Contents Part I Second-Order Linear Partial Differential Equations 1 The Wave Equation 3 1.1 Equation for String Vibration 4 1.1.1 Derivation of the Equation for String Vibration 4 1.1.2 Initial and Boundary Conditions 9 1.1.3 Terminology 11 1.2 D’Alembert’s Formula 12 1.2.1 D’Alembert’s Formula 13 1.2.2 Characteristics 15 1.2.3 The Case of a Semi-infinitely Long String 17 1.2.4 DuhamePs Principle 22 1.3 Method of Separation of Variables 24 1.3.1 Making Boundary Conditions Homogeneous 24 1.3.2 Method of Separation of Variables—Its Procedure 25 1.3.3 Physical Implications 31 1.3.4 Inhomogeneous Governing Equations 32 1.4 Wave Equation in Higher Dimensions 35 1.4.1 Small and Transverse Vibration of a Membrane 35 1.4.2 Definite Problems 40 1.4.3 Solutions for Cauchy Problems 41 1.5 Solution Properties 46 1.5.1 The Energy of a Vibrating Membrane 47 1.5.2 Solution Uniqueness of Problems for the Wave Equation 48 2 The Heat Equation 57 2.1 Modelling Heat Conduction 57 2.1.1 Derivation of the Heat Equation 57 2.1.2 Initial and Boundary Conditions 61 2.1.3 Physical Analogies 62 2.2 Method of Integral Transform 63 2.2.1 Convolution and Fourier Transform 64 2.2.2 Solution for Cauchy Problems 67 2.2.3 Solution Properties 70 2.2.4 Inhomogeneous Governing Equations 72 2.3 A Revisit to the Method of Separation of Variables 75 2.3.1 An Example with the Heat Equation 75 2.3.2 Sturm-Liouville System 81 2.3.3 Inhomogeneous Governing Equations 88 2.4 Solution Properties 89 2.4.1 Maximum Principle 89 2.4.2 Solution Uniqueness 91 2.4.3 Stability 92 3 Poisson’s Equation 99 3.1 Poisson’s Equation and Harmonic Equation 99 3.1.1 Definitions 99 3.1.2 Motivation from Physics 101 3.1.3 Boundary Conditions 106 3.2 Variational Principle 108 3.3 Harmonic Functions in Polar System 112 3.3.1 Laplace’s Equation in Polar System 113 3.3.2 Radial Solutions to Laplace’s Equation 115 3.4 The Method of Green^ Function 117 3.4.1 Green’s Formulae Related to Laplacian Operator 117 3.4.2 Fundamental Solution 118 3.4.3 Derivation of Green’s Function 120 3.4.4 Properties of Green’s Function 122 3.4.5 Problems for Poisson’s Equation 123 3.4.6 Final Remarks 125 3.5 Image Method for Electric Potentials 125 3.5.1 The Case in Three-Dimensional Half Space 126 3.5.2 The Case in a Spherical Domain 127 3.6 Solution Uniqueness 130 3.6.1 Mean-Value Formula 130 3.6.2 Maximum Principle 132 3.6.3 Strong Maximum Principle 133 3.6.4 Energy Method 134 4 Summary over Second-Order Linear Partial Differential Equation 141 4.1 Classification of Second-Order Linear Partial Differential Equations 141 4.1.1 Cases with Two Variables 141 4.1.2 Summary and Examples 148 4.1.3 Multivariable Situations 150 4.2 Topical Discussion 155 Part II Special Functions 5 Bessel Functions 165 5.1 Bessel Equation and Bessel Functions 165 5.1.1 Physical Motivation 165 5.1.2 Bessel Function of the First Kind 169 5.1.3 Bessel Function of the Second Kind 173 5.2 Properties of Bessel Functions 177 5.2.1 Recurrence Formulae 177 5.2.2 Zeros 179 5.2.3 Approximating Formula 181 5.2.4 Orthogonality 181 5.2.5 Analogies with Sinusoidal Functions 184 5.3 Solving PDE Problems with Bessel Functions 186 5.4 Generalisation 195 5.4.1 Hankel Functions 195 5.4.2 Modified Bessel Functions 195 6 Legendre Polynomial 199 6.1 Legendre Equation 199 6.2 Legendre Polynomial 204 6.2.1 Series Solution to the Legendre Equation 204 6.2.2 Legendre Polynomial 207 6.3 Properties of Legendre Polynomial 210 6.3.1 Rodrigues Formula 210 6.3.2 Key Properties at a Glance 210 6.3.3 Orthogonal Systems 212 6.3.4 Discussion on General Orthogonal Polynomials 218 6.4 Applications with Legendre Polynomials 220 6.4.1 Solving PDEs Defined in a Sphere 220 6.4.2 Legendre-Gauss Quadrature 222 6.5 The Associated Legendre Functions 225 7 Introduction of Hypergeometric Function 231 7.1 Commonalities of Bessel and Legendre Functions 231 7.2 Gauss Hypergeometric Function 234 7.2.1 Definition 234 7.2.2 Hypergeometric Differential Equation 235 7.2.3 Legendre Function and Legendre PolynomialA Revisit 239 7.3 Confluent Hypergeometric Function 241 7.3.1 Kummer Differential Equation and Its Solutions 241 7.3.2 Special Cases 242 7.4 Final Remarks on Hypergeometric Functions 244 Bibliography 247 Index 249
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