Partial differential equations analytical and numerical methods

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Title:

Personal Author:

Gockenbach, Mark S.

Publication Information:

Philadelphia, PA : Society for Industrial and Applied Mathematics, 2002

Physical Description:

1 CD-ROM ; 12 cm.

ISBN:

9780898715187

General Note:

Accompanies text of the same title : QA377 G62 2002

Subject Term:

Differential equations, Partial

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
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This introductory text on partial differential equations is the first to integrate modern and classical techniques for solving PDEs at a level suitable for undergraduates. The author successfully complements the classical topic of Fourier series with modern finite element methods. The result is an up-to-date, powerful, and flexible approach to solving PDEs, which both faculty and students will find refreshing, challenging, and rewarding. Linear algebra is a key component of the text, providing a framework both for computing solutions and for understanding the theoretical basis of the methods. Although techniques are emphasized over theory, the methods are presented in a mathematically sound fashion to develop a strong foundation for further study. Numerous exercises and examples involve meaningful experiments with realistic physical parameters, allowing students to use physical intuition to understand the qualitative features of the solutions.

Author Notes

Mark S. Gockenbach is an Associate Professor of Mathematical Sciences at Michigan Technological University

Foreword	p. xiii
Preface	p. xvii
1 Classification of differential equations	p. 1
2 Models in one dimension	p. 9
2.1 Heat flow in a bar; Fourier's law	p. 9
2.1.1 Boundary and initial conditions for the heat equation	p. 13
2.1.2 Steady-state heat flow	p. 14
2.1.3 Diffusion	p. 16
2.2 The hanging bar	p. 21
2.2.1 Boundary conditions for the hanging bar	p. 24
2.3 The wave equation for a vibrating string	p. 27
2.4 Suggestions for further reading	p. 30
3 Essential linear algebra	p. 31
3.1 Linear systems as linear operator equations	p. 31
3.2 Existence and uniqueness of solutions to Ax = b	p. 38
3.2.1 Existence	p. 38
3.2.2 Uniqueness	p. 42
3.2.3 The Fredholm alternative	p. 45
3.3 Basis and dimension	p. 50
3.4 Orthogonal bases and projections	p. 55
3.4.1 The L[superscript 2] inner product	p. 58
3.4.2 The projection theorem	p. 61
3.5 Eigenvalues and eigenvectors of a symmetric matrix	p. 68
3.5.1 The transpose of a matrix and the dot product	p. 71
3.5.2 Special properties of symmetric matrices	p. 72
3.5.3 The spectral method for solving Ax = b	p. 74
3.6 Preview of methods for solving ODEs and PDEs	p. 77
3.7 Suggestions for further reading	p. 78
4 Essential ordinary differential equations	p. 79
4.1 Converting a higher-order equation to a first-order system	p. 79
4.2 Solutions to some simple ODEs	p. 82
4.2.1 The general solution of a second-order homogeneous ODE with constant coefficients	p. 82
4.2.2 A special inhomogeneous second-order linear ODE	p. 85
4.2.3 First-order linear ODEs	p. 87
4.3 Linear systems with constant coefficients	p. 91
4.3.1 Homogeneous systems	p. 92
4.3.2 Inhomogeneous systems and variation of parameters	p. 96
4.4 Numerical methods for initial value problems	p. 101
4.4.1 Euler's method	p. 102
4.4.2 Improving on Euler's method: Runge-Kutta methods	p. 104
4.4.3 Numerical methods for systems of ODEs	p. 108
4.4.4 Automatic step control and Runge-Kutta-Fehlberg methods	p. 110
4.5 Stiff systems of ODEs	p. 115
4.5.1 A simple example of a stiff system	p. 117
4.5.2 The backward Euler method	p. 118
4.6 Green's functions	p. 123
4.6.1 The Green's function for a first-order linear ODE	p. 123
4.6.2 The Dirac delta function	p. 125
4.6.3 The Green's function for a second-order IVP	p. 126
4.6.4 Green's functions for PDEs	p. 127
4.7 Suggestions for further reading	p. 128
5 Boundary value problems in statics	p. 131
5.1 The analogy between BVPs and linear algebraic systems	p. 131
5.1.1 A note about direct integration	p. 141
5.2 Introduction to the spectral method; eigenfunctions	p. 144
5.2.1 Eigenpairs of -d[superscript 2]/dx[superscript 2] under Dirichlet conditions	p. 144
5.2.2 Representing functions in terms of eigenfunctions	p. 146
5.2.3 Eigenfunctions under other boundary conditions; other Fourier series	p. 150
5.3 Solving the BVP using Fourier series	p. 155
5.3.1 A special case	p. 155
5.3.2 The general case	p. 156
5.3.3 Other boundary conditions	p. 161
5.3.4 Inhomogeneous boundary conditions	p. 164
5.3.5 Summary	p. 166
5.4 Finite element methods for BVPs	p. 172
5.4.1 The principle of virtual work and the weak form of a BVP	p. 173
5.4.2 The equivalence of the strong and weak forms of the BVP	p. 177
5.5 The Galerkin method	p. 180
5.6 Piecewise polynomials and the finite element method	p. 188
5.6.1 Examples using piecewise linear finite elements	p. 193
5.6.2 Inhomogeneous Dirichlet conditions	p. 197
5.7 Green's functions for BVPs	p. 202
5.7.1 The Green's function and the inverse of a differential operator	p. 207
5.8 Suggestions for further reading	p. 210
6 Heat flow and diffusion	p. 211
6.1 Fourier series methods for the heat equation	p. 211
6.1.1 The homogeneous heat equation	p. 214
6.1.2 Nondimensionalization	p. 217
6.1.3 The inhomogeneous heat equation	p. 220
6.1.4 Inhomogeneous boundary conditions	p. 222
6.1.5 Steady-state heat flow and diffusion	p. 224
6.1.6 Separation of variables	p. 225
6.2 Pure Neumann conditions and the Fourier cosine series	p. 229
6.2.1 One end insulated; mixed boundary conditions	p. 229
6.2.2 Both ends insulated; Neumann boundary conditions	p. 231
6.2.3 Pure Neumann conditions in a steady-state BVP	p. 237
6.3 Periodic boundary conditions and the full Fourier series	p. 245
6.3.1 Eigenpairs of -d[superscript 2]/dx[superscript 2] under periodic boundary conditions	p. 247
6.3.2 Solving the BVP using the full Fourier series	p. 249
6.3.3 Solving the IBVP using the full Fourier series	p. 252
6.4 Finite element methods for the heat equation	p. 256
6.4.1 The method of lines for the heat equation	p. 260
6.5 Finite elements and Neumann conditions	p. 266
6.5.1 The weak form of a BVP with Neumann conditions	p. 266
6.5.2 Equivalence of the strong and weak forms of a BVP with Neumann conditions	p. 267
6.5.3 Piecewise linear finite elements with Neumann conditions	p. 269
6.5.4 Inhomogeneous Neumann conditions	p. 273
6.5.5 The finite element method for an IBVP with Neumann conditions	p. 274
6.6 Green's functions for the heat equation	p. 279
6.6.1 The Green's function for the one-dimensional heat equation under Dirichlet conditions	p. 280
6.6.2 Green's functions under other boundary conditions	p. 281
6.7 Suggestions for further reading	p. 283
7 Waves	p. 285
7.1 The homogeneous wave equation without boundaries	p. 285
7.2 Fourier series methods for the wave equation	p. 291
7.2.1 Fourier series solutions of the homogeneous wave equation	p. 293
7.2.2 Fourier series solutions of the inhomogeneous wave equation	p. 296
7.2.3 Other boundary conditions	p. 301
7.3 Finite element methods for the wave equation	p. 305
7.3.1 The wave equation with Dirichlet conditions	p. 306
7.3.2 The wave equation under other boundary conditions	p. 312
7.4 Point sources and resonance	p. 318
7.4.1 The wave equation with a point source	p. 318
7.4.2 Another experiment leading to resonance	p. 321
7.5 Suggestions for further reading	p. 324
8 Problems in multiple spatial dimensions	p. 327
8.1 Physical models in two or three spatial dimensions	p. 327
8.1.1 The divergence theorem	p. 328
8.1.2 The heat equation for a three-dimensional domain	p. 330
8.1.3 Boundary conditions for the three-dimensional heat equation	p. 332
8.1.4 The heat equation in a bar	p. 333
8.1.5 The heat equation in two dimensions	p. 334
8.1.6 The wave equation for a three-dimensional domain	p. 334
8.1.7 The wave equation in two dimensions	p. 334
8.1.8 Equilibrium problems and Laplace's equation	p. 335
8.1.9 Green's identities and the symmetry of the Laplacian	p. 336
8.2 Fourier series on a rectangular domain	p. 339
8.2.1 Dirichlet boundary conditions	p. 339
8.2.2 Solving a boundary value problem	p. 345
8.2.3 Time-dependent problems	p. 346
8.2.4 Other boundary conditions for the rectangle	p. 348
8.2.5 Neumann boundary conditions	p. 349
8.2.6 Dirichlet and Neumann problems for Laplace's equation	p. 352
8.2.7 Fourier series methods for a rectangular box in three dimensions	p. 354
8.3 Fourier series on a disk	p. 359
8.3.1 The Laplacian in polar coordinates	p. 360
8.3.2 Separation of variables in polar coordinates	p. 362
8.3.3 Bessel's equation	p. 363
8.3.4 Properties of the Bessel functions	p. 366
8.3.5 The eigenfunctions of the negative Laplacian on the disk	p. 368
8.3.6 Solving PDEs on a disk	p. 372
8.4 Finite elements in two dimensions	p. 377
8.4.1 The weak form of a BVP in multiple dimensions	p. 377
8.4.2 Galerkin's method	p. 378
8.4.3 Piecewise linear finite elements in two dimensions	p. 379
8.4.4 Finite elements and Neumann conditions	p. 388
8.4.5 Inhomogeneous boundary conditions	p. 389
8.5 Suggestions for further reading	p. 392
9 More about Fourier series	p. 393
9.1 The complex Fourier series	p. 394
9.1.1 Complex inner products	p. 395
9.1.2 Orthogonality of the complex exponentials	p. 396
9.1.3 Representing functions with complex Fourier series	p. 397
9.1.4 The complex Fourier series of a real-valued function	p. 398
9.2 Fourier series and the FFT	p. 401
9.2.1 Using the trapezoidal rule to estimate Fourier coefficients	p. 402
9.2.2 The discrete Fourier transform	p. 404
9.2.3 A note about using packaged FFT routines	p. 409
9.2.4 Fast transforms and other boundary conditions; the discrete sine transform	p. 410
9.2.5 Computing the DST using the FFT	p. 411
9.3 Relationship of sine and cosine series to the full Fourier series	p. 415
9.4 Pointwise convergence of Fourier series	p. 419
9.4.1 Modes of convergence for sequences of functions	p. 419
9.4.2 Pointwise convergence of the complex Fourier series	p. 422
9.5 Uniform convergence of Fourier series	p. 436
9.5.1 Rate of decay of Fourier coefficients	p. 436
9.5.2 Uniform convergence	p. 439
9.5.3 A note about Gibbs's phenomenon	p. 443
9.6 Mean-square convergence of Fourier series	p. 444
9.6.1 The space L[superscript 2](-l, l)	p. 445
9.6.2 Mean-square convergence of Fourier series	p. 448
9.6.3 Cauchy sequences and completeness	p. 450
9.7 A note about general eigenvalue problems	p. 455
9.8 Suggestions for further reading	p. 459
10 More about finite element methods	p. 461
10.1 Implementation of finite element methods	p. 461
10.1.1 Describing a triangulation	p. 462
10.1.2 Computing the stiffness matrix	p. 465
10.1.3 Computing the load vector	p. 467
10.1.4 Quadrature	p. 467
10.2 Solving sparse linear systems	p. 473
10.2.1 Gaussian elimination for dense systems	p. 473
10.2.2 Direct solution of banded systems	p. 475
10.2.3 Direct solution of general sparse systems	p. 477
10.2.4 Iterative solution of sparse linear systems	p. 478
10.2.5 The conjugate gradient algorithm	p. 482
10.2.6 Convergence of the CG algorithm	p. 485
10.2.7 Preconditioned CG	p. 486
10.3 An outline of the convergence theory for finite element methods	p. 488
10.3.1 The Sobolev space H[superscript 1 subscript 0]([Omega])	p. 489
10.3.2 Best approximation in the energy norm	p. 491
10.3.3 Approximation by piecewise polynomials	p. 491
10.3.4 Elliptic regularity and L[superscript 2] estimates	p. 492
10.4 Finite element methods for eigenvalue problems	p. 494
10.5 Suggestions for further reading	p. 499
A Proof of Theorem 3.47	p. 501
B Shifting the data in two dimensions	p. 505
B.0.1 Inhomogeneous Dirichlet conditions on a rectangle	p. 505
B.0.2 Inhomogeneous Neumann conditions on a rectangle	p. 508
C Solutions to odd-numbered exercises	p. 515
Bibliography	p. 603
Index	p. 607

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