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E-raamat: Fourier Series and Numerical Methods for Partial Differential Equations [Wiley Online]

(Luther College)
  • Formaat: 332 pages, Drawings: 15 B&W, 0 Color; Graphs: 85 B&W, 0 Color
  • Ilmumisaeg: 20-Aug-2010
  • Kirjastus: John Wiley & Sons Inc
  • ISBN-10: 470651385
  • ISBN-13: 9780470651384
Teised raamatud teemal:
  • Wiley Online
  • Hind: 122,65 €*
  • * hind, mis tagab piiramatu üheaegsete kasutajate arvuga ligipääsu piiramatuks ajaks
Fourier Series and Numerical Methods for Partial Differential EquationsSuurem pilt
  • Formaat: 332 pages, Drawings: 15 B&W, 0 Color; Graphs: 85 B&W, 0 Color
  • Ilmumisaeg: 20-Aug-2010
  • Kirjastus: John Wiley & Sons Inc
  • ISBN-10: 470651385
  • ISBN-13: 9780470651384
Teised raamatud teemal:
Wiley Online e-raamatudRohkem infot Wiley Online kohta
Raamatu kodulehekülg: https://onlinelibrary.wiley.com/doi/book/10.1002/9780470651384
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

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 multidimensional 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.



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.

Preface xi
Acknowledgments xiii
1 Introduction
1(18)
1.1 Terminology and Notation
1(1)
1.2 Classification
2(1)
1.3 Canonical Forms
3(1)
1.4 Common PDEs
4(1)
1.5 Cauchy-Kowalevski Theorem
5(2)
1.6 Initial Boundary Value Problems
7(1)
1.7 Solution Techniques
8(1)
1.8 Separation of Variables
9(10)
Exercises
15(4)
2 Fourier Series
19(50)
2.1 Vector Spaces
19(4)
2.1.1 Subspaces
21(1)
2.1.2 Basis and Dimension
21(1)
2.1.3 Inner Products
22(1)
2.2 The Integral as an Inner Product
23(3)
2.2.1 Piecewise Continuous Functions
24(1)
2.2.2 Inner Product on Cp (a, b)
25(1)
2.3 Principle of Superposition
26(4)
2.3.1 Finite Case
26(2)
2.3.2 Infinite Case
28(2)
2.3.3 Hubert Spaces
30(1)
2.4 General Fourier Series
30(1)
2.5 Fourier Sine Series on (0, c)
31(4)
2.5.1 Odd, Periodic Extensions
33(2)
2.6 Fourier Cosine Series on (0, c)
35(2)
2.6.1 Even, Periodic Extensions
36(1)
2.7 Fourier Series on (-c, c)
37(3)
2.7.1 2c-Periodic Extensions
40(1)
2.8 Best Approximation
40(5)
2.9 Bessel's Inequality
45(1)
2.10 Piecewise Smooth Functions
46(4)
2.11 Fourier Series Convergence
50(8)
2.11.1 Alternate Form
50(1)
2.11.2 Riemann-Lebesgue Lemma
51(1)
2.11.3 A Dirichlet Kernel Lemma
52(2)
2.11.4 A Fourier Theorem
54(4)
2.12 2c-Periodic Functions
58(3)
2.13 Concluding Remarks
61(8)
Exercises
62(7)
3 Sturm-Liouville Problems
69(28)
3.1 Basic Examples
69(1)
3.2 Regular Sturm-Liouville Problems
70(1)
3.3 Properties
71(8)
3.3.1 Eigenfunction Orthogonality
72(2)
3.3.2 Real Eigenvalues
74(1)
3.3.3 Eigenfunction Uniqueness
75(2)
3.3.4 Non-negative Eigenvalues
77(2)
3.4 Examples
79(6)
3.4.1 Neumann Boundary Conditions on [ 0, c]
79(1)
3.4.2 Robin and Neumann BCs
80(3)
3.4.3 Periodic Boundary Conditions
83(2)
3.5 Bessel's Equation
85(5)
3.6 Legendre's Equation
90(7)
Exercises
93(4)
4 Heat Equation
97(16)
4.1 Heat Equation in 1D
97(3)
4.2 Boundary Conditions
100(1)
4.3 Heat Equation in 2D
101(2)
4.4 Heat Equation in 3D
103(2)
4.5 Polar-Cylindrical Coordinates
105(3)
4.6 Spherical Coordinates
108(5)
Exercises
108(5)
5 Heat Transfer in 1D
113(26)
5.1 Homogeneous IBVP
113(3)
5.1.1 Example: Insulated Ends
114(2)
5.2 Semihomogeneous PDE
116(4)
5.2.1 Variation of Parameters
117(2)
5.2.2 Example: Semihomogeneous IBVP
119(1)
5.3 Nonhomogeneous Boundary Conditions
120(11)
5.3.1 Example: Nonhomogeneous Boundary Condition
122(3)
5.3.2 Example: Time-Dependent Boundary Condition
125(2)
5.3.3 Laplace Transforms
127(1)
5.3.4 Duhamel's Theorem
128(3)
5.4 Spherical Coordinate Example
131(8)
Exercises
133(6)
6 Heat Transfer in 2D and 3D
139(42)
6.1 Homogeneous 2D IBVP
139(4)
6.1.1 Example: Homogeneous IBVP
142(1)
6.2 Semihomogeneous 2D IBVP
143(4)
6.2.1 Example: Internal Source or Sink
146(1)
6.3 Nonhomogeneous 2D IBVP
147(3)
6.4 2D BVP: Laplace and Poisson Equations
150(19)
6.4.1 Dirichlet Problems
150(4)
6.4.2 Dirichlet Example
154(2)
6.4.3 Neumann Problems
156(3)
6.4.4 Neumann Example
159(4)
6.4.5 Dirichlet, Neumann BC Example
163(3)
6.4.6 Poisson Problems
166(3)
6.5 Nonhomogeneous 2D Example
169(1)
6.6 Time-Dependent BCs
170(3)
6.7 Homogeneous 3D IBVP
173(8)
Exercises
176(5)
7 Wave Equation
181(26)
7.1 Wave Equation in 1D
181(18)
7.1.1 d'Alembert's Solution
184(3)
7.1.2 Homogeneous IBVP: Series Solution
187(3)
7.1.3 Semihomogeneous IBVP
190(3)
7.1.4 Nonhomogeneous IBVP
193(2)
7.1.5 Homogeneous IBVP in Polar Coordinates
195(4)
7.2 Wave Equation in 2D
199(8)
7.2.1 2D Homogeneous Solution
199(3)
Exercises
202(5)
8 Numerical Methods: an Overview
207(12)
8.1 Grid Generation
208(6)
8.1.1 Adaptive Grids
210(2)
8.1.2 Multilevel Methods
212(2)
8.2 Numerical Methods
214(4)
8.2.1 Finite Difference Method
214(2)
8.2.2 Finite Element Method
216(1)
8.2.3 Finite Analytic Method
217(1)
8.3 Consistency and Convergence
218(1)
9 The Finite Difference Method
219(30)
9.1 Discretization
219(3)
9.2 Finite Difference Formulas
222(1)
9.2.1 First Partials
222(1)
9.2.2 Second Partials
222(1)
9.3 ID Heat Equation
223(3)
9.3.1 Explicit Formulation
223(1)
9.3.2 Implicit Formulation
224(2)
9.4 Crank-Nicolson Method
226(1)
9.5 Error and Stability
226(5)
9.5.1 Error Types
226(1)
9.5.2 Stability
227(4)
9.6 Convergence in Practice
231(1)
9.7 1D Wave Equation
231(3)
9.7.1 Implicit Formulation
231(2)
9.7.2 Initial Conditions
233(1)
9.8 2D Heat Equation in Cartesian Coordinates
234(5)
9.9 Two-Dimensional Wave Equation
239(1)
9.10 2D Heat Equation in Polar Coordinates
239(10)
Exercises
244(5)
10 Finite Element Method
249(22)
10.1 General Framework
250(2)
10.2 1D Elliptical Example
252(5)
10.2.1 Reformulations
252(1)
10.2.2 Equivalence in Forms
253(2)
10.2.3 Finite Element Solution
255(2)
10.3 2D Elliptical Example
257(4)
10.3.1 Weak Formulation
257(1)
10.3.2 Finite Element Approximation
258(3)
10.4 Error Analysis
261(3)
10.5 1D Parabolic Example
264(7)
10.5.1 Weak Formulation
264(1)
10.5.2 Method of Lines
265(1)
10.5.3 Backward Euler's Method
266(2)
Exercises
268(3)
11 Finite Analytic Method
271(24)
11.1 1D Transport Equation
272(8)
11.1.1 Finite Analytic Solution
273(2)
11.1.2 FA and FD Coefficient Comparison
275(4)
11.1.3 Hybrid Finite Analytic Solution
279(1)
11.2 2D Transport Equation
280(10)
11.2.1 FA Solution on Uniform Grids
282(5)
11.2.2 The Poisson Equation
287(3)
11.3 Convergence and Accuracy
290(5)
Exercises
291(4)
Appendix A FA 1D Case
295(8)
Appendix B FA 2D Case
303(8)
B.1 The Case θ = 1
308(1)
B.2 The Case θ = Bx + Ay
309(2)
References 311(4)
Index 315
RICHARD A. BERNATZ, PhD, is Professor in the Department of Mathematics at Luther College. Dr. Bernatz is the author of numerous journal articles in his areas of research interest, which include climatology, mathematical models of watersheds, and computational fluid dynamics with applications in meteorology.
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