This text is written for the standard, one-semester, undergraduate course in elementary partial differential equations. The topics include derivations of some of the standard equations of mathematical physics (including the heat equation, the wave equation, and Laplace's equation) and methods for solving those equations on bounded and unbounded domains. Methods include eigenfunction expansions, or separation of variables, and methods based on Fourier and Laplace transforms.
Author(s): John David Logan
Series: Undergraduate Texts in Mathematics
Edition: 2nd
Publisher: Springer
Year: 2004
Language: English
Pages: 221
Tags: Математика;Дифференциальные уравнения;Дифференциальные уравнения в частных производных;
Preface to the Second Edition......Page 6
To the Student......Page 9
Contents......Page 10
1.1 Mathematical Models......Page 12
1.2 Conservation Laws......Page 20
1.3 Diffusion......Page 27
1.4 P DEs in Biology......Page 33
1.5 Vibrations and Acoustics......Page 43
1.6 Quantum Mechanics *......Page 50
1.7 Heat Flow in Three Dimensions......Page 53
1.8 Laplace's Equation......Page 58
1.9 Classification of PD Es......Page 63
2.1 Cauchy Problem for the Heat Equation......Page 69
2.2 Cauchy Problem for the Wave Equation......Page 75
2.3 Ill-Posed Problems......Page 80
2.4 Semi-Infinite Domains......Page 83
2.5 Sources and Duhamel's Principle......Page 87
2.6 Laplace 1tansforms......Page 92
2.7 Fourier 1tansforms......Page 97
2.8 Solving PDEs Using Computer Algebra Systems*......Page 103
3.1 The Fourier Method......Page 107
3.2 Orthogonal Expansions......Page 109
3.3 Classical Fourier Series......Page 118
3.4 Sturm-Liouville Problems......Page 123
4.1 Separation of Variables......Page 132
4.2 Flux and Radiation Conditions......Page 140
4.3 Laplace's Equation......Page 147
4.4 Cooling of a Sphere......Page 154
4.5 Diffusion in a Disk......Page 159
4.6 Sources on Bounded Domains......Page 164
4.7 Parameter Identification Problems *......Page 167
4.8 Finite Difference Methods *......Page 172
5.1 Age-Structured Models......Page 183
5.2 1taveling Wave Fronts......Page 192
5.3 Equilibria and Stability......Page 198
Appendix: Ordinary Differential Equations......Page 208
Table of Laplace Thansforms......Page 214
References......Page 215
Index......Page 217
