Partial Differential Equations and Boundary-Value Problems with Applications

Mark A. Pinsky

ISBN: 9781470409142 | Year: 2013 | Paperback | Pages: 544 | Language : English

Book Size: 180 x 240 mm | Territorial Rights: Restricted

Price: 1750.00

Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d’Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green’s functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase).

With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations.

Mark A. Pinsky, Northwestern University, Evanston, IL

Preliminaries

Partial Differential Equations

    What is a partial differential equation?
    Superposition principle and subtraction principle
    Sources of PDEs in classical physics
    The one-dimensional heat equation
    Classification of second-order PDEs
Separation of Variables
    What is a separated solution?
    Separated solutions of Laplace’s equation
    Real and complex separated solutions
    Separated solutions with boundary conditions
Orthogonal Functions
    Inner product space of functions
    Projection of a function onto an orthogonal set
    Orthonormal sets of functions
    Parseval’s equality, completeness, and mean square convergence
    Weighted inner product
    Gram-Schmidt orthogonalization
    Complex inner product

Fourier Series

Definitions and Examples

    Orthogonality relations
    Definition of Fourier coefficients
    Even functions and odd functions
    Periodic functions
    Implementation with Mathematica
    Fourier sine and cosine series
Convergence of Fourier Series
    Piecewise smooth functions
    Dirichlet kernel
    Proof of convergence
Uniform Convergence and the Gibbs Phenomenon
    Example of Gibbs overshoot
    Implementation with Mathematica
    Uniform and nonuniform convergence
    Two criteria for uniform convergence
    Differentiation of Fourier series
    Integration of Fourier series
    A continuous function with a divergent Fourier series
Parseval’s Theorem and Mean Square Error
    Statement and proof of Parseval’s theorem
    Application to mean square error
    Application to the isoperimetric theorem

Complex Form of Fourier Series

    Fourier series and Fourier coefficients
    Parseval’s theorem in complex form
    Applications and examples
    Fourier series of mass distributions
Sturm-Liouville Eigenvalue Problems
    Examples of Sturm-Liouville eigenvalue problems
    Some general properties of S-L eigenvalue problems
    Example of transcendental eigenvalues
    Further properties: completeness and positivity
    General Sturm-Liouville problems
    Complex-valued eigenfunctions and eigenvalues

Boundary-Value Problems In Rectangular Coordinates

The Heat Equation

    Fourier’s law of heat conduction
    Derivation of the heat equation
    Boundary conditions
    Steady-state solutions in a slab
    Time-periodic solutions
    Applications to geophysics
    Implementation with Mathematica
Homogeneous Boundary Conditions on a Slab
    Separated solutions with boundary conditions
    Solution of the initial-value problem in a slab
    Asymptotic behavior and relaxation time
    Uniqueness of solutions
    Examples of transcendental eigenvalues
Nonhomogeneous Boundary Conditions
    Statement of problem
    Five-stage method of solution
    Temporally nonhomogeneous problems
The Vibrating String
    Derivation of the equation
    Linearized model
    Motion of the plucked string
    Acoustic interpretation
    Explicit (d’Alembert) representation
    Motion of the struck string
    d’Alembert’s general solution
    Vibrating string with external forcing
Applications of Multiple Fourier Series
    The heat equation (homogeneous boundary conditions)
    Laplace’s equation
    The heat equation (nonhomogeneous boundary conditions)
    The wave equation (nodal lines)
    Multiplicities of the eigenvalues
    Implementation with Mathematica
    Application to Poisson’s equation

Boundary-Value Problems in Cylindrical Coordinates

Laplace’s Equation and Applications

    Laplacian in cylindrical coordinates
    Separated solutions of Laplace’s equation in p, cp
    Application to boundary-value problems
    Regularity
    Uniqueness of solutions
    Exterior problems
    Wedge domains
    Neumann problems
    Explicit representation by Poisson’s formula
Bessel Functions
    Bessel’s equation
    The power series solution of Bessel’s equation
    Integral representation of Bessel functions
    The second solution of Bessel’s equation
    Zeros of the Bessel function 0•
    Asymptotic behavior and zeros of Bessel functions
    Fourier-Bessel series
    Implementation with Mathematica
The Vibrating Drumhead
    Wave equation in polar coordinates
    Solution of initial-value problems
    Implementation with Mathematica
Heat Flow in the Infinite Cylinder
    Separated solutions
    Initial-value problems in a cylinder
    Initial-value problems between two cylinders
    Implementation with Mathematica
    Time-periodic heat flow in the cylinder
Heat Flow in the Finite Cylinder
    Separated solutions
    Solution of Laplace’s equation
    Solutions of the heat equation with zero boundary conditions
    General initial-value problems for the heat equation

Boundary-Value Problems in Spherical Coordinates

Spherically Symmetric Solutions

    Laplacian in spherical coordinates
    Time-periodic heat flow: Applications to geophysics
    Initial-value problem for heat flow in a sphere
    The three-dimensional wave equation
    Convergence of series in three dimensions
Legendre Functions and Spherical Bessel Functions
    Separated solutions in spherical coordinates
    Legendre polynomials
    Legendre polynomial expansions
    Implementation with Mathematica
    Associated Legendre functions
    Spherical Bessel functions
Laplace’s Equation in Spherical Coordinates
    Boundary-value problems in a sphere
    Boundary-value problems exterior to a sphere
    Applications to potential theory


Fourier Transforms and Applications

Basic Properties of the Fourier Transform

    Passage from Fourier series to Fourier integrals
    Definition and properties of the Fourier transform
    Fourier sine and cosine transforms
    Generalized h-transform
    Fourier transforms in several variables
    The uncertainty principle
    Proof of convergence
Solution of the Heat Equation for an Infinite Rod
    First method: Fourier series and passage to the limit
    Second method: Direct solution by Fourier transform
    Verification of the solution
    Explicit representation by the Gauss-Weierstrass kernel
    Some explicit formulas
    Solutions on a half-line: The method of images
    The Black-Scholes model
    Hermite polynomials
Solutions of the Wave Equation and Laplace’s Equation
    One-dimensional wave equation and d’Alembert’s formula
    General solution of the wave equation
    Three-dimensional wave equation and Huygens’ principle
    Extended validity of the explicit representation
    Application to one- and two-dimensional wave equations
    Laplace’s equation in a half-space: Poisson’s formula
Solution of the Telegraph Equation
    Fourier representation of the solution
    Uniqueness of the solution
    Time-periodic solutions of the telegraph equation

Asymptotic Analys

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