Separation of Variables for Partial Differential Equations: An Eigenfunction Approach

by Cain; George

• ISBN13: 9781584884200
• ISBN10: 1584884207
• Format: Hardcover
• Copyright: 2005-11-21
• Publisher: Chapman & Hall/

Summary

Adopting the view common in the finite element analysis, the authors of Separation of Variables for Partial Differential Equations: An Eigenfunction Approach introduce a computable separation of variables solution as an analytic approximate solution. At the heart of the text, they consider a general partial differential equation in two independent variables with a source term and subject to boundary and initial conditions. They give an algorithm for approximating and solving the problem and illustrate the application of this approach to the heat, wave, and potential equations. They illustrate the power of the technique by solving a variety of practical problems, many of which go well beyond the usual textbook examples.Written at the advanced undergraduate level, the book will serve equally well as a text for students and as a reference for instructors and users of separation of variables. It requires a background in engineering mathematics, but no prior exposure to separation of variables. The abundant worked examples provide guidance for deciding whether and how to apply the method to any given problem, help in interpreting computed solutions, and give insight into cases in which formal answers may be useless.

Table of Contents

Acknowledgments	v
Preface	vii
Potential, Heat, and Wave Equation	1	(24)
Overview	1	(1)
Classification of second order equations	2	(1)
Laplace's and Poisson's equation	3	(8)
The heat equation	11	(7)
The wave equation	18	(7)
Basic Approximation Theory	25	(20)
Norms and inner products	26	(3)
Projection and best approximation	29	(5)
Important function spaces	34	(11)
Sturm--Liouville Problems	45	(22)
Sturm--Liouville problems for φ = μφ	45	(8)
Sturm--Liouville problems for Lφ = μφ	53	(6)
A Sturm--Liouville problem with an interface	59	(8)
Fourier Series	67	(28)
Introduction	67	(1)
Convergence	68	(6)
Convergence of Fourier series	74	(3)
Cosine and sine series	77	(3)
Operations on Fourier series	80	(4)
Partial sums of the Fourier series and the Gibbs phenomenon	84	(11)
Eigenfunction Expansions for Equations in Two Independent Variables	95	(20)
One-Dimensional Diffusion Equation	115	(46)
Applications of the eigenfunction expansion method	115	(36)
Example 6.1 How many terms of the series solution are enough?	115	(4)
Example 6.2 Determination of an unknown diffusivity from measured data	119	(1)
Example 6.3 Thermal waves	120	(5)
Example 6.4 Matching a temperature history	125	(4)
Example 6.5 Phase shift for a thermal wave	129	(2)
Example 6.6 Dynamic determination of a convective heat transfer coefficient from measured data	131	(3)
Example 6.7 Radial heat flow in a sphere	134	(3)
Example 6.8 A boundary layer problem	137	(2)
Example 6.9 The Black--Scholes equation	139	(3)
Example 6.10 Radial heat flow in a disk	142	(4)
Example 6.11 Heat flow in a composite slab	146	(2)
Example 6.12 Reaction-diffusion with blowup	148	(3)
Convergence of uN (x, t) to the analytic solution	151	(4)
Influence of the boundary conditions and Duhamel's solution	155	(6)
One-Dimensional Wave Equation	161	(34)
Applications of the eigenfunction expansion method	161	(27)
Example 7.1 A vibrating string with initial displacement	161	(5)
Example 7.2 A vibrating string with initial velocity	166	(2)
Example 7.3 A forced wave and resonance	168	(3)
Example 7.4 Wave propagation in a resistive medium	171	(4)
Example 7.5 Oscillations of a hanging chain	175	(2)
Example 7.6 Symmetric pressure wave in a sphere	177	(3)
Example 7.7 Controlling the shape of a wave	180	(2)
Example 7.8 The natural frequencies of a uniform beam	182	(3)
Example 7.9 A system of wave equations	185	(3)
Convergence of uN (x, t) to the analytic solution	188	(2)
Eigenfunction expansions and Duhamel's principle	190	(5)
Potential Problems in the Plane	195	(60)
Applications of the eigenfunction expansion method	195	(42)
Example 8.1 The Dirichlet problem for the Laplacian on a rectangle	195	(6)
Example 8.2 Preconditioning for general boundary data	201	(12)
Example 8.3 Poisson's equation with Neumann boundary data	213	(2)
Example 8.4 A discontinuous potential	215	(3)
Example 8.5 Lubrication of a plane slider bearing	218	(2)
Example 8.6 Lubrication of a step bearing	220	(1)
Example 8.7 The Dirichlet problem on an L-shaped domain	221	(4)
Example 8.8 Poisson's equation in polar coordinates	225	(5)
Example 8.9 Steady-state heat flow around an insulated pipe I	230	(2)
Example 8.10 Steady-state heat flow around an insulated pipe II	232	(2)
Example 8.11 Poisson's equation on a triangle	234	(3)
Eigenvalue problem for the two-dimensional Laplacian	237	(10)
Example 8.12 The eigenvalue problem for the Laplacian on a rectangle	237	(2)
Example 8.13 The Green's function for the Laplacian on a square	239	(4)
Example 8.14 The eigenvalue problem for the Laplacian on a disk	243	(1)
Example 8.15 The eigenvalue problem for the Laplacian on the surface of a sphere	244	(3)
Convergence of uN(x, y) to the analytic solution	247	(8)
Multidimensional Problems	255	(22)
Applications of the eigenfunction expansion method	255	(10)
Example 9.1 A diffusive pulse test	255	(3)
Example 9.2 Standing waves on a circular membrane	258	(2)
Example 9.3 The potential inside a charged sphere	260	(1)
Example 9.4 Pressure in a porous slider bearing	261	(4)
The eigenvalue problem for the Laplacian in R3	265	(12)
Example 9.5 An eigenvalue problem for quadrilaterals	265	(1)
Example 9.6 An eigenvalue problem for the Laplacian in a cylinder	266	(1)
Example 9.7 Periodic heat flow in a cylinder	267	(2)
Example 9.8 An eigenvalue problem for the Laplacian in a sphere	269	(2)
Example 9.9 The eigenvalue problem for Schrodinger's equation with a spherically symmetric potential well	271	(6)
Bibliography	277	(2)
Index	279