Partial Differential Equations Solution Manual
© Leon van Dommelen
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Contents
SOLUTION MANUAL
Partial Differential Equations
Leon van Dommelen
Contents
1
. Introduction
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Basic Concepts
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The prevalence of partial differential equations
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Definitions
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Typical boundary conditions
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The Standard Examples
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The Laplace equation
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Solution stanexl-a
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Solution stanexl-b
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Solution stanexl-b1
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Solution stanexl-b2
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Solution stanexl-b3
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Solution stanexl-b5
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Solution stanexl-c
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Solution stanexl-d
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Solution stanexl-e
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The heat equation
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Solution stanexh-a
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Solution stanexh-b
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The wave equation
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Solution stanexw-a
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Solution stanexw-b
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Solution stanexw-c
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Solution stanexw-e
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Solution stanexw-f
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Classification
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Introduction
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Scalar second order equations
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Solution clasnd-a
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Changes of Coordinates
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Introduction
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The formulae for coordinate transformations
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Rotation of coordinates
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Solution rotcoor-a
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Explanation of the classification
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Solution expclass-a
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Two-Dimensional Coordinate Transforms
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Characteristic Coordinates
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Parabolic equations in two dimensions
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Elliptic equations in two dimensions
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Solution 2dcanel-a
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Properly Posedness
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The conditions for properly posedness
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Solution ppc-a
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Solution ppc-b
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Solution ppc-c
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An improperly posed parabolic problem
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An improperly posed elliptic problem
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Solution ppe-a
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Solution ppe-b
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Solution ppe-c
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Solution ppe-d
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Solution ppe-e
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Solution ppe-f
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Improperly posed hyperbolic problems
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Solution pph-a
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Solution pph-b
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Energy methods
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The Poisson equation
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Solution emp-a
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Solution emp-b
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The heat equation
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The wave equation
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Variational methods [None]
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. Green’s Functions
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Introduction
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The one-dimensional Poisson equation
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Solution gf1d-a
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Solution gf1d-b
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More on delta and Green’s functions
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The Poisson equation in infinite space
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Overview
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Loose derivation
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Solution pninfl-a
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Rigorous derivation
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The Poisson or Laplace equation in a finite region
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Overview
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Intro to the solution procedure
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Derivation of the integral solution
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Solution pnfd-a
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Boundary integral (panel) methods
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Poisson’s integral formulae
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Derivation
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Solution pnifd-a
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Solution pnifd-b
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The integral formula for the Neumann problem
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Smoothness of the solution
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. First Order Equations
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Classification and characteristics
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Numerical solution
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Analytical solution
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Application of the boundary or initial condition
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The inviscid Burgers’ equation
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Wave steepening
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Shocks
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Conservation laws
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Shock relation
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The entropy condition
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First order equations in more dimensions
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Systems of First Order Equations (None)
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. D'Alembert Solution of the Wave equation
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Introduction
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Extension to finite regions
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The physical problem
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The mathematical problem
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Dealing with the boundary conditions
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The final solution
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. Separation of Variables
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A simple example
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The physical problem
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The mathematical problem
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Outline of the procedure
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Step 1: Find the eigenfunctions
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Should we solve the other equation?
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Step 2: Solve the problem
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Comparison with D'Alembert
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Understanding the Procedure
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An ordinary differential equation as a model
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Vectors versus functions
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The inner product
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Matrices versus operators
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Some limitations
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An Example with Periodic Boundary Conditions
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The physical problem
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The mathematical problem
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Outline of the procedure
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Step 1: Find the eigenfunctions
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Step 2: Solve the problem
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Summary of the solution
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Finding the Green's function
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Inhomogeneous boundary conditions
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The physical problem
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The mathematical problem
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Outline of the procedure
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Step 0: Fix the boundary conditions
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Step 1: Find the eigenfunctions
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Step 2: Solve the problem
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Summary of the solution
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Finding the Green's functions
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An alternate procedure
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The physical problem
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The mathematical problem
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Step 0: Fix the boundary conditions
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Step 1: Find the eigenfunctions
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Step 2: Solve the problem
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Summary of the solution
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A Summary of Separation of Variables
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The form of the solution
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Limitations of the method
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The procedure
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More general eigenvalue problems
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More general eigenfunctions
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The physical problem
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The mathematical problem
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Step 0: Fix the boundary conditions
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Step 1: Find the eigenfunctions
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Step 2: Solve the problem
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Summary of the solution
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An alternative procedure
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A Problem in Three Independent Variables
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The physical problem
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The mathematical problem
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Step 1: Find the eigenfunctions
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Step 2: Solve the problem
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Summary of the solution
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. Fourier Transforms [None]
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. Laplace Transforms
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Overview of the Procedure
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Typical procedure
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About the coordinate to be transformed
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A parabolic example
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The physical problem
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The mathematical problem
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Transform the problem
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Solve the transformed problem
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Transform back
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A hyperbolic example
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The physical problem
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The mathematical problem
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Transform the problem
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Solve the transformed problem
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Transform back
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An alternate procedure
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