(26)
E List of Computer Programs	527	(6)
Index	533