目录 ContentsPart I Second-Order Linear Partial Differential Equations1 The Wave Equation 31.1 Equation for String Vibration 41.1.1 Derivation of the Equation for String Vibration 41.1.2 Initial and Boundary Conditions 91.1.3 Terminology 111.2 DAlemberts Formula 121.2.1 DAlemberts Formula 131.2.2 Characteristics 151.2.3 The Case of a Semi-infinitely Long String 171.2.4 DuhamePs Principle 221.3 Method of Separation of Variables 241.3.1 Making Boundary Conditions Homogeneous 241.3.2 Method of Separation of Variables—Its Procedure 251.3.3 Physical Implications 311.3.4 Inhomogeneous Governing Equations 321.4 Wave Equation in Higher Dimensions 351.4.1 Small and Transverse Vibration of a Membrane 351.4.2 Definite Problems 401.4.3 Solutions for Cauchy Problems 411.5 Solution Properties 461.5.1 The Energy of a Vibrating Membrane 471.5.2 Solution Uniqueness of Problems for the Wave Equation 482 The Heat Equation 572.1 Modelling Heat Conduction 572.1.1 Derivation of the Heat Equation 572.1.2 Initial and Boundary Conditions 612.1.3 Physical Analogies 622.2 Method of Integral Transform 632.2.1 Convolution and Fourier Transform 642.2.2 Solution for Cauchy Problems 672.2.3 Solution Properties 702.2.4 Inhomogeneous Governing Equations 722.3 A Revisit to the Method of Separation of Variables 752.3.1 An Example with the Heat Equation 752.3.2 Sturm-Liouville System 812.3.3 Inhomogeneous Governing Equations 882.4 Solution Properties 892.4.1 Maximum Principle 892.4.2 Solution Uniqueness 912.4.3 Stability 923 Poissons Equation 993.1 Poissons Equation and Harmonic Equation 993.1.1 Definitions 993.1.2 Motivation from Physics 1013.1.3 Boundary Conditions 1063.2 Variational Principle 1083.3 Harmonic Functions in Polar System 1123.3.1 Laplaces Equation in Polar System 1133.3.2 Radial Solutions to Laplaces Equation 1153.4 The Method of Green^ Function 1173.4.1 Greens Formulae Related to Laplacian Operator 1173.4.2 Fundamental Solution 1183.4.3 Derivation of Greens Function 1203.4.4 Properties of Greens Function 1223.4.5 Problems for Poissons Equation 1233.4.6 Final Remarks 1253.5 Image Method for Electric Potentials 1253.5.1 The Case in Three-Dimensional Half Space 1263.5.2 The Case in a Spherical Domain 1273.6 Solution Uniqueness 1303.6.1 Mean-Value Formula 1303.6.2 Maximum Principle 1323.6.3 Strong Maximum Principle 1333.6.4 Energy Method 1344 Summary over Second-Order Linear Partial Differential Equation 1414.1 Classification of Second-Order Linear Partial Differential Equations 1414.1.1 Cases with Two Variables 1414.1.2 Summary and Examples 1484.1.3 Multivariable Situations 1504.2 Topical Discussion 155Part II Special Functions5 Bessel Functions 1655.1 Bessel Equation and Bessel Functions 1655.1.1 Physical Motivation 1655.1.2 Bessel Function of the First Kind 1695.1.3 Bessel Function of the Second Kind 1735.2 Properties of Bessel Functions 1775.2.1 Recurrence Formulae 1775.2.2 Zeros 1795.2.3 Approximating Formula 1815.2.4 Orthogonality 1815.2.5 Analogies with Sinusoidal Functions 1845.3 Solving PDE Problems with Bessel Functions 1865.4 Generalisation 1955.4.1 Hankel Functions 1955.4.2 Modified Bessel Functions 1956 Legendre Polynomial 1996.1 Legendre Equation 1996.2 Legendre Polynomial 2046.2.1 Series Solution to the Legendre Equation 2046.2.2 Legendre Polynomial 2076.3 Properties of Legendre Polynomial 2106.3.1 Rodrigues Formula 2106.3.2 Key Properties at a Glance 2106.3.3 Orthogonal Systems 2126.3.4 Discussion on General Orthogonal Polynomials 2186.4 Applications with Legendre Polynomials 2206.4.1 Solving PDEs Defined in a Sphere 2206.4.2 Legendre-Gauss Quadrature 2226.5 The Associated Legendre Functions 2257 Introduction of Hypergeometric Function 2317.1 Commonalities of Bessel and Legendre Functions 2317.2 Gauss Hypergeometric Function 2347.2.1 Definition 2347.2.2 Hypergeometric Differential Equation 2357.2.3 Legendre Function and Legendre PolynomialA Revisit 2397.3 Confluent Hypergeometric Function 2417.3.1 Kummer Differential Equation and Its Solutions 2417.3.2 Special Cases 2427.4 Final Remarks on Hypergeometric Functions 244Bibliography 247Index 249
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