This text introduces and promotes practice of necessary problem-solving skills. The presentation is concise and friendly to the reader. The teaching-by-examples approach provides numerous carefully chosen examples that guide step-by-step learning of concepts and techniques. Fourier series, Sturm-Liouville problem, Fourier transform, and Laplace transform are included. The book's level of presentation and structure is well suited for use in engineering, physics and applied mathematics courses.
Table of Contents
Introduction
Basic definitions
Examples
First-order equations
Linear first-order equations
General solution
Initial condition
Quasilinear first-order equations
Characteristic curves
Examples
Second-order equations
Classification of second-order equations
Canonical forms
Hyperbolic equations
Elliptic equations
Parabolic equations
The Sturm-Liouville Problem
General consideration
Examples of Sturm-Liouville Problems
One-Dimensional Hyperbolic Equations
Wave Equation
Boundary and Initial Conditions
Longitudinal Vibrations of a Rod and Electrical Oscillations
Rod oscillations: Equations and boundary conditions
Electrical Oscillations in a Circuit
Traveling Waves: D'Alembert Method
Cauchy problem for nonhomogeneous wave equation
D'Alembert's formula
The Green's function
Well-posedness of the Cauchy problem
Finite intervals: The Fourier Method for Homogeneous Equations
The Fourier Method for Nonhomogeneous Equations
The Laplace Transform Method: simple cases
Equations with Nonhomogeneous Boundary Conditions
The Consistency Conditions and Generalized Solutions
Energy in the Harmonics
Dispersion of waves
Cauchy problem in an infinite region
Propagation of a wave train
One-Dimensional Parabolic Equations
Heat Conduction and Diffusion: Boundary Value Problems
Heat conduction
Diffusion equation
One-dimensional parabolic equations and initial and boundary conditions
The Fourier Method for Homogeneous Equations
Nonhomogeneous Equations
The Green's function and Duhamel's principle
The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions
Large time behavior of solutions
Maximum principle
The heat equation in an infinite region
Elliptic equations
Elliptic differential equations and related physical problems
Harmonic functions
Boundary conditions
Example of an ill-posed problem
Well-posed boundary value problems
Maximum principle and its consequences
Laplace equation in polar coordinates
Laplace equation and interior BVP for circular domain
Laplace equation and exterior BVP for circular domain
Poisson equation: general notes and a simple case
Poisson Integral
Application of Bessel functions for the solution of Poisson equations in a circle
Three-dimensional Laplace equation for a cylinder
Three-dimensional Laplace equation for a ball
Axisymmetric case
Non-axisymmetric case
BVP for Laplace Equation in a Rectangular Domain
The Poisson Equation with Homogeneous Boundary Conditions
Green's function for Poisson equations
Homogeneous boundary conditions
Nonhomogeneous boundary conditions
Some other important equations
Helmholtz equation
Schrӧdinger equation
Two Dimensional Hyperbolic Equations
Derivation of the Equations of Motion
Boundary and Initial Conditions
Oscillations of a Rectangular Membrane
The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions
The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions
The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions
Small Transverse Oscillations of a Circular Membrane
The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions
Axisymmetric Oscillations of a Membrane
The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions
Forced Axisymmetric Oscillations
The Fourier Method for Equations with Nonhomogeneous Boundary Conditions
Two-Dimensional Parabolic Equations
Heat Conduction within a Finite Rectangular Domain
The Fourier Method for the Homogeneous Heat Equation (Free Heat Exchange)
The Fourier Method for Nonhomogeneous Heat Equation with Homogeneous Boundary conditions
Heat Conduction within a Circular Domain
The Fourier Method for the Homogeneous Heat Equation
The Fourier Method for the Nonhomogeneous Heat Equation
Heat conduction in an Infinite Medium
Heat Conduction in a Semi-Infinite Medium
Nonlinear equations
Burgers equation
Kink solution
Symmetries of the Burgers equation
General solution of the Cauchy problem.
Interaction of kinks
Korteweg-de Vries equation
Symmetry properties of the KdV equation
Cnoidal waves
Solitons
Bilinear formulation of the KdV equation
Hirota's method
Multisoliton solutions
Nonlinear Schrӧdinger equation
Symmetry properties of NSE
Solitary waves
Appendix A. Fourier Series, Fourier and Laplace Transforms
Appendix B. Bessel and Legendre Functions
Appendix C. Sturm-Liouville problem and auxiliary functions for one and two dimensions
Appendix D.
D1. The Sturm-Liouville problem for a circle
D2. The Sturm-Liouville problem for the rectangle
Appendix E.
E1. The Laplace and Poisson equations for a rectangular domain with nonhomogeneous boundary conditions.
E2. The heat conduction equations with nonhomogeneous boundary conditions.
