Liqiu Wang • Xuesheng Zhou • Xiaohao Wei

Heat Conduction

Mathematical Models and Analytical Solutions

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Contents

1 Introduction 1 1.1 Partial Differential Equations 1

1.1.1 Partial Differential Equations and Their Orders 1 1.1.2 Linear, Nonlinear and Quasi-Linear Equations 2 1.1.3 Solutions of Partial Differential Equations 3 1.1.4 Classification of Linear Second-Order Equations 5 1.1.5 Canonical Forms 7

1.2 Three Basic Equations of Mathematical Physics 11 1.2.1 Physical Laws and Equations of Mathematical Physics 11 1.2.2 Approaches of Developing Equations

of Mathematical Physics 12 1.2.3 Wave Equations 13 1.2.4 Heat-Conduction Equations 15 1.2.5 Potential Equations 17

1.3 Theory of Heat Conduction And Three Types of Heat-Conduction Equations 18 1.3.1 Constitutive Relations of Heat Flux 18 1.3.2 The Boltzmann Transport Equation

and Dual-Phase-Lagging Heat Conduction 25 1.3.3 Three Types of Heat-Conduction Equations 30

1.4 Conditions and Problems for Determining Solutions 32 1.4.1 Initial Conditions 32 1.4.2 Boundary Conditions 33 1.4.3 Problems for Determining Solutions 37 1.4.4 Well-Posedness of PDS 38 1.4.5 Exampleof Developing PDS 39

2 Wave Equations 41 2.1 The Solution Structure Theorem for Mixed Problems

and its Application 41 2.2 Fourier Method for One-Dimensional Mixed Problems 45

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2.2.1 Boundary Condition of the First Kind 45 2.2.2 Boundary Condition of the Second Kind 50

2.3 Method of Separation of Variables for One-Dimensional Mixed Problems 51 2.3.1 Method of Separation of Variables 51 2.3.2 Generalized Fourier Method of Expansion 54 2.3.3 Important Properties of Eigenvalue Problems (2.19) 56

2.4 Well-Posedness and Generalized Solution 58 2.4.1 Existence 58 2.4.2 Uniqueness 60 2.4.3 Stability 61 2.4.4 Generalized Solution 62 2.4.5 PDS with Variable Coefficients 64

2.5 Two-Dimensional Mixed Problems 67 2.5.1 Rectangular Domain 67 2.5.2 Circular Domain 69

2.6 Three-Dimensional Mixed Problems 76 2.6.1 Cuboid Domain 76 2.6.2 Spherical Domain 78

2.7 Methods of Solving One-Dimensional Cauchy Problems 83 2.7.1 Method of Fourier Transformation 83 2.7.2 Method of Characteristics 85 2.7.3 Physical Meaning 86 2.7.4 Domains of Dependence, Determinacy and Influence 89 2.7.5 Problems in a Semi-Infinite Domain

and the Method of Continuation 92 2.8 Two- and Three-Dimensional Cauchy Problems 96

2.8.1 Method of Fourier Transformation 96 2.8.2 Method of Spherical Means 99 2.8.3 Method of Descent 101 2.8.4 Physical Meanings of the Poisson and Kirchhoff Formulas.. 109

3 Heat-Conduction Equations 113 3.1 The Solution Structure Theorem For Mixed Problems 113 3.2 Solutions of Mixed Problems 116

3.2.1 One-Dimensional Mixed Problems 116 3.2.2 Two-Dimensional Mixed Problems 118 3.2.3 Three-Dimensional Mixed Problems 120

3.3 Well-Posedness of PDS 122 3.3.1 Existence 123 3.3.2 Uniqueness 124 3.3.3 Stability 125

3.4 One-Dimensional Cauchy Problems: Fundamental Solution 126 3.4.1 One-Dimensional Cauchy Problems 126

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3.4.2 Fundamental Solution of the One-Dimensional Heat-Conduction Equation 128

3.4.3 Problems in Semi-Infmite Domain and the Method of Continuation 130

3.4.4 PDS with Variable Thermal Conductivity 133 3.5 Multiple Fourier Transformations for Two- and Three-

Dimensional Cauchy Problems 136 3.5.1 Two-Dimensional Case 136 3.5.2 Three-Dimensional Case 137

3.6 Typical PDS of Diffusion 137 3.6.1 Fick's Law of Diffusion and Diffusion Equation 138 3.6.2 Diffusion from a Constant Source 139 3.6.3 Diffusion from an Instant Plane Source 140 3.6.4 Diffusion Between Two Semi-Infmite Domains 141

4 Mixed Problems of Hyperbolic Heat-Conduction Equations 143 4.1 Solution Structure Theorem 143 4.2 One-Dimensional Mixed Problems 147

4.2.1 Mixed Boundary Conditions of the First and the Third Kind 147 4.2.2 Mixed Boundary Conditions of the Second

and the Third Kind 150 4.3 Two-Dimensional Mixed Problems 152

4.3.1 Rectangular Domain 152 4.3.2 Circular Domain 161

4.4 Three-Dimensional Mixed Problems 163 4.4.1 Cuboid Domain 163 4.4.2 Cylindrical Domain 166 4.4.3 Spherical Domain 167

5 Cauchy Problems of Hyperbolic Heat-Conduction Equations 171 5.1 Riemann Method for Cauchy Problems 171

5.1.1 Conjugate Operator and Green Formula 171 5.1.2 Cauchy Problems and Riemann Functions 172 5.1.3 Example 175

5.2 Riemann Method and Method of Laplace Transformation for One-Dimensional Cauchy Problems 177 5.2.1 Riemann Method 178 5.2.2 Method of Laplace Transformation 184 5.2.3 Some Properties of Solutions 186

5.3 Verification of Solutions, Physics and Measurement of To 188 5.3.1 Verify the Solution for u(x,0) = 0 and ut(x,0) = 1 188 5.3.2 Verify the Solution for u(x, 0) = 1 and ut(x, 0) = 0 191

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5.3.3 Verify the Solution for f(x, t) = 1, u(x, 0) = 0 and ut(x,0) = 0 193

5.3.4 Physics and Measurement of To 194 5.3.5 Measuring To by Characteristic Curves 195 5.3.6 Measuring To by a Unit Impulsive Source 5(x — xo,t — to). . 196

5.4 Method of Descent for Two-Dimensional Problems and Discussion Of Solutions 197 5.4.1 Transform to Three-Dimensional Wave Equations 197 5.4.2 Solution of PDS (5.62) 198 5.4.3 Solution of PDS (5.61) 199 5.4.4 Verification of CDS 200 5.4.5 Special Cases 203

5.5 Domains of Dependence and Influence, Measuring TB by Characteristic Cones 205 5.5.1 Domain of Dependence 205 5.5.2 Domain of Influence 206 5.5.3 Measuring To by Characteristic Cones 207

5.6 Comparison of Fundamental Solutions of Classical and Hyperbolic Heat-Conduction Equations 209 5.6.1 Fundamental S olutions of Two Kinds

of Heat-Conduction Equations 209 5.6.2 Common Properties 210 5.6.3 Different Properties 211

5.7 Methods for Solving Axially Symmetrie and Spherically-Symmetric Cauchy Problems 212 5.7.1 The Hankel Transformation for Two-Dimensional Axially

Symmetrie Problems 212 5.7.2 Spherical Bessel Transformation for Spherically-

Symmetric Cauchy Problems 214 5.7.3 Method of Continuation for Spherically-

Symmetric Problems 217 5.7.4 Discussion of Solution (5.98) 219

5.8 Methods of Fourier Transformation and Spherical Means for Three-Dimensional Cauchy Problems 222 5.8.1 An Integral Formula of Bessel Function 222 5.8.2 Fourier Transformation for Three-Dimensional Problems... 224 5.8.3 Method of Spherical Means for PDS (5.115) 226 5.8.4 Discussion 230

6 Dual-Phase-Lagging Heat-Conduction Equations 233 6.1 Solution Structure Theorem for Mixed Problems 233

6.1.1 Notes on Dual-Phase-Lagging Heat-Conduction Equations . 233 6.1.2 Solution Structure Theorem 234

6.2 Fourier Method of Expansion for One-Dimensional Mixed Problems 239

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6.2.1 Fourier Method of Expansion 239 6.2.2 Existence 246

6.3 Separation ofVariables for One-Dimensional Mixed Problems 249 6.3.1 Eigenvalue Problems 249 6.3.2 Eigenvalues and Eigenfunctions 250 6.3.3 Solve Mixed Problems with Table 2.1 253

6.4 Solution Structure Theorem: Another Form and Application 257 6.4.1 One-Dimensional Mixed Problems 257 6.4.2 Two-Dimensional Mixed Problems 261 6.4.3 Three-Dimensional Mixed Problems 266 6.4.4 Summary and Remarks 271

6.5 Mixed Problems in a Circular Domain 273 6.5.1 Solution from y(r, 0) 274 6.5.2 Solution from <p(r, 0) 276 6.5.3 Solution from f(r, 0,t) 278

6.6 Mixed Problems in a Cylindrical Domain 281 6.6.1 Solution from y(r, 0,z) 281 6.6.2 Solution from <p(r, 0,z) 284 6.6.3 Solution from f(r, 0,z,t) 286 6.6.4 Green Function of the Dual-Phase-Lagging

Heat-Conduction Equation 288 6.7 Mixed Problems in a Spherical Domain 289

6.7.1 Solution from i//(r, 0, (p) 289 6.7.2 Solution from 3>(r, 0, <p) 291 6.7.3 Solution from f(r, 0, <p,t) 293

6.8 Cauchy Problems 296 6.9 Perturbation Method for Cauchy Problems 300

6.9.1 Introduction 301 6.9.2 The Perturbation Method for Solving

Hyperbolic Heat-Conduction Equations 302 6.9.3 Perturbation Solutions of Dual-Phase-Lagging

Heat-Conduction Equations 304 6.9.4 Solutions for Initial-Value of Lower-Order Polynomials . . . . 310 6.9.5 Perturbation Method for Two-

and Three-dimensional Problems 313 6.10 Thermal Waves and Resonance 314

6.10.1 Thermal Waves 315 6.10.2 Resonance 322

6.11 Heat Conducüon in Two-Phase Systems 326 6.11.1 One- and Two-Equation Models 326 6.11.2 Equivalence with Dual-Phase-Lagging Heat Conducüon . . .333

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7 Potential Equations 335 7.1 Fourier Method of Expansion 335 7.2 Separation of Variables and Fourier Sin/Cos Transformation 339

7.2.1 Separation of Variables 339 7.2.2 Fourier Sine/Cosine Transformation in a Finite Region 346

7.3 Methods for Solving Nonhomogeneous Potential Equations 353 7.3.1 Equation Homogenization by Function Transformation . . . . 353 7.3.2 Extremum Principle 355 7.3.3 Four Examples of Applications 357

7.4 Fundamental Solution and the Harmonie Function 363 7.4.1 Fundamental Solution 363 7.4.2 Green Function 365 7.4.3 Harmonie Functions 369

7.5 Well-Posedness of Boundary-Value Problems 376 7.6 Green Functions 382

7.6.1 Green Function 382 7.6.2 Properties of Green Functions of the Dirichlet Problems.... 385

7.7 Method of Green Functions for Boundary-Value Problems of the First Kind 388 7.7.1 Mirror Image Method for Finding Green Functions 388 7.7.2 Examples Using the Method of Green Functions 388 7.7.3 Boundary-Value Problems in Unbounded Domains 397

7.8 Potential Theory 403 7.8.1 Potentials 403 7.8.2 Generalized Integrals with Parameters 406 7.8.3 Solid Angle and Russin Surface 409 7.8.4 Properties of Surface Potentials 411

7.9 Transformation of Boundary-Value Problems of Laplace Equations to Integral Equations 414 7.9.1 Integral Equations 414 7.9.2 Transformation of Boundary-Value Problems

into Integral Equations 416 7.9.3 Boundary-Value Problems of Poisson Equations 419 7.9.4 Two-Dimensional Potential Equations 420

A Special Functions 425 A.l Bessel and Legendre Equations 425 A.2 Bessel Functions 427 A.3 Properties of Bessel Functions 432 A.4 Legendre Polynomials 436 A.5 Properties of Legendre Polynomials 438 A.6 Associated Legendre Polynomials 440

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B Integral Transformations 447 B.l Fourier Integral Transformation 447

B.l.l Fourier Integral 447 B.1.2 Fourier Transformation 450 B.1.3 Generalized Functions and the ö-function 455 B.l.4 Generalized Fourier Transformation 458 B.1.5 The Multiple Fourier Transformation 462

B.2 Laplace Transformation 464 B.2.1 Laplace Transformation 464 B.2.2 Properties of Laplace Transformation 466 B.2.3 Determine Inverse Image Functions

by Calculating Residues 469 B.2.4 Convolution Theorem 471

C Tables of Integral Transformations 475

D Eigenvalue Problems 483 D.l Regulär Sturm-Liouville Problems 483 D.2 The Lagrange Equality and Self-Conjugate

Boundary-Value Problems 484 D.3 Properties of S-L Problems 485 D.4 Singular S-L Problems 487

References 491

Index 511