Partial Differential Equations Analytical Methods and Applications Textbooks in Mathematics 1st Edition by Victor Henner,Tatyana Belozerova,Alexander Nepomnyashchy ISBN 1138339830 978-1138339835

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Product details:

ISBN 10: 1138339830 

ISBN 13: 978-1138339835

Author: Victor Henner, Tatyana Belozerova, Alexander Nepomnyashchy

Table of contents:

1. Introduction

1.1 Basic Definitions
1.2 Examples

2. First-Order Equations

2.1 Linear First-Order Equations
2.2 General Solution
2.3 Initial Condition
2.4 Quasilinear First-Order Equations
2.5 Characteristic Curves
2.6 Examples

3. Second-Order Equations

3.1 Classification of Second-Order Equations
3.2 Canonical Forms
3.3 Hyperbolic Equations
3.4 Elliptic Equations
3.5 Parabolic Equations

4. The Sturm-Liouville Problem

4.1 General Consideration
4.2 Examples of Sturm-Liouville Problems

5. One-Dimensional Hyperbolic Equations

5.1 Wave Equation
5.2 Boundary and Initial Conditions
5.3 Longitudinal Vibrations of a Rod and Electrical Oscillations
5.4 Traveling Waves: D’Alembert Method
5.5 Cauchy Problem for Nonhomogeneous Wave Equation
5.6 The Green’s Function
5.7 Well-Posedness of the Cauchy Problem
5.8 The Fourier Method for Homogeneous Equations
5.9 The Fourier Method for Nonhomogeneous Equations
5.10 The Laplace Transform Method: Simple Cases
5.11 Equations with Nonhomogeneous Boundary Conditions
5.12 Energy in the Harmonics
5.13 Dispersion of Waves
5.14 Cauchy Problem in an Infinite Region
5.15 Propagation of a Wave Train

6. One-Dimensional Parabolic Equations

6.1 Heat Conduction and Diffusion: Boundary Value Problems
6.2 The Fourier Method for Homogeneous Equations
6.3 Nonhomogeneous Equations
6.4 The Green’s Function and Duhamel’s Principle
6.5 Large Time Behavior of Solutions
6.6 Maximum Principle
6.7 The Heat Equation in an Infinite Region

7. Elliptic Equations

7.1 Elliptic Differential Equations and Related Physical Problems
7.2 Harmonic Functions
7.3 Boundary Conditions
7.4 Well-Posed Boundary Value Problems
7.5 Maximum Principle and Its Consequences
7.6 Laplace and Poisson Equations
7.7 Three-Dimensional Laplace Equation
7.8 Helmholtz and Schrӧdinger Equations

8. Two-Dimensional Hyperbolic Equations

8.1 Derivation of the Equations of Motion
8.2 Boundary and Initial Conditions
8.3 Oscillations of a Rectangular and Circular Membrane
8.4 The Fourier Method for Various Boundary Conditions

9. Two-Dimensional Parabolic Equations

9.1 Heat Conduction in Rectangular and Circular Domains
9.2 The Fourier Method for Homogeneous and Nonhomogeneous Heat Equations
9.3 Heat Conduction in Infinite and Semi-Infinite Medium

10. Nonlinear Equations

10.1 Burgers Equation
10.2 Kink Solution and Symmetries
10.3 General Solution of the Cauchy Problem
10.4 Korteweg-de Vries (KdV) Equation
10.5 Cnoidal Waves and Solitons
10.6 Nonlinear Schrӧdinger Equation

Appendices

Appendix A: Fourier Series, Fourier and Laplace Transforms
Appendix B: Bessel and Legendre Functions
Appendix C: Sturm-Liouville Problem and Auxiliary Functions
Appendix D: Sturm-Liouville Problems for Circular and Rectangular Domains
Appendix E: Laplace, Poisson, and Heat Conduction Equations with Nonhomogeneous Boundary Conditions

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