Partial Differential Equations: Analytical Solution Techniques(The Wadsworth & Brooks/Cole Mathematics Series)

This is a text for a two-semester or three-quarter sequence of courses in partial differential equations. It is assumed that the student has a good background in vector calculus and ordinary differential equations and has been introduced to such elementary aspects of partial differential equations as separation of variables, Fourier series, and eigenfunction expansions. Some familiarity is also assumed with the application of complex variable techniques, including conformal map- ping, integration in the complex plane, and the use of integral transforms. Linear theory is developed in the first half of the book and quasilinear and nonlinear problems are covered in the second half, but the material is presented in a manner that allows flexibility in selecting and ordering topics. For example, it is possible to start with the scalar first-order equation in Chapter 5, to include or delete the nonlinear equation in Chapter 6, and then to move on to the second- order equations, selecting and omitting topics as dictated by the course. At the University of Washington, the material in Chapters 1-4 is covered during the third quarter of a three-quarter sequence that is part of the required program for first-year graduate students in Applied Mathematics. We offer the material in Chapters 5-8 to more advanced students in a two-quarter sequence.

Table of Contents:
1 The Diffusion Equation.- 1.1 Heat Conduction.- 1.2 Fundamental Solution.- 1.2.1 Similarity (Invariance).- 1.2.2 Qualitative Behavior; Diffusion.- 1.2.3 Laplace Transforms.- 1.2.4 Fourier Transforms.- 1.3 Initial-Value Problem (Cauchy Problem) on the Infinite Domain; Superposition.- 1.4 Initial- and Boundary-Value Problems on the Semi-Infinite Domain; Green's Functions.- 1.4.1 Green's Function of the First Kind.- 1.4.2 Homogeneous Boundary-Value Problems.- 1.4.3 Inhomogeneous Boundary Condition u = g(t).- 1.4.4 Green's Function of the Second Kind.- 1.4.5 Homogeneous Boundary-Value Problems.- 1.4.6 Inhomogeneous Boundary Condition.- 1.4.7 The General Boundary-Value Problem.- 1.5 Initial- and Boundary-Value Problems on the Finite Domain; Green's Functions.- 1.5.1 Green's Function of the First Kind.- 1.5.2 Connection with Separation of Variables.- 1.5.3 Connection with Laplace Transform Solution.- 1.5.4 Uniqueness of Solutions.- 1.5.5 Inhomogeneous Boundary Conditions.- 1.5.6 Higher-Dimensional Problems.- 1.6 Burgers' Equation.- 1.6.1 The Cole-Hopf Transformation.- 1.6.2 Initial-Value Problem on ? ? ?.- 1.6.3 Boundary-Value Problem on 0 ?.- Review Problems.- Problems.- References.- 2 Laplace's Equation.- 2.1 Applications.- 2.1.1 Incompressible Irrotational Flow.- 2.1.2 Two-Dimensional Incompressible Flow.- 2.2 The Two-Dimensional Problem; Conformai Mapping.- 2.2.1 Mapping of Harmonic Functions.- 2.2.2 Transformation of Boundary Conditions.- 2.2.3 Example, Solution in a "Simpler" Transformed Domain.- 2.3 Fundamental Solution; Dipole Potential.- 2.3.1 Point Source in Three Dimensions.- 2.3.2 Fundamental Solution in Two-Dimensions; Descent.- 2.3.3 Effect of Lower Derivative Terms.- 2.3.4 Potential Due to a Dipole.- 2.4 Potential Due to Volume, Surface, and Line Distribution of Sources and Dipoles.- 2.4.1 Volume Distribution of Sources.- 2.4.2 Surface and Line Distribution of Sources or Dipoles.- 2.4.3 An Example: Flow Over a Nonlifting Body of Revolution.- 2.4.4 Limiting Surface Values for Source and Dipole Distributions.- 2.5 Green's Formula and Applications.- 2.5.1 Gauss' Integral Theorem.- 2.5.2 Energy Theorem and Corollaries.- 2.5.3 Uniqueness Theorems.- 2.5.4 Mean-Value Theorem.- 2.5.5 Surface Distribution of Sources and Dipoles.- 2.5.6 Potential Due to Dipole Distribution of Unit Strength.- 2.6 Green's and Neumann's Functions.- 2.6.1 Green's function.- 2.6.2 Neumann's function.- 2.7 Dirichlet's and Neumann's Problems.- 2.8 Examples of Green's and Neumann's Functions.- 2.8.1 Upper Half-Plane, y ? 0 (Two Dimensions).- 2.8.2 Upper Half-Space, z ? 0 (Three Dimensions).- 2.8.3 Interior (Exterior) of Unit Sphere or Circle.- 2.9 Estimates; Harnack's Inequality.- 2.10 Connection between Green's Function and Conformai Mapping (Two Dimensions); Dipole-Green's Functions.- 2.11 Series Representations; Connection with Separation of Variables.- 2.12 Solutions in Terms of Integral Equations.- 2.12.1 Dirichlet's Problem.- 2.12.2 Neumann's Problem.- Review Problems.- Problems.- References.- 3 The Wave Equation.- 3.1 The Vibrating String.- 3.2 Shallow-Water Waves.- 3.2.1 Assumptions.- 3.2.2 Hydrostatic Balance.- 3.2.3 Conservation of Mass.- 3.2.4 Conservation of Momentum in the X direction.- 3.2.5 Smooth Solutions.- 3.2.6 Energy Conservation.- 3.2.7 Initial-Value Problem.- 3.2.8 Signaling Problem.- 3.2.9 Small-Amplitude Theory.- 3.3 Compressible Flow.- 3.3.1 Conservation Laws.- 3.3.2 One-Dimensional Ideal Gas.- 3.3.3 Signaling Problem for One-Dimensional Flow.- 3.3.4 Inviscid, Non-Heat-Conducting Gas; Analogy with Shallow-Water Waves.- 3.3.5 Small-Disturbance Theory in One-Dimensional Flow (Signaling Problem).- 3.3.6 Small Disturbance Theory in Three Dimensional, Inviscid Non-Heat-Conducting Flow.- 3.4 The One-Dimensional Problem in the Infinite Domain.- 3.4.1 Fundamental Solution.- 3.4.2 General Initial- Value Problem on ? ? ?.- 3.4.3 An Example.- 3.5 Initial- and Boundary-Value Problems on the Semi-Infinite Interval; Green's Functions.- 3.5.1 Green's Function of the First Kind.- 3.5.2 Homogeneous Boundary Condition, Nonzero Initial Conditions.- 3.5.3 Inhomogeneous Boundary Condition u(0,t) = g(t).- 3.5.4 An Example.- 3.5.5 A Second Example: Solutions with a Fixed Interface; Reflected and Transmitted Waves.- 3.5.6 Green's Function of the Second Kind.- 3.6 Initial- and Boundary-Value Problems on the Finite Interval; Green's Functions.- 3.6.1 Green's Function of the First Kind on 0 ? x ? 1.- 3.6.2 The Inhomogeneous Problem, Nonzero Initial Conditions.- 3.6.3 Inhomogeneous Boundary Conditions.- 3.6.4 Uniqueness of the General Initial- and Boundary-Value Problem of the First Kind.- 3.7 Effect of Lower-Derivative Terms.- 3.7.1 Transformation to D'Alembert Form: Removal of Lower-Derivative Terms.- 3.7.2 Fundamental Solution; Stability.- 3.7.3 Green's Functions; Initial- and Boundary-Value Problems.- 3.8 Dispersive Waves on the Infinite Interval.- 3.8.1 Uniform Waves.- 3.8.2 General Initial-Value Problem.- 3.8.3 Group Velocity.- 3.8.4 Dispersion.- 3.9 The Three-Dimensional Wave Equation; Acoustics.- 3.9.1 Fundamental Solution.- 3.9.2 Arbitrary Source Distribution.- 3.9.3 Initial-Value Problems for the Homogeneous Equation.- 3.10 Examples in Acoustics and Aerodynamics.- 3.10.1 The Bursting Balloon.- 3.10.2 Source 0; A = C = 0.- 4.2.2 Hyperbolic Examples.- 4.2.3 The Parabolic Problem, ? = 0; C = 0.- 4.2.4 The Elliptic Problem, ? < 0; B = 0, A = C.- 4.3 The Role of Characteristics in Hyperbolic Equations.- 4.3.1 Cauchy's Problem.- 4.3.2 Characteristics as Carriers of Discontinuities in the Second Derivative.- 4.4 Solution of Hyperbolic Equations in Terms of Characteristics.- 4.4.1 Cauchy Data on a Spacelike Arc.- 4.4.2 Cauchy Problem; the Numerical Method of Characteristics.- 4.4.3 Goursat's Problem; Boundary Conditions on a Timelike Arc.- 4.4.4 Characteristic Boundary-Value Problem.- 4.4.5 Well-Posedness.- 4.4.6 The General Solution of Cauchy's Problem; the Riemann function.- 4.4.7 Weak Solutions; Propagation of Discontinuities in P and Q; Stability.- 4.5 Hyperbolic Systems of Two First-Order Equations.- 4.5.1 The Perturbation of a Quasilinear System near a Known Solution.- 4.5.2 Characteristics.- 4.5.3 Transformation to Characteristic Variables.- 4.5.4 Numerical Solutions; Propagation of Discontinuities.- 4.5.5 Connection with the Second-Order Equation.- 4.5.6 Perturbation of the Dam-Breaking Problem.- Problems.- References.- 5 Quasilinear First-Order Equations.- 5.1 The Scalar Conservation Law; Quasilinear Equations.- 5.1.1 Flow of Water in a Conduit with Friction.- 5.1.2 Traffic Flow.- 5.2 Continuously Differentiable Solution of the Quasilinear Equation in Two Independent Variables.- 5.2.1 Geometrical Aspects of Solutions.- 5.2.2 Characteristic Curves; the Solution Surface.- 6 Nonlinear First-Order Equations.- 6.1 Geometrical Optics: A Nonlinear Equation.- 6.1.1 Huyghens' Construction; the Eikonal Equation.- 6.1.2 The Equation for Light Rays.- 6.1.3 Fermat's Principle.- 6.2 Applications Leading to the Hamilton-Jacobi Equation.- 6.2.1 The Variation of a Functional.- 6.2.2 A Variational Principle; The Euler-Lagrange Equations.- 6.2.3 Hamiltonian Form of the Variational Problem.- 6.2.4 Field of Extremals from a Point; The Hamilton-Jacobi Equation.- 6.2.5 Extremals from a Manifold; Transversality.- 6.2.6 Canonical Transformations.- 6.3 The Nonlinear Equation.- 6.3.1 The Geometry of Solutions.- 6.3.2 Focal Strips and Characteristic Strips.- 6.3.3 The Inital-Value Problem.- 6.3.4 Example Problems for the Eikonal Equation.- 6.4 The Complete Integral; Solutions by Envelope Formation.- 6.4.1 Envelope Surfaces Associated with the Complete Integral.- 6.4.2 Relationship between Characteristic Strips and the Complete Integral.- 6.4.3 The Complete Integral of the Hamilton-Jacobi Equation.- Problems.- References.- 7 Quasilinear Hyperbolic Systems.- 7.1 The Quasilinear Second-Order Hyperbolic Equation.- 7.1.1 Transformation to Characteristic Variables.- 7.1.2 The Cauchy Problem; the Numerical Method of Characteristics.- 7.2 Systems of n First-Order Equations.- 7.2.1 Characteristic Curves and the Normal Form.- 7.2.2 Unsteady Nonisentropic Flow.- 7.2.3 A Semilinear Example.- 7.3 Systems of Two First-Order Equations.- 7.3.1 Characteristic Coordinates.- 7.3.2 The Hodograph Transformation.- 7.3.3 The Riemann Invariants.- 7.3.4 Applications of the Riemann Invariants.- 1.4 Shallow-Water Waves.- 7.4.1 Characteristic Coordinates; Riemann Invariants.- 7.4.2 Simple Wave Solutions.- 7.4.3 Solutions with Bores.- 7.5 Compressible Flow Problems.- 7.5.1 One-Dimensional Unsteady Flow.- 7.5.2 Steady Irrotational Two-Dimensional Flow.- Problems.- References.- 8 Perturbation Solutions.- 8.1 Asymptotic Expansions.- 8.1.1 Order Symbols.- 8.1.2 Definition of an Asymptotic Expansion.- 8.1.3 Asymptotic Expansion of a Given function.- 8.1.4 Asymptotic Expansion of the Root of an Algebraic Equation.- 8.1.5 Asymptotic Expansion of a Definite Integral.- 8.2 Regular Perturbations.- 8.2.1 Green's Function for an Ordinary Differential Equation.- 8.2.2 Eigenvalues and Eigenfunctions of a Perturbed Self-Adjoint Operator.- 8.2.3 A Boundary Perturbation Problem.- 8.3 Matched Asymptotic Expansions.- 8.3.1 An Ordinary Differential Equation.- 8.3.2 A Second Example.- 8.3.3 Interior Dirichlet Problems for Elliptic Equations.- 8.3.4 Slender Body Theory; a Problem with a Boundary Singularity.- 8.3.5 Burgers' Equation for ? ? 1.- 8.4 Cumulative Perturbations; Solution Valid in the Far Field.- 8.4.1 The Oscillator with a Weak Nonlinear Damping; Regular Expansion.- 8.4.2 The Multiple Scale Expansion.- 8.4.3 Near-Identity Averaging Transformations.- 8.4.4 Evolution Equations for a Weakly Nonlinear Problem.- Problems.- References.