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Fourier Series and Numerical Methods for Partial Differential Equations

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The importance of partial differential equations (PDEs) in modeling phenomena in engineering as well as in the physical, natural, and social sciences is well known by students and practitioners in these fields. Striking a balance between theory and applications, Fourier Series and Numerical Methods for Partial Differential Equations presents an introduction to the analytical and numerical methods that are essential for working with partial differential equations. Combining methodologies from calculus, introductory linear algebra, and ordinary differential equations (ODEs), the book strengthens and extends readers' knowledge of the power of linear spaces and linear transformations for purposes of understanding and solving a wide range of PDEs.

The book begins with an introduction to the general terminology and topics related to PDEs, including the notion of initial and boundary value problems and also various solution techniques. Subsequent chapters explore: 

  • The solution process for Sturm-Liouville boundary value ODE problems and a Fourier series representation of the solution of initial boundary value problems in PDEs
  • The concept of completeness, which introduces readers to Hilbert spaces 
  • The application of Laplace transforms and Duhamel's theorem to solve time-dependent boundary conditions
  •  The finite element method, using finite dimensional subspaces
  •  The finite analytic method with applications of the Fourier series methodology to linear version of non-linear PDEs

 Throughout the book, the author incorporates his own class-tested material, ensuring an accessible and easy-to-follow presentation that helps readers connect presented objectives with relevant applications to their own work. Maple is used throughout to solve many exercises, and a related Web site features Maple worksheets for readers to use when working with the book's one- and multi-dimensional problems.

Fourier Series and Numerical Methods for Partial Differential Equations is an ideal book for courses on applied mathematics and partial differential equations at the upper-undergraduate and graduate levels. It is also a reliable resource for researchers and practitioners in the fields of mathematics, science, and engineering who work with mathematical modeling of physical phenomena, including diffusion and wave aspects.

Author(s): Richard Bernatz
Edition: 1
Publisher: Wiley
Year: 2010

Language: English
Pages: 337
Tags: Математика;Дифференциальные уравнения;Дифференциальные уравнения в частных производных;

FOURIER SERIES AND NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS......Page 6
CONTENTS......Page 10
Preface......Page 16
Acknowledgments......Page 18
1.1 Terminology and Notation......Page 20
1.2 Classification......Page 21
1.3 Canonical Forms......Page 22
1.4 Common PDEs......Page 23
1.5 Cauchy–Kowalevski Theorem......Page 24
1.6 Initial Boundary Value Problems......Page 26
1.7 Solution Techniques......Page 27
1.8 Separation of Variables......Page 28
Exercises......Page 34
2.1 Vector Spaces......Page 38
2.1.2 Basis and Dimension......Page 40
2.1.3 Inner Products......Page 41
2.2 The Integral as an Inner Product......Page 42
2.2.1 Piecewise Continuous Functions......Page 43
2.2.2 Inner Product on Cp(a, b)......Page 44
2.3.1 Finite Case......Page 45
2.3.2 Infinite Case......Page 47
2.4 General Fourier Series......Page 49
2.5 Fourier Sine Series on (0, c)......Page 50
2.5.1 Odd, Periodic Extensions......Page 52
2.6 Fourier Cosine Series on (0, c)......Page 54
2.6.1 Even, Periodic Extensions......Page 55
2.7 Fourier Series on (–c,c)......Page 56
2.8 Best Approximation......Page 59
2.9 Bessel's Inequality......Page 64
2.10 Piecewise Smooth Functions......Page 65
2.11.1 Alternate Form......Page 69
2.11.2 Riemann–Lebesgue Lemma......Page 70
2.11.3 A Dirichlet Kernel Lemma......Page 71
2.11.4 A Fourier Theorem......Page 73
2.12 2c-Periodic Functions......Page 77
2.13 Concluding Remarks......Page 80
Exercises......Page 81
3.1 Basic Examples......Page 88
3.2 Regular Sturm–Liouville Problems......Page 89
3.3 Properties......Page 90
3.3.1 Eigenfunction Orthogonality......Page 91
3.3.2 Real Eigenvalues......Page 93
3.3.3 Eigenfunction Uniqueness......Page 94
3.3.4 Non-negative Eigenvalues......Page 96
3.4.1 Neumann Boundary Conditions [0, c]......Page 98
3.4.2 Robin and Neumann BCs......Page 99
3.4.3 Periodic Boundary Conditions......Page 102
3.5 Bessel's Equation......Page 104
3.6 Legendre's Equation......Page 109
Exercises......Page 112
4.1 Heat Equation in 1D......Page 116
4.2 Boundary Conditions......Page 119
4.3 Heat Equation in 2D......Page 120
4.4 Heat Equation in 3D......Page 122
4.5 Polar-Cylindrical Coordinates......Page 124
Exercises......Page 127
5.1 Homogeneous IBVP......Page 132
5.1.1 Example: Insulated Ends......Page 133
5.2 Semihomogeneous PDE......Page 135
5.2.1 Variation of Parameters......Page 136
5.2.2 Example: Semihomogeneous IBVP......Page 138
5.3 Nonhomogeneous Boundary Conditions......Page 139
5.3.1 Example: Nonhomogeneous Boundary Condition......Page 141
5.3.2 Example: Time-Dependent Boundary Condition......Page 144
5.3.3 Laplace Transforms......Page 146
5.3.4 Duhamel's Theorem......Page 147
5.4 Spherical Coordinate Example......Page 150
Exercises......Page 152
6.1 Homogeneous 2D IBVP......Page 158
6.1.1 Example: Homogeneous IBVP......Page 161
6.2 Semihomogeneous 2D IBVP......Page 162
6.2.1 Example: Internal Source or Sink......Page 165
6.3 Nonhomogeneous 2D IBVP......Page 166
6.4.1 Dirichlet Problems......Page 169
6.4.2 Dirichlet Example......Page 173
6.4.3 Neumann Problems......Page 175
6.4.4 Neumann Example......Page 178
6.4.5 Dirichlet, Neumann BC Example......Page 182
6.4.6 Poisson Problems......Page 185
6.5 Nonhomogeneous 2D Example......Page 188
6.6 Time-Dependent BCs......Page 189
6.7 Homogeneous 3D IBVP......Page 192
Exercises......Page 195
7.1 Wave Equation in ID......Page 200
7.1.1 d'Alembert's Solution......Page 203
7.1.2 Homogeneous IBVP: Series Solution......Page 206
7.1.3 Semihomogeneous IBVP......Page 209
7.1.4 Nonhomogeneous IBVP......Page 212
7.1.5 Homogeneous IBVP in Polar Coordinates......Page 214
7.2.1 2D Homogeneous Solution......Page 218
Exercises......Page 221
8 Numerical Methods: an Overview......Page 226
8.1 Grid Generation......Page 227
8.1.1 Adaptive Grids......Page 229
8.1.2 Multilevel Methods......Page 231
8.2.1 Finite Difference Method......Page 233
8.2.2 Finite Element Method......Page 235
8.2.3 Finite Analytic Method......Page 236
8.3 Consistency and Convergence......Page 237
9.1 Discretization......Page 238
9.2.2 Second Partíais......Page 241
9.3.1 Explicit Formulation......Page 242
9.3.2 Implicit Formulation......Page 243
9.5.1 Error Types......Page 245
9.5.2 Stability......Page 246
9.7.1 Implicit Formulation......Page 250
9.7.2 Initial Conditions......Page 252
9.8 2D Heat Equation in Cartesian Coordinates......Page 253
9.10 2D Heat Equation in Polar Coordinates......Page 258
Exercises......Page 263
10 Finite Element Method......Page 268
10.1 General Framework......Page 269
10.2.1 Reformulations......Page 271
10.2.2 Equivalence in Forms......Page 272
10.2.3 Finite Element Solution......Page 274
10.3.1 Weak Formulation......Page 276
10.3.2 Finite Element Approximation......Page 277
10.4 Error Analysis......Page 280
10.5.1 Weak Formulation......Page 283
10.5.2 Method of Lines......Page 284
10.5.3 Backward Euler's Method......Page 285
Exercises......Page 287
11 Finite Analytic Method......Page 290
11.1 1D Transport Equation......Page 291
11.1.1 Finite Analytic Solution......Page 292
11.1.2 FA and FD Coefficient Comparison......Page 294
11.1.3 Hybrid Finite Analytic Solution......Page 298
11.2 2D Transport Equation......Page 299
11.2.1 FA Solution on Uniform Grids......Page 301
11.2.2 The Poisson Equation......Page 306
11.3 Convergence and Accuracy......Page 309
Exercises......Page 310
Appendix A: FA 1D Case......Page 314
Appendix B: FA 2D Case......Page 322
B.l The Case ø = 1......Page 327
B.2 The Case ø = –Bx + Ay......Page 328
References......Page 330
Index......Page 334