Partial Differential Equations for Scientists and Engineers

By: Stanley J. Farlow

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Paperback | 1 September 1993

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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.

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Introduction
Introduction to Partial Differential Equations
Diffusion-Type Problems
Diffusion-Type Problems (Parabolic Equations)
Boundary Conditions for Diffusion-Type Problems
Derivation of the Heat Equation
Separation of Variables
Transforming Nonhomogeneous BCs into Homogeneous Ones
Solving More Complicated Problems by Separation of Variables
Transforming Hard Equations into Easier Ones
Solving Nonhomogeneous PDEs (Eigenfunction Expansions)
Integral Transforms (Sine and Cosine Transforms)
The Fourier Series and Transform
The Fourier Transform and its Application to PDEs
The Laplace Transform
Duhamel's Principle
The Convection Term u subscript x in Diffusion Problems
Hyperbolic-Type Problems
The One Dimensional Wave Equation (Hyperbolic Equations)
The D'Alembert Solution of the Wave Equation
More on the D'Alembert Solution
Boundary Conditions Associated with the Wave Equation
The Finite Vibrating String (Standing Waves)
The Vibrating Beam (Fourth-Order PDE)
Dimensionless Problems
Classification of PDEs (Canonical Form of the Hyperbolic Equation)
The Wave Equation in Two and Three Dimensions (Free Space)
The Finite Fourier Transforms (Sine and Cosine Transforms)
Superposition (The Backbone of Linear Systems)
First-Order Equations (Method of Characteristics)
Nonlinear First-Order Equations (Conservation Equations)
Systems of PDEs
The Vibrating Drumhead (Wave Equation in Polar Coordinates)
Elliptic-Type Problems
The Laplacian (an intuitive description)
General Nature of Boundary-Value Problems
Interior Dirichlet Problem for a Circle
The Dirichlet Problem in an Annulus
Laplace's Equation in Spherical Coordinates (Spherical Harmonics)
A Nonhomogeneous Dirichlet Problem (Green's Functions)
Numerical and Approximate Methods
Numerical Solutions (Elliptic Problems)
An Explicit Finite-Difference Method
An Implicit Finite-Difference Method (Crank-Nicolson Method)
Analytic versus Numerical Solutions
Classification of PDEs (Parabolic and Elliptic Equations)
Monte Carlo Methods (An Introduction)
Monte Carlo Solutions of Partial Differential Equations)
Calculus of Variations (Euler-Lagrange Equations)
Variational Methods for Solving PDEs (Method of Ritz)
Perturbation method for Solving PDEs
Conformal-Mapping Solution of PDEs
Answers to Selected Problems
Integral Transform Tables
PDE Crossword Puzzle
Laplacian in Different Coordinate Systems
Types of Partial Differential Equations
Index
Table of Contents provided by Publisher. All Rights Reserved.

Partial Differential Equations for Scientists and Engineers

At a Glance

Paperback

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