Partial Differential Equations for Scientists and Engineers

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.

Table of Contents

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 Int roduction)

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