Most physical phenomena, whether in the domain of fluid dynamics, electricity, magnetism, mechanics, optics, or heat flow, can be described in general by partial differential equations. Indeed, such equations are crucial to mathematical physics. Although simplifications can be made that reduce these equations to ordinary differential equations, nevertheless the complete description of physical systems resides in the general area of partial differential equations.
This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing the mathematical model) and how to solve the equation (along with initial and boundary conditions).
Written for advanced undergraduate and graduate students, as well as professionals working in the applied sciences, this clearly written book offers realistic, practical coverage of diffusion-type problems, hyperbolic-type problems, elliptic-type problems, and numerical and approximate methods. Each chapter contains a selection of relevant problems (answers are provided) and suggestions for further reading.
Author(s): Stanley J. Farlow
Series: Dover Books on Mathematics
Publisher: Dover Publications
Year: 1993
Language: English
Pages: 448
PART 1 - Introduction
LESSON 1 - Introduction to Partial Differential Equations
PART 2 - Diffusion-Type Problems
LESSON 2 - Diffusion-Type Problems (Parabolic Equations)
LESSON 3 - Boundary Conditions for Diffusion-Type Problems
LESSON 4 - Derivation of the Heat Equation
LESSON 5 - Separation of Variables
LESSON 6 - Transforming Nonhomogeneous BCs into Homogeneous Ones
LESSON 7 - Solving More Complicated Problems by Separation of Variables
LESSON 8 - Transforming Hard Equations into Easier Ones
LESSON 9 - Solving Nonhomogeneous PDEs (Eigenfunction Expansions)
LESSON 10 - Integral Transforms (Sine and Cosine Transforms)
LESSON 11 - The Fourier Series and Transform
LESSON 12 - The Fourier Transform and Its Application to PDEs
LESSON 13 - The Laplace Transform
LESSON 14 - Duhamel’s Principle
LESSON 15 - The Convection Term ux in the Diffusion Problems
PART 3 - Hyperbolic-Type Problems
LESSON 16 - The One-Dimensional Wave Equation (Hyperbolic Equations)
LESSON 17 - The D’Alembert Solution of the Wave Equation
LESSON 18 - More on the D’Alembert Solution
LESSON 19 - Boundary Conditions Associated with the Wave Equation
LESSON 20 - The Finite Vibrating String (Standing Waves)
LESSON 21 - The Vibrating Beam (Fourth-Order PDE)
LESSON 22 - Dimensionless Problems
LESSON 23 - Classification of PDEs (Canonical Form of the Hyperbolic Equation)
LESSON 24 - The Wave Equation in Two and Three Dimensions (Free Space)
LESSON 25 - The Finite Fourier Transforms (Sine and Cosine Transforms)
LESSON 26 - Superposition (The Backbone of Linear Systems)
LESSON 27 - First-Order Equations (Method of Characteristics)
LESSON 28 - Nonlinear First-Order Equations (Conservation Equations)
LESSON 29 - Systems of PDEs
LESSON 30 - The Vibrating Drumhead (Wave Equation in Polar Coordinates)
PART 4 - Elliptic-Type Problems
LESSON 31 - The Laplacian (an Intuitive Description)
LESSON 32 - General Nature of Boundary-Value Problems
LESSON 33 - Interior Dirichlet Problem for a Circle
LESSON 34 - The Dirichlet Problem in an Annulus
LESSON 35 - Laplace’s Equation in Spherical Coordinates (Spherical Harmonics)
LESSON 36 - A Nonhomogeneous Dirichlet Problem (Green’s Function)
PART 5 - Numerical and Approximate Methods
LESSON 37 - Numerical Solutions (Elliptic Problems)
LESSON 38 - An Explicit Finite-Difference Method
LESSON 39 - An Implicit Finite-Difference Method (Crank-Nicolson Method)
LESSON 40 - Analytic versus Numerical Solutions
LESSON 41 - Classification of PDEs (Parabolic and Elliptic Equations)
LESSON 42 - Monte Carlo Methods (an Introduction)
LESSON 43 - Monte Carlo Solution of Partial Differential Equations
LESSON 44 - Calculus of Variations (Euler-Lagrange Equations)
LESSON 45 - Variational Methods for Solving PDEs (Method of Ritz)
LESSON 46 - Perturbation Methods for Solving PDEs
LESSON 47 - Conformal-Mapping Solutions of PDEs
ANSWERS TO SELECTED PROBLEMS
APPENDIX 1 - Integral Transform Tables
APPENDIX 2 - PDE Crossword Puzzle
APPENDIX 3 - Laplacian in Different Coordinate Systems
APPENDIX 4 - Types of Partial Differential Equations
Index
