Principles of Partial Differential Equations

✍ Scribed by Alexander Komech, Andrew Komech

• Publisher: Springer
• Year: 2009
• Tongue: English
• Leaves: 172
• Series: Problem books in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This concise book covers the classical tools of PDE theory used in today's science and engineering: characteristics, the wave propagation, the Fourier method, distributions, Sobolev spaces, fundamental solutions, and Green's functions. The approach is problem-oriented, giving the reader an opportunity to master solution techniques. The theoretical part is rigorous and with important details presented with care. Hints are provided to help the reader restore the arguments to their full rigor. Many examples from physics are intended to keep the book intuitive and to illustrate the applied nature of the subject. The book is useful for a higher-level undergraduate course and for self-study.

✦ Table of Contents

Cover......Page 1
Series: Problem Books in Mathematics......Page 2
Principles of Partial Differential Equations......Page 4
Copyright......Page 5
Preface......Page 6
Acknowledgements......Page 8
Contents......Page 10
1 Derivation of the d’Alembert equation......Page 12
2 The d’Alembert method for infinite string......Page 18
3 Analysis of the d’Alembert formula......Page 23
4 Second-order hyperbolic equations in the plane......Page 30
5 Semi-infinite string......Page 41
6 Finite string......Page 55
7 Wave equation with many independent variables......Page 57
8 General hyperbolic equations......Page 67
9 Derivation of the heat equation......Page 76
10 Mixed problem for the heat equation......Page 78
11 The Sturm – Liouville problem......Page 79
12 Eigenfunction expansions......Page 85
13 The Fourier method for the heat equation......Page 89
14 Mixed problem for the d’Alembert equation......Page 94
15 The Fourier method for nonhomogeneous equations......Page 97
16 The Fourier method for nonhomogeneous boundary conditions......Page 104
17 The Fourier method for the Laplace equation......Page 106
18 Motivation......Page 116
19 Distributions......Page 120
20 Operations on distributions......Page 121
21 Differentiation of jumps and the product rule......Page 126
22 Fundamental solutions of ordinary differential equations......Page 129
23 Green’s function on an interval......Page 132
24 Solvability condition for the boundary value problems......Page 136
25 The Sobolev functional spaces......Page 139
26 Well-posedness of the wave equation in the Sobolev spaces......Page 141
27 Solutions to the wave equation in the sense of distributions......Page 142
28 Fundamental solutions of the Laplace operator in \mathbb{R}_n......Page 144
29 Potentials and their properties......Page 148
30 Computing potentials via the Gauss theorem......Page 154
31 Method of reflections......Page 155
32 Green’s functions in 2D via conformal mappings......Page 160
A. Classification of the second-order equations......Page 166
References......Page 170
Index......Page 172