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Elementary Applied Partial Differential Equations: With Fourier Series and Boundary Value Probl...

ISBN: 9780132638074 | 013263807X
Format: Hardcover
Publisher: Prentice Hall
Pub. Date: 9/1/1997

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SummaryTable of Contents
Appropriate for an elementary undergraduate first course of varying lengths. Its in-depth elementary presentation is intended primarily for students in science, engineering, and applied mathematics. Emphasizing the physical interpretation of mathematical solutions, this book introduces applied mathematics while presenting partial differential equations
Prefacexiii
1 Heat Equation
1(31)
1.1 Introduction
1(1)
... MORE
1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod
2(8)
1.3 Boundary Conditions
10(3)
1.4 Equilibrium Temperature Distribution
13(5)
1.4.1 Prescribed Temperature
13(2)
1.4.2 Insulated Boundaries
15(3)
1.5 Derivation of the Heat Equation in Two or Three Dimensions
18(14)
2 Method of Separation of Variables
32(54)
2.1 Introduction
32(1)
2.2 Linearity
33(2)
2.3 Heat Equation with Zero Temperatures at Finite Ends
35(21)
2.3.1 Introduction
35(1)
2.3.2 Separation of Variables
36(2)
2.3.3 Time-Dependent Equation
38(1)
2.3.4 Boundary Value Problem
39(5)
2.3.5 Product Solutions and the Principle of Superposition
44(3)
2.3.6 Orthogonality of Sines
47(1)
2.3.7 Formulation, Solution, and Interpretation of an Example
48(2)
2.3.8 Summary
50(6)
2.4 Worked Examples with the Heat Equation (Other Boundary Value Problems)
56(11)
2.4.1 Heat Conduction in a Rod with Insulated Ends
56(4)
2.4.2 Heat Conduction in a Thin Circular Ring
60(5)
2.4.3 Summary of Boundary Value Problems
65(2)
2.5 Laplace's Equation: Solutions and Qualitative Properties
67(19)
2.5.1 Laplace's Equation inside a Rectangle
67(6)
2.5.2 Laplace's Equation for a Circular Disk
73(4)
2.5.3 Fluid Flow Past a Circular Cylinder (Lift)
77(3)
2.5.4 Qualitative Properties of Laplace's Equation
80(6)
3 Fourier Series
86(44)
3.1 Introduction
86(2)
3.2 Statement of Convergence Theorem
88(4)
3.3 Fourier Cosine and Sine Series
92(20)
3.3.1 Fourier Sine Series
93(9)
3.3.2 Fourier Cosine Series
102(3)
3.3.3 Representing f(x) by Both a Sine and Cosine Series
105(1)
3.3.4 Even and Odd Parts
105(2)
3.3.5 Continuous Fourier Series
107(5)
3.4 Term-by-Term Differentiation of Fourier Series
112(11)
3.5 Term-By-Term Integration of Fourier Series
123(4)
3.6 Complex Form of Fourier Series
127(3)
4 Vibrating Strings and Membranes
130(21)
4.1 Introduction
130(1)
4.2 Derivation of a Vertically Vibrating String
130(3)
4.3 Boundary Conditions
133(4)
4.4 Vibrating String with Fixed Ends
137(7)
4.5 Vibrating Membrane
144(2)
4.6 Reflection and Refraction of Electromagnetic (Light) and Acoustic (Sound) Waves
146(5)
4.6.1 Snell's Law of Refraction
147(1)
4.6.2 Intensity (Amplitude) of Reflected and Refracted Waves
148(1)
4.6.3 Total Internal Reflection
149(2)
5 Sturm-Liouville Eigenvalue Problems
151(64)
5.1 Introduction
151(1)
5.2 Examples
152(3)
5.2.1 Heat Flow in a Nonuniform Rod
152(1)
5.2.2 Circularly Symmetric Heat Flow
153(2)
5.3 Sturm-Liouville Eigenvalue Problems
155(8)
5.3.1 General Classification
155(1)
5.3.2 Regular Sturm-Liouville Eigenvalue Problem
156(1)
5.3.3 Example and Illustration of Theorems
157(6)
5.4 Worked Example -- Heat Flow in a Nonuniform Rod without Sources
163(4)
5.5 Self-Adjoint Operators and Sturm-Liouville Eigenvalue Problems
167(16)
5.6 Rayleigh Quotient
183(5)
5.7 Worked Example -- Vibrations of a Nonuniform String
188(3)
5.8 Boundary Conditions of the Third Kind
191(14)
5.9 Large Eigenvalues (Asymptotic Behavior)
205(4)
5.10 Approximation Properties
209(6)
6 An Elementary Discussion of Finite Difference Numerical Methods for Partial Differential Equations
215(51)
6.1 Introduction
215(1)
6.2 Finite Differences and Truncated Taylor Series
216(6)
6.3 Heat Equation
222(24)
6.3.1 Introduction
222(1)
6.3.2 A Partial Difference Equation
222(2)
6.3.3 Computations
224(3)
6.3.4 Fourier-von Neumann Stability Analysis
227(7)
6.3.5 Separation of Variables for Partial Difference Equations and Analytic Solutions of Ordinary Difference Equations
234(2)
6.3.6 Matrix Notation
236(4)
6.3.7 Nonhomogeneous Problems
240(1)
6.3.8 Other Numerical Schemes
240(2)
6.3.9 Other Types of Boundary Conditions
242(4)
6.4 Two-Dimensional Heat Equation
246(2)
6.5 Wave Equation
248(4)
6.6 Laplace's Equation
252(7)
6.7 Finite Element Method
259(7)
6.7.1 Approximation with Nonorthogonal Functions (Weak Form of the Partial Differential Equation)
259(3)
6.7.2 The Simplest Triangular Finite Elements
262(4)
7 Partial Differential Equations with At Least Three Independent Variables
266(71)
7.1 Introduction
266(1)
7.2 Separation of the Time Variable
267(4)
7.2.1 Vibrating Membrane -- Any Shape
267(2)
7.2.2 Heat Conduction -- Any Region
269(1)
7.2.3 Summary
270(1)
7.3 Vibrating Rectangular Membrane
271(9)
7.4 Statements and Illustrations of Theorems for the Eigenvalue Problem XXX(2)XXX + XXX = 0
280(5)
7.5 Self-Adjoint Operators and Multidimensional Eigenvalue Problems
285(5)
7.6 Rayleigh Quotient
290(2)
7.7 Vibrating Circular Membrane and Bessel Functions
292(15)
7.7.1 Introduction
292(1)
7.7.2 Separation of Variables
293(1)
7.7.3 Eigenvalue Problems (One Dimensional)
294(2)
7.7.4 Bessel's Differential Equation
296(1)
7.7.5 Singular Points and Bessel's Differential Equation
297(1)
7.7.6 Bessel Functions and Their Asymptotic Properties (near z = 0)
298(1)
7.7.7 Eigenvalue Problem Involving Bessel Functions
299(2)
7.7.8 Initial Value Problem for a Vibrating Circular Membrane
301(1)
7.7.9 Circularly Symmetric Case
302(5)
7.8 More on Bessel Functions
307(8)
7.8.1 Qualitative Properties of Bessel Functions
307(2)
7.8.2 Asymptotic Formulas for the Eigenvalues
309(1)
7.8.3 Zeros of Bessel Functions and Nodal Curves
310(1)
7.8.4 Series Representation of Bessel Functions
311(4)
7.9 Laplace's Equation in a Circular Cylinder
315(11)
7.9.1 Introduction
315(1)
7.9.2 Separation of Variables
316(2)
7.9.3 Zero Temperature on the Lateral Sides and on the Bottom or Top
318(1)
7.9.4 Zero Temperature on the Top and Bottom
319(3)
7.9.5 Modified Bessel Functions
322(4)
7.10 Spherical Problems and Legendre Polynomials
326(11)
7.10.1 Introduction
326(1)
7.10.2 Separation of Variables and One-Dimensional Eigenvalue Problems
326(2)
7.10.3 Associated Legendre Functions and Legendre Polynomials
328(3)
7.10.4 Radial Eigenvalue Problems
331(1)
7.10.5 Product Solutions, Modes of Vibration, and the Initial Value Problem
332(1)
7.10.6 Laplace's Equation inside a Spherical Cavity
332(5)
8 Nonhomogeneous Problems
337(33)
8.1 Introduction
337(1)
8.2 Heat Flow with Sources and Nonhomogeneous Boundary Conditions
337(6)
8.3 Method of Eigenfunction Expansion with Homogeneous Boundary Conditions (Differentiating Series of Eigenfunctions)
343(6)
8.4 Method of Eigenfunction Expansion Using Green's Formula (with or without Homogeneous Boundary Conditions)
349(5)
8.5 Forced Vibrating Membranes and Resonance
354(8)
8.6 Poisson's Equation
362(8)
9 Green's Functions for Time-Independent Problems
370(64)
9.1 Introduction
370(1)
9.2 One-dimensional Heat Equation
370(5)
9.3 Green's Functions for Boundary Value Problems for Ordinary Differential Equations
375(20)
9.3.1 One-dimensional Steady-State Heat Equation
375(1)
9.3.2 The Method of Variation of Parameters
376(3)
9.3.3 The Method of Eigenfunction Expansion for Green's Functions
379(2)
9.3.4 The Dirac Delta Function and Its Relationship to Green's Functions
381(6)
9.3.5 Nonhomogeneous Boundary Conditions
387(2)
9.3.6 Summary
389(6)
9.4 Fredholm Alternative and Modified Green's Functions
395(10)
9.4.1 Introduction
395(1)
9.4.2 Fredholm Alternative
396(3)
9.4.3 Modified Green's Functions
399(6)
9.5 Green's Functions for Poisson's Equation
405(22)
9.5.1 Introduction
405(1)
9.5.2 Multidimensional Dirac Delta Function and Green's Functions
406(2)
9.5.3 Green's Functions by the Method of Eigenfunction Expansion (Multidimensional)
408(2)
9.5.4 Direct Solution of Green's Functions (One-dimensional Eigenfunctions)
410(1)
9.5.5 Using Green's Functions for Problems with Nonhomogeneous Boundary Conditions
411(2)
9.5.6 Infinite Space Green's Functions
413(3)
9.5.7 Green's Functions for Bounded Domains Using Infinite Space Green's Functions
416(1)
9.5.8 Green's Functions for a Semi-Infinite Plane (y > 0) Using Infinite Space Green's Functions -- the Method of Images
417(2)
9.5.9 Green's Functions for a Circle -- The Method of Images
419(8)
9.6 Perturbed Eigenvalue Problems
427(6)
9.6.1 Introduction
427(1)
9.6.2 Mathematical Example
427(2)
9.6.3 Vibrating Nearly Circular Membrane
429(4)
9.7 Summary
433(1)
10 Infinite Domain Problems--Fourier Transform Solutions of Partial Differential Equations
434(61)
10.1 Introduction
434(1)
10.2 Heat Equation on an Infinite Domain
434(4)
10.3 Fourier Transform Pair
438(9)
10.3.1 Motivation from Fourier Series Identity
438(1)
10.3.2 Fourier Integral
439(1)
10.3.3 Inverse Fourier Transform of a Gaussian
440(7)
10.4 Fourier Transform and the Heat Equation
447(11)
10.4.1 Heat Equation
447(4)
10.4.2 Fourier Transforming the Heat Equation -- Transforms of Derivatives
451(2)
10.4.3 Convolution Theorem
453(2)
10.4.4 Summary of Properties of the Fourier Transform
455(3)
10.5 Fourier Sine and Cosine Transforms -- the Heat Equation on Semi-Infinite Intervals
458(11)
10.5.1 Introduction
458(1)
10.5.2 Heat Equation on a Semi-Infinite Interval I
459(1)
10.5.3 Fourier Sine and Cosine Transforms
460(2)
10.5.4 Transforms of Derivatives
462(1)
10.5.5 Heat Equation on a Semi-Infinite Interval II
463(6)
10.6 Worked Examples Using Transforms
469(21)
10.6.1 One-Dimensional Wave Equation on an Infinite Interval
469(2)
10.6.2 Laplace's Equation in a Semi-Infinite Strip
471(3)
10.6.3 Laplace's Equation in a Half-Plane
474(4)
10.6.4 Laplace's Equation in a Quarter-Plane
478(3)
10.6.5 Heat Equation in a Plane (Two-Dimensional Fourier Transforms)
481(9)
10.7 Scattering and Inverse Scattering
490(5)
11 Green's Functions for Time-Dependent Problems
495(30)
11.1 Introduction
495(1)
11.2 Green's Functions for the Wave Equation
495(17)
11.2.1 Introduction
495(1)
11.2.2 Green's Formula
496(2)
11.2.3 Reciprocity
498(2)
11.2.4 Using the Green's Function
500(2)
11.2.5 Infinite Space Green's Functions
502(2)
11.2.6 One-Dimensional Infinite Space Green's Function (d'Alembert's Solution)
504(2)
11.2.7 Three-Dimensional Infinite Space Green's Function (Huygens' Principle)
506(2)
11.2.8 Summary
508(4)
11.3 Green's Functions for the Heat Equation
512(13)
11.3.1 Introduction
512(1)
11.3.2 Nonself-Adjoint Nature of the Heat Equation
513(1)
11.3.3 Green's Formula
514(1)
11.3.4 Adjoint Green's Function
515(1)
11.3.5 Reciprocity
516(1)
11.3.6 Representation of the Solution Using Green's Functions
516(2)
11.3.7 Green's Function for the Heat Equation on an Infinite Domain
518(2)
11.3.8 Green's Function for the Heat Equation (Semi-Infinite Domain)
520(1)
11.3.9 Green's Function for the Heat Equation (on a Finite Region)
521(4)
12 The Method of Characteristics for Linear and Quasi-Linear Wave Equations
525(43)
12.1 Introduction
525(1)
12.2 Characteristics For First-Order Wave Equations
526(5)
12.2.1 Introduction
526(1)
12.2.2 Method of Characteristics for First-Order Partial Differential Equations
527(4)
12.3 Method of Characteristics for the One-Dimensional Wave Equation
531(7)
12.3.1 Introduction
531(1)
12.3.2 Initial Value Problem (Infinite Domain)
532(4)
12.3.3 d'Alembert's Solution
536(2)
12.4 Semi-Infinite Strings and Reflections
538(5)
12.5 Method of Characteristics for a Vibrating String of Fixed Length
543(3)
12.6 The Method of Characteristics for Quasi-linear Partial Differential Equations
546(16)
12.6.1 Method of Characteristics
546(2)
12.6.2 Traffic Flow
548(1)
12.6.3 Method of Characteristics (Q = 0)
549(2)
12.6.4 Shock Waves
551(6)
12.6.5 Quasi-Linear Example
557(5)
12.7 First-Order Nonlinear Partial Differential Equations
562(6)
12.7.1 Derive Eikonal Equation from Wave Equation
562(2)
12.7.2 Solving the Eikonal Equation in Uniform Media and Reflected Waves
564(2)
12.7.3 First-Order Nonlinear Partial Differential Equations
566(2)
13 A Brief Introduction to Laplace Transform Solution of Partial Differential Equations
568(30)
13.1 Introduction
568(1)
13.2 Elementary Properties of the Laplace Transform
569(9)
13.2.1 Introduction
569(1)
13.2.2 Singularities of the Laplace Transform
569(4)
13.2.3 Transforms of Derivatives
573(1)
13.2.4 Convolution Theorem
574(4)
13.3 Green's Functions for Initial Value Problems for Ordinary Differential Equations
578(2)
13.4 An Elementary Signal Problem for the Wave Equation
580(4)
13.5 A Signal Problem for a Vibrating String of Finite Length
584(3)
13.6 The Wave Equation and Its Green's Function
587(3)
13.7 Inversion of Laplace Transforms Using Contour Integrals in the Complex Plane
590(5)
13.8 Solving the Wave Equation Using Laplace Transforms (with Complex Variables)
595(3)
14 Topics: Dispersive Waves, Stability, Nonlinearity, and Perturbation Methods
598(93)
14.1 Introduction
598(1)
14.2 Dispersive Waves and Group Velocity
599(6)
14.2.1 Traveling Waves and the Dispersion Relation
599(3)
14.2.2 Group Velocity I
602(3)
14.3 Wave Guides
605(6)
14.3.1 Response to Concentrated Periodic Sources with Frequency (Wf)
607(1)
14.3.2 Green's Function if Mode Propagates
608(1)
14.3.3 Green's Function if Mode Does Not Propagate
609(1)
14.3.4 Design Considerations
609(2)
14.4 Fiber Optics
611(4)
14.5 Group Velocity II and the Method of Stationary Phase
615(6)
14.5.1 Method of Stationary Phase
615(3)
14.5.2 Application to Linear Dispersive Waves
618(3)
14.6 Slowly Varying Dispersive Waves (Group Velocity and Caustics)
621(9)
14.6.1 Approximate Solutions of Dispersive Partial Differential Equations
621(2)
14.6.2 Formation of a Caustic
623(7)
14.7 Envelope Equations (Concentrated Wave Numbers)
630(13)
14.7.1 Schrodinger Equation
631(1)
14.7.2 Linearized Korteweg-de Vries Equation
632(2)
14.7.3 Nonlinear Dispersive Waves: Korteweg-deVries Equation
634(2)
14.7.4 Solitons and Inverse Scattering
636(3)
14.7.5 Nonlinear Schrodinger Equation
639(4)
14.8 Stability and Instability
643(17)
14.8.1 Brief Ordinary Differential Equations and Bifurcation Theory
643(4)
14.8.2 Elementary Example of a Stable Equilibrium for a Partial Differential Equation
647(1)
14.8.3 Typical Unstable Equilibrium for a Partial Differential Equation and Pattern Formation
648(3)
14.8.4 Ill-Posed Problems
651(1)
14.8.5 Slightly Unstable Dispersive Waves and the Linearized Complex Ginzburg-Landau Equation
652(1)
14.8.6 Nonlinear Complex Ginzburg-Landau Equation
653(7)
14.9 Singular Perturbation Methods: Multiply Scaled Variables
660(17)
14.9.1 Ordinary Differential Equation: Weakly Nonlinearly Damped Oscillator
661(3)
14.9.2 Ordinary Differential Equation: Slowly Varying Oscillator
664(3)
14.9.3 Slightly Unstable Partial Differential Equation on Fixed Spatial Domain
667(3)
14.9.4 Slowly Varying Media for the Wave Equation
670(2)
14.9.5 Slowly Varying Linear Dispersive Waves (Including Weak Non-linear Effects)
672(5)
14.10 Singular Perturbation Methods: Boundary Layers
677(14)
14.10.1 Boundary Layer in an Ordinary Differential Equation
678(5)
14.10.2 Diffusion of a Pollutant Dominated by Convection
683(8)
Bibliography691(4)
Selected Answers to Starred Exercises695(19)
Index714

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