(Students Solutions Manual - Nakhle H. Asmar - pp.209 ) ... Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems - PrenticeHall - R.Haberman(1987).pdf

Students Solutions Manual
PARTIAL DIFFERENTIAL
EQUATIONS
with FOURIER SERIES and
BOUNDARY VALUE PROBLEMS
Second Edition
NAKHLE´ H. ASMAR
University of Missouri
Contents
Preface v
Errata vi
1 A Preview of Applications and Techniques 1
What Is a Partial Differential Equation? 1
Solving and Interpreting a Partial Differential Equation 2
2 Fourier Series 4
Periodic Functions 4
Fourier Series 6
Fourier Series of Functions with Arbitrary Periods 10
Half-Range Expansions: The Cosine and Sine Series 14
Mean Square Approximation and Parseval’s Identity 16
Form of Fourier Series 18
Forced Oscillations 21
Supplement on Convergence
Uniform Convergence and Fourier Series 27
Dirichlet Test and Convergence of Fourier Series 28
3 Partial Differential Equations in Rectangular Coordinates 29
Partial Differential Equations in Physics and Engineering 29
Solution of the One Dimensional Wave Equation:
The Method of Separation of Variables 31
D’Alembert’s Method 35
The One Dimensional Heat Equation 41
Heat Conduction in Bars: Varying the Boundary Conditions 43
The Two Dimensional Wave and Heat Equations 48
Laplace’s Equation in Rectangular Coordinates 49
Poisson’s Equation: The Method of Eigenfunction Expansions 50
Neumann and Robin Conditions 52
Contents iii
4 Partial Differential Equations in
Polar and Cylindrical Coordinates 54
The Laplacian in Various Coordinate Systems 54
Vibrations of a Circular Membrane: Symmetric Case 79
Vibrations of a Circular Membrane: General Case 56
Laplace’s Equation in Circular Regions 59
Laplace’s Equation in a Cylinder 63
The Helmholtz and Poisson Equations 65
Supplement on Bessel Functions
Bessel’s Equation and Bessel Functions 68
Bessel Series Expansions 74
Integral Formulas and Asymptotics for Bessel Functions 79
5 Partial Differential Equations in Sp