Heat Conduction (eBook)

Preface 5
Contents 7
Introduction 14
1.1 Partial Differential Equations 14
1.2 Three Basic Equations of Mathematical Physics 24
1.3 Theory of Heat Conduction And Three Types of Heat- Conduction Equations 31
1.4 Conditions and Problems for Determining Solutions 45
1.4.1 Initial Conditions 45
1.4.2 Boundary Conditions 46
1.4.3 Problems for Determining Solutions 50
1.4.4 Well-Posedness of PDS 51
1.4.5 Example of Developing PDS 52
Wave Equations 54
2.1 The Solution Structure Theorem for Mixed Problems and its Application 54
2.2 Fourier Method for One-Dimensional Mixed Problems 58
2.2.1 Boundary Condition of the First Kind 58
2.2.2 Boundary Condition of the Second Kind 63
2.3 Method of Separation of Variables for One- Dimensional Mixed Problems 64
2.3.1 Method of Separation of Variables 64
2.3.2 Generalized Fourier Method of Expansion 67
2.3.3 Important Properties of Eigenvalue Problems (2.19) 69
2.4 Well-Posedness and Generalized Solution 71
2.4.1 Existence 71
2.4.2 Uniqueness 73
2.4.3 Stability 74
2.4.4 Generalized Solution 75
2.4.5 PDS with Variable Coefficients 77
2.5 Two-Dimensional Mixed Problems 80
2.5.1 Rectangular Domain 80
2.5.2 Circular Domain 82
2.6 Three-Dimensional Mixed Problems 89
2.6.1 Cuboid Domain 89
2.6.2 Spherical Domain 91
2.7 Methods of Solving One-Dimensional Cauchy Problems 96
2.7.1 Method of Fourier Transformation 96
2.7.2 Method of Characteristics 98
2.7.3 Physical Meaning 99
2.7.4 Domains of Dependence, Determinacy and Influence 102
2.7.5 Problems in a Semi-Infinite Domain and the Method of Continuation 105
2.8 Two- and Three-Dimensional Cauchy Problems 109
2.8.1 Method of Fourier Transformation 109
2.8.2 Method of Spherical Means 112
2.8.3 Method of Descent 114
2.8.4 Physical Meanings of the Poisson and Kirchhoff Formulas 122
Heat-Conduction Equations 126
3.1 The Solution Structure Theorem For Mixed Problems 126
3.2 Solutions of Mixed Problems 129
3.2.1 One-Dimensional Mixed Problems 129
3.2.2 Two-Dimensional Mixed Problems 131
3.2.3 Three-Dimensional Mixed Problems 133
3.3 Well-Posedness of PDS 135
3.3.1 Existence 136
3.3.2 Uniqueness 137
3.3.3 Stability 138
3.4 One-Dimensional Cauchy Problems: Fundamental Solution 139
3.4.1 One-Dimensional Cauchy Problems 139
3.4.2 Fundamental Solution of the One-Dimensional Heat- Conduction Equation 141
3.4.3 Problems in Semi-Infinite Domain and the Method of Continuation 143
3.4.4 PDS with Variable Thermal Conductivity 146
3.5 Multiple Fourier Transformations for Two-and Three- Dimensional Cauchy Problems 149
3.6 Typical PDS of Diffusion 150
3.6.1 FickÌs Law of Diffusion and Diffusion Equation 151
3.6.2 Diffusion from a Constant Source 152
3.6.3 Diffusion from an Instant Plane Source 153
3.6.4 Diffusion Between Two Semi-Infinite Domains 154
Mixed Problems of Hyperbolic Heat- Conduction Equations 156
4.1 Solution Structure Theorem 156
4.2 One-Dimensional Mixed Problems 160
4.2.1 Mixed Boundary Conditions of the First and the Third Kind 160
4.2.2 Mixed Boundary Conditions of the Second and the Third Kind 163
4.3 Two-Dimensional Mixed Problems 165
4.3.1 Rectangular Domain 165
4.3.2 Circular Domain 174
4.4 Three-Dimensional Mixed Problems 176
4.4.1 Cuboid Domain 176
4.4.2 Cylindrical Domain 179
4.4.3 Spherical Domain 180
Cauchy Problems of Hyperbolic Heat- Conduction Equations 184
5.1 Riemann Method for Cauchy Problems 184
5.1.1 Conjugate Operator and Green Formula 184
5.1.2 Cauchy Problems and Riemann Functions 185
5.1.3 Example 188
5.2 Riemann Method and Method of Laplace Transformation for One- Dimensional Cauchy Problems 190
5.2.1 Riemann Method 191
5.2.2 Method of Laplace Transformation 197
5.2.3 Some Properties of Solutions 199
5.3 Verification of Solutions, Physics and Measurement of 201
5.3.1 Verify the Solution for u(x, 0) 201
5.3.4 Physics and Measurement of 207
5.3.5 Measuring 208
5.3.6 Measuring 209
5.4 Method of Descent for Two-Dimensional Problems and Discussion Of Solutions 210
5.4.1 Transform to Three-Dimensional Wave Equations 210
5.4.2 Solution of PDS (5.62) 211
5.4.3 Solution of PDS (5.61) 212
5.4.4 Verification of CDS 213
5.4.5 Special Cases 216
5.5 Domains of Dependence and Influence, Measuring by Characteristic Cones 218
5.5.1 Domain of Dependence 218
5.5.2 Domain of Influence 219
5.5.3 Measuring 220
5.6 Comparison of Fundamental Solutions of Classical and Hyperbolic Heat- Conduction Equations 222
5.6.1 Fundamental Solutions of Two Kinds of Heat- Conduction Equations 222
5.6.2 Common Properties 223
5.6.3 Different Properties 224
5.7 Methods for Solving Axially Symmetric and Spherically- Symmetric Cauchy Problems 225
5.7.1 The Hankel Transformation for Two-Dimensional Axially Symmetric Problems 225
5.7.2 Spherical Bessel Transformation for Spherically-Symmetric Cauchy Problems 227
5.7.3 Method of Continuation for Spherically-Symmetric Problems 230
5.7.4 Discussion of Solution (5.98) 232
5.8 Methods of Fourier Transformation and Spherical Means for Three- Dimensional Cauchy Problems 235
5.8.1 An Integral Formula of Bessel Function 235
5.8.2 Fourier Transformation for Three-Dimensional Problems 237
5.8.3 Method of Spherical Means for PDS (5.115) 239
5.8.4 Discussion 243
Dual-Phase-Lagging Heat-Conduction Equations 246
6.1 Solution Structure Theorem for Mixed Problems 246
6.1.1 Notes on Dual-Phase-Lagging Heat-Conduction Equations 246
6.1.2 Solution Structure Theorem 247
6.2 Fourier Method of Expansion for One- Dimensional Mixed Problems 252
6.2.1 Fourier Method of Expansion 252
6.2.2 Existence 259
6.3 Separation of Variables for One- Dimensional Mixed Problems 262
6.3.1 Eigenvalue Problems 262
6.3.2 Eigenvalues and Eigenfunctions 263
6.3.3 Solve Mixed Problems with Table 2.1 266
6.4 Solution Structure Theorem: Another Form and Application 270
6.4.1 One-Dimensional Mixed Problems 270
6.4.2 Two-Dimensional Mixed Problems 274
6.4.3 Three-Dimensional Mixed Problems 279
6.4.4 Summary and Remarks 284
6.5 Mixed Problems in a Circular Domain 286
6.6 Mixed Problems in a Cylindrical Domain 294
6.7 Mixed Problems in a Spherical Domain 302
6.8 Cauchy Problems 309
6.9 Perturbation Method for Cauchy Problems 313
6.9.1 Introduction 314
6.9.2 The Perturbation Method for Solving Hyperbolic Heat- Conduction Equations 315
6.9.3 Perturbation Solutions of Dual-Phase-Lagging Heat- Conduction Equations 317
6.9.4 Solutions for Initial-Value of Lower-Order Polynomials 323
6.9.5 Perturbation Method for Two-and Three- dimensional Problems 326
6.10 Thermal Waves and Resonance 327
6.10.1 Thermal Waves 328
6.10.2 Resonance 335
6.11 Heat Conduction in Two-Phase Systems 339
6.11.1 One- and Two-Equation Models 339
6.11.2 Equivalence with Dual-Phase-Lagging Heat Conduction 346
Potential Equations 348
7.1 Fourier Method of Expansion 348
7.2 Separation of Variables and Fourier Sin/Cos Transformation 352
7.2.1 Separation of Variables 352
7.2.2 Fourier Sine/Cosine Transformation in a Finite Region 359
7.3 Methods for Solving Nonhomogeneous Potential Equations 366
7.3.1 Equation Homogenization by Function Transformation 366
7.3.2 Extremum Principle 368
7.3.3 Four Examples of Applications 370
7.4 Fundamental Solution and the Harmonic Function 376
7.4.1 Fundamental Solution 376
7.4.2 Green Function 378
7.4.3 Harmonic Functions 382
7.5 Well-Posedness of Boundary-Value Problems 389
7.6 Green Functions 395
7.6.1 Green Function 395
7.6.2 Properties of Green Functions of the Dirichlet Problems 398
7.7 Method of Green Functions for Boundary-Value Problems of the First Kind 401
7.7.1 Mirror Image Method for Finding Green Functions 401
7.7.2 Examples Using the Method of Green Functions 401
7.7.3 Boundary-Value Problems in Unbounded Domains 410
7.8 Potential Theory 416
7.8.1 Potentials 416
7.8.2 Generalized Integrals with Parameters 419
7.8.3 Solid Angle and Russin Surface 422
7.8.4 Properties of Surface Potentials 424
7.9 Transformation of Boundary-Value Problems of Laplace Equations to Integral Equations 427
7.9.1 Integral Equations 427
7.9.2 Transformation of Boundary-Value Problems into Integral Equations 429
7.9.3 Boundary-Value Problems of Poisson Equations 432
7.9.4 Two-Dimensional Potential Equations 433
Special Functions 438
A.1 Bessel and Legendre Equations 438
A.2 Bessel Functions 440
A.3 Properties of Bessel Functions 445
A.4 Legendre Polynomials 449
A.5 Properties of Legendre Polynomials 451
A.6 Associated Legendre Polynomials 453
Integral Transformations 459
B.1 Fourier Integral Transformation 459
B.2 Laplace Transformation 476
Tables of Integral Transformations 486
Eigenvalue Problems 494
D.1 Regular Sturm-Liouville Problems 494
D.2 The Lagrange Equality and Self-Conjugate Boundary- Value Problems 495
D.3 Properties of S-L Problems 496
D.4 Singular S-L Problems 498
References 501
Index 520