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Full Description
This book emphasizes the physical interpretation of mathematical solutions and introduces applied mathematics while presenting differential equations. Coverage includes Fourier series, orthogonal functions, boundary value problems, Green's functions, and transform methods.This text is ideal for readers interested in science, engineering, and applied mathematics.
Contents
1. Heat Equation1.1 Introduction1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod1.3 Boundary Conditions1.4 Equilibrium Temperature Distribution1.4.1 Prescribed Temperature1.4.2 Insulated Boundaries1.5 Derivation of the Heat Equation in Two or Three Dimensions2. Method of Separation of Variables2.1 Introduction2.2 Linearity2.3 Heat Equation with Zero Temperatures at Finite Ends2.3.1 Introduction2.3.2 Separation of Variables2.3.3 Time-Dependent Equation2.3.4 Boundary Value Problem2.3.5 Product Solutions and the Principle of Superposition2.3.6 Orthogonality of Sines2.3.7 Formulation, Solution, and Interpretation of an Example2.3.8 Summary2.4 Worked Examples with the Heat Equation: Other Boundary Value Problems2.4.1 Heat Conduction in a Rod with Insulated Ends2.4.2 Heat Conduction in a Thin Circular Ring2.4.3 Summary of Boundary Value Problems2.5 Laplace's Equation: Solutions and Qualitative Properties2.5.1 Laplace's Equation Inside a Rectangle2.5.2 Laplace's Equation for a Circular Disk2.5.3 Fluid Flow Past a Circular Cylinder (Lift)2.5.4 Qualitative Properties of Laplace's Equation3. Fourier Series3.1 Introduction3.2 Statement of Convergence Theorem3.3 Fourier Cosine and Sine Series3.3.1 Fourier Sine Series3.3.2 Fourier Cosine Series3.3.3 Representing f(x) by Both a Sine and Cosine Series3.3.4 Even and Odd Parts3.3.5 Continuous Fourier Series3.4 Term-by-Term Differentiation of Fourier Series3.5 Term-By-Term Integration of Fourier Series3.6 Complex Form of Fourier Series4. Wave Equation: Vibrating Strings and Membranes4.1 Introduction4.2 Derivation of a Vertically Vibrating String4.3 Boundary Conditions4.4 Vibrating String with Fixed Ends4.5 Vibrating Membrane4.6 Reflection and Refraction of Electromagnetic (Light) and Acoustic (Sound) Waves4.6.1 Snell's Law of Refraction4.6.2 Intensity (Amplitude) of Reflected and Refracted Waves4.6.3 Total Internal Reflection5. Sturm-Liouville Eigenvalue Problems5.1 Introduction5.2 Examples5.2.1 Heat Flow in a Nonuniform Rod5.2.2 Circularly Symmetric Heat Flow5.3 Sturm-Liouville Eigenvalue Problems5.3.1 General Classification5.3.2 Regular Sturm-Liouville Eigenvalue Problem5.3.3 Example and Illustration of Theorems5.4 Worked Example: Heat Flow in a Nonuniform Rod without Sources5.5 Self-Adjoint Operators and Sturm-Liouville Eigenvalue Problems5.6 Rayleigh Quotient5.7 Worked Example: Vibrations of a Nonuniform String5.8 Boundary Conditions of the Third Kind5.9 Large Eigenvalues (Asymptotic Behavior)5.10 Approximation Properties6. Finite Difference Numerical Methods for Partial Differential Equations6.1 Introduction6.2 Finite Differences and Truncated Taylor Series6.3 Heat Equation6.3.1 Introduction6.3.2 A Partial Difference Equation6.3.3 Computations6.3.4 Fourier-von Neumann Stability Analysis6.3.5 Separation of Variables for Partial Difference Equations and Analytic Solutions of Ordinary Difference Equations6.3.6 Matrix Notation6.3.7 Nonhomogeneous Problems6.3.8 Other Numerical Schemes6.3.9 Other Types of Boundary Conditions6.4 Two-Dimensional Heat Equation6.5 Wave Equation6.6 Laplace's Equation6.7 Finite Element Method6.7.1 Approximation with Nonorthogonal Functions (Weak Form of the Partial Differential Equation)6.7.2 The Simplest Triangular Finite Elements7. Higher Dimensional Partial Differential Equations7.1 Introduction7.2 Separation of the Time Variable7.2.1 Vibrating Membrane: Any Shape7.2.2 Heat Conduction: Any Region7.2.3 Summary7.3 Vibrating Rectangular Membrane7.4 Statements and Illustrations of Theorems for the Eigenvalue Problem 2 + = 07.5 Green's Formula, Self-Adjoint Operators and Multidimensional Eigenvalue Problems7.6 Rayleigh Quotient and Laplace's Equation7.6.1 Rayleigh Quotient7.6.2 Time-Dependent Heat Equation and Laplace's Equation7.7 Vibrating Circular Membrane and Bessel Functions7.7.1 Introduction7.7.2 Separation of Variables7.7.3 Eigenvalue Problems (One Dimensional)7.7.4 Bessel's Differential Equation7.7.5 Singular Points and Bessel's Differential Equation7.7.6 Bessel Functions and Their Asymptotic Properties (near z = 0)7.7.7 Eigenvalue Problem Involving Bessel Functions7.7.8 Initial Value Problem for a Vibrating Circular Membrane7.7.9 Circularly Symmetric Case7.8 More on Bessel Functions7.8.1 Qualitative Properties of Bessel Functions7.8.2 Asymptotic Formulas for the Eigenvalues7.8.3 Zeros of Bessel Functions and Nodal Curves7.8.4 Series Representation of Bessel Functions7.9 Laplace's Equation in a Circular Cylinder7.9.1 Introduction7.9.2 Separation of Variables7.9.3 Zero Temperature on the Lateral Sides and on the Bottom or Top7.9.4 Zero Temperature on the Top and Bottom7.9.5 Modified Bessel Functions7.10 Spherical Problems and Legendre Polynomials7.10.1 Introduction7.10.2 Separation of Variables and One-Dimensional Eigenvalue Problems7.10.3 Associated Legendre Functions and Legendre Polynomials7.10.4 Radial Eigenvalue Problems7.10.5 Product Solutions, Modes of Vibration, and the Initial Value Problem7.10.6 Laplace's Equation Inside a Spherical Cavity8. Nonhomogeneous Problems8.1 Introduction8.2 Heat Flow with Sources and Nonhomogeneous Boundary Conditions8.3 Method of Eigenfunction Expansion with Homogeneous Boundary Conditions (Differentiating Series of Eigenfunctions)8.4 Method of Eigenfunction Expansion Using Green's Formula (With or Without Homogeneous Boundary Conditions)8.5 Forced Vibrating Membranes and Resonance8.6 Poisson's Equation9. Green's Functions for Time-Independent Problems9.1 Introduction9.2 One-dimensional Heat Equation9.3 Green's Functions for Boundary Value Problems for Ordinary Differential Equations9.3.1 One-Dimensional Steady-State Heat Equation9.3.2 The Method of Variation of Parameters9.3.3 The Method of Eigenfunction Expansion for Green's Functions9.3.4 The Dirac Delta Function and Its Relationship to Green's Functions9.3.5 Nonhomogeneous Boundary Conditions9.3.6 Summary9.4 Fredholm Alternative and Generalized Green's Functions9.4.1 Introduction9.4.2 Fredholm Alternative9.4.3 Generalized Green's Functions9.5 Green's Functions for Poisson's Equation9.5.1 Introduction9.5.2 Multidimensional Dirac Delta Function and Green's Functions9.5.3 Green's Functions by the Method of Eigenfunction Expansion and the Fredholm Alternative9.5.4 Direct Solution of Green's Functions (One-Dimensional Eigenfunctions)9.5.5 Using Green's Functions for Problems with Nonhomogeneous Boundary Conditions9.5.6 Infinite Space Green's Functions9.5.7 Green's Functions for Bounded Domains Using Infinite Space Green's Functions9.5.8 Green's Functions for a Semi-Infinite Plane (y > 0) Using Infinite Space Green's Functions: The Method of Images9.5.9 Green's Functions for a Circle: The Method of Images9.6 Perturbed Eigenvalue Problems9.6.1 Introduction9.6.2 Mathematical Example9.6.3 Vibrating Nearly Circular Membrane9.7 Summary10. Infinite Domain Problems: Fourier Transform Solutions of Partial Differential Equations10.1 Introduction10.2 Heat Equation on an Infinite Domain10.3 Fourier Transform Pair10.3.1 Motivation from Fourier Series Identity10.3.2 Fourier Transform10.3.3 Inverse Fourier Transform of a Gaussian10.4 Fourier Transform and the Heat Equation10.4.1 Heat Equation10.4.2 Fourier Transforming the Heat Equation: Transforms of Derivatives10.4.3 Convolution Theorem10.4.4 Summary of Properties of the Fourier Transform10.5 Fourier Sine and Cosine Transforms: The Heat Equation on Semi-Infinite Intervals10.5.1 Introduction10.5.2 Heat Equation on a Semi-Infinite Interval I10.5.3 Fourier Sine and Cosine Transforms10.5.4 Transforms of Derivatives10.5.5 Heat Equation on a Semi-Infinite Interval II10.5.6 Tables of Fourier Sine and Cosine Transforms10.6 Worked Examples Using Transforms10.6.1 One-Dimensional Wave Equation on an Infinite Interval10.6.2 Laplace's Equation in a Semi-Infinite Strip10.6.3 Laplace's Equation in a Half-Plane10.6.4 Laplace's Equation in a Quarter-Plane10.6.5 Heat Equation in a Plane (Two-Dimensional Fourier Transforms)10.6.6 Table of Double-Fourier Transforms10.7 Scattering and Inverse Scattering11. Green's Functions for Wave and Heat Equations11.1 Introduction11.2 Green's Functions for the Wave Equation11.2.1 Introduction11.2.2 Green's Formula11.2.3 Reciprocity11.2.4 Using the Green's Function11.2.5 Green's Function for the Wave Equation11.2.6 Alternate Differential Equation for the Green's Function11.2.7 Infinite Space Green's Function for the One-Dimensional Wave Equation and d'Alembert's Solution11.2.8 Infinite Space Green's Function for the Three-Dimensional Wave Equation (Huygens' Principle)11.2.9 Two-Dimensional Infinite Space Green's Function11.2.10 Summary11.3 Green's Functions for the Heat Equation11.3.1 Introduction11.3.2 Non-Self-Adjoint Nature of the Heat Equation11.3.3 Green's Formula11.3.4 Adjoint Green's Function11.3.5 Reciprocity11.3.6 Representation of the Solution Using Green's Functions11.3.7 Alternate Differential Equation for the Green's Function11.3.8 Infinite Space Green's Function for the Diffusion Equation11.3.9 Green's Function for the Heat Equation (Semi-Infinite Domain)11.3.10 Green's Function for the Heat Equation (on a Finite Region)12. The Method of Characteristics for Linear and Quasilinear Wave Equations12.1 Introduction12.2 Characteristics for First-Order Wave Equations12.2.1 Introduction12.2.2 Method of Characteristics for First-Order Partial Differential Equations12.3 Method of Characteristics for the One-Dimensional Wave Equation12.3.1 General Solution12.3.2 Initial Value Problem (Infinite Domain)12.3.3 D'alembert's Solution12.4 Semi-Infinite Strings and Reflections12.5 Method of Characteristics for a Vibrating String of Fixed Length12.6 The Method of Characteristics for Quasilinear Partial Differential Equations12.6.1 Method of Characteristics12.6.2 Traffic Flow12.6.3 Method of Characteristics (Q=0)12.6.4 Shock Waves12.6.5 Quasilinear Example12.7 First-Order Nonlinear Partial Differential Equations12.7.1 Eikonal Equation Derived from the Wave Equation12.7.2 Solving the Eikonal Equation in Uniform Media and Reflected Waves12.7.3 First-Order Nonlinear Partial Differential Equations13. Laplace Transform Solution of Partial Differential Equations13.1 Introduction13.2 Properties of the Laplace Transform13.2.1 Introduction13.2.2 Singularities of the Laplace Transform13.2.3 Transforms of Derivatives13.2.4 Convolution Theorem13.3 Green's Functions for Initial Value Problems for Ordinary Differential Equations13.4 A Signal Problem for the Wave Equation13.5 A Signal Problem for a Vibrating String of Finite Length13.6 The Wave Equation and its Green's Function13.7 Inversion of Laplace Transforms Using Contour Integrals in the Complex Plane13.8 Solving the Wave Equation Using Laplace Transforms (with Complex Variables)BibliographyAnswers to Starred ExercisesIndex
