Heat Conduction: Mathematical Models and Analytical Solutions / Edition 1

Heat Conduction: Mathematical Models and Analytical Solutions / Edition 1

by Liqiu Wang, Xuesheng Zhou, Xiaohao Wei

ISBN-10: 3540740287

ISBN-13: 9783540740285

Pub. Date: 02/13/2008

Publisher: Springer Berlin Heidelberg

Many phenomena in social, natural and engineering fields are governed by wave, potential, parabolic heat conduction, hyperbolic heat-conduction and dual-phase-lagging heat-conduction equations. These equations are not only appropriate for describing heat conduction at various scales, but also the most important mathematical equations in physics. The focus of the

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Many phenomena in social, natural and engineering fields are governed by wave, potential, parabolic heat conduction, hyperbolic heat-conduction and dual-phase-lagging heat-conduction equations. These equations are not only appropriate for describing heat conduction at various scales, but also the most important mathematical equations in physics. The focus of the present monograph is on these equations: their solution structures, methods of finding their solutions under various supplementary conditions, as well as the physical implication and applications of their solutions.

Therefore, the present monograph can serve as a reference for researchers working on heat conduction of macro- and micro-scales as well as on mathematical methods of physics. It can also serve as a text for graduate courses on heat conduction or on mathematical equations in physics.

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Springer Berlin Heidelberg
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Table of Contents

Introduction     1
Partial Differential Equations     1
Partial Differential Equations and Their Orders     1
Linear, Nonlinear and Quasi-Linear Equations     2
Solutions of Partial Differential Equations     3
Classification of Linear Second-Order Equations     5
Canonical Forms     7
Three Basic Equations of Mathematical Physics     11
Physical Laws and Equations of Mathematical Physics     11
Approaches of Developing Equations of Mathematical Physics     12
Wave Equations     13
Heat-Conduction Equations     15
Potential Equations     17
Theory of Heat Conduction And Three Types of Heat-Conduction Equations     18
Constitutive Relations of Heat Flux     18
The Boltzmann Transport Equation and Dual-Phase-Lagging Heat Conduction     25
Three Types of Heat-Conduction Equations     30
Conditions and Problems for Determining Solutions     32
Initial Conditions     32
Boundary Conditions     33
Problems for Determining Solutions     37
Well-Posedness of PDS     38
Example of Developing PDS     39
Wave Equations     41
The SolutionStructure Theorem for Mixed Problems and its Application     41
Fourier Method for One-Dimensional Mixed Problems     45
Boundary Condition of the First Kind     45
Boundary Condition of the Second Kind     50
Method of Separation of Variables for One-Dimensional Mixed Problems     51
Method of Separation of Variables     51
Generalized Fourier Method of Expansion     54
Important Properties of Eigenvalue Problems (2.19)     56
Well-Posedness and Generalized Solution     58
Existence     58
Uniqueness     60
Stability     61
Generalized Solution     62
PDS with Variable Coefficients     64
Two-Dimensional Mixed Problems     67
Rectangular Domain     67
Circular Domain     69
Three-Dimensional Mixed Problems     76
Cuboid Domain     76
Spherical Domain     78
Methods of Solving One-Dimensional Cauchy Problems     83
Method of Fourier Transformation     83
Method of Characteristics     85
Physical Meaning     86
Domains of Dependence, Determinacy and Influence     89
Problems in a Semi-Infinite Domain and the Method of Continuation     92
Two- and Three-Dimensional Cauchy Problems     96
Method of Fourier Transformation     96
Method of Spherical Means     99
Method of Descent     101
Physical Meanings of the Poisson and Kirchhoff Formulas     109
Heat-Conduction Equations     113
The Solution Structure Theorem For Mixed Problems     113
Solutions of Mixed Problems     116
One-Dimensional Mixed Problems     116
Two-Dimensional Mixed Problems     118
Three-Dimensional Mixed Problems     120
Well-Posedness of PDS     122
Existence     123
Uniqueness     124
Stability     125
One-Dimensional Cauchy Problems: Fundamental Solution     126
One-Dimensional Cauchy Problems     126
Fundamental Solution of the One-Dimensional Heat-Conduction Equation     128
Problems in Semi-Infinite Domain and the Method of Continuation     130
PDS with Variable Thermal Conductivity     133
Multiple Fourier Transformations for Two- and Three- Dimensional Cauchy Problems     136
Two-Dimensional Case     136
Three-Dimensional Case     137
Typical PDS of Diffusion      137
Fick's Law of Diffusion and Diffusion Equation     138
Diffusion from a Constant Source     139
Diffusion from an Instant Plane Source     140
Diffusion Between Two Semi-Infinite Domains     141
Mixed Problems of Hyperbolic Heat-Conduction Equations     143
Solution Structure Theorem     143
One-Dimensional Mixed Problems     147
Mixed Boundary Conditions of the First and the Third Kind     147
Mixed Boundary Conditions of the Second and the Third Kind     150
Two-Dimensional Mixed Problems     152
Rectangular Domain     152
Circular Domain     161
Three-Dimensional Mixed Problems     163
Cuboid Domain     163
Cylindrical Domain     166
Spherical Domain     167
Cauchy Problems of Hyperbolic Heat-Conduction Equations     171
Riemann Method for Cauchy Problems     171
Conjugate Operator and Green Formula     171
Cauchy Problems and Riemann Functions     172
Example     175
Riemann Method and Method of Laplace Transformation for One-Dimensional Cauchy Problems     177
Riemann Method     178
Method of Laplace Transformation      184
Some Properties of Solutions     186
Verification of Solutions, Physics and Measurement of [tau subscript 0]     188
Verify the Solution for u(x, 0) = 0 and u[subscript t](x, 0) = 1     188
Verify the Solution for u(x, 0) = 1 and u[subscript t](x, 0) = 0     191
Verify the Solution for f(x, t) = 1, u(x, 0) = 0 and u[subscript t](x, 0) = 0     193
Physics and Measurement of [tau subscript 0]     194
Measuring [tau subscript 0] by Characteristic Curves     195
Measuring [tau subscript 0] by a Unit Impulsive Source [delta](x - x[subscript 0], t - t[subscript 0])     196
Method of Descent for Two-Dimensional Problems and Discussion Of Solutions     197
Transform to Three-Dimensional Wave Equations     197
Solution of PDS (5.62)     198
Solution of PDS (5.61)     199
Verification of CDS     200
Special Cases     203
Domains of Dependence and Influence, Measuring [tau subscript 0] by Characteristic Cones     205
Domain of Dependence     205
Domain of Influence     206
Measuring [tau subscript 0] by Characteristic Cones     207
Comparison of Fundamental Solutions of Classical and Hyperbolic Heat-Conduction Equations     209
Fundamental Solutions of Two Kinds of Heat-Conduction Equations     209
Common Properties     210
Different Properties     211
Methods for Solving Axially Symmetric and Spherically-Symmetric Cauchy Problems     212
The Hankel Transformation for Two-Dimensional Axially Symmetric Problems     212
Spherical Bessel Transformation for Spherically-Symmetric Cauchy Problems     214
Method of Continuation for Spherically-Symmetric Problems     217
Discussion of Solution (5.98)     219
Methods of Fourier Transformation and Spherical Means for Three-Dimensional Cauchy Problems     222
An Integral Formula of Bessel Function     222
Fourier Transformation for Three-Dimensional Problems     224
Method of Spherical Means for PDS (5.115)     226
Discussion     230
Dual-Phase-Lagging Heat-Conduction Equations     233
Solution Structure Theorem for Mixed Problems     233
Notes on Dual-Phase-Lagging Heat-Conduction Equations     233
Solution Structure Theorem     234
Fourier Method of Expansion for One-Dimensional Mixed Problems     239
Fourier Method of Expansion     239
Existence     246
Separation of Variables for One-Dimensional Mixed Problems     249
Eigenvalue Problems     249
Eigenvalues and Eigenfunctions     250
Solve Mixed Problems with Table 2.1     253
Solution Structure Theorem: Another Form and Application     257
One-Dimensional Mixed Problems     257
Two-Dimensional Mixed Problems     261
Three-Dimensional Mixed Problems     266
Summary and Remarks     271
Mixed Problems in a Circular Domain     273
Solution from [psi](r, [theta])     274
Solution from [open phi] (r, [theta])     276
Solution from f(r, [theta], t)     278
Mixed Problems in a Cylindrical Domain     281
Solution from [psi](r, [theta], z)     281
Solution from [open phi](r, [theta], z)     284
Solution from f(r, [theta], z, t)     286
Green Function of the Dual-Phase-Lagging Heat-Conduction Equation     288
Mixed Problems in a Spherical Domain     289
Solution from [psi](r, [theta], [open phi])     289
Solution from [Phi](r, [theta], [open phi])     291
Solution from f(r, [theta], [open phi], t)     293
Cauchy Problems     296
Perturbation Method for Cauchy Problems     300
Introduction     301
The Perturbation Method for Solving Hyperbolic Heat-Conduction Equations     302
Perturbation Solutions of Dual-Phase-Lagging Heat-Conduction Equations     304
Solutions for Initial-Value of Lower-Order Polynomials     310
Perturbation Method for Two- and Three-dimensional Problems     313
Thermal Waves and Resonance     314
Thermal Waves     315
Resonance     322
Heat Conduction in Two-Phase Systems     326
One- and Two-Equation Models     326
Equivalence with Dual-Phase-Lagging Heat Conduction     333
Potential Equations     335
Fourier Method of Expansion     335
Separation of Variables and Fourier Sin/Cos Transformation     339
Separation of Variables     339
Fourier Sine/Cosine Transformation in a Finite Region     346
Methods for Solving Nonhomogeneous Potential Equations     353
Equation Homogenization by Function Transformation     353
Extremum Principle     355
Four Examples of Applications     357
Fundamental Solution and the Harmonic Function     363
Fundamental Solution     363
Green Function     365
Harmonic Functions     369
Well-Posedness of Boundary-Value Problems      376
Green Functions     382
Green Function     382
Properties of Green Functions of the Dirichlet Problems     385
Method of Green Functions for Boundary-Value Problems of the First Kind     388
Mirror Image Method for Finding Green Functions     388
Examples Using the Method of Green Functions     388
Boundary-Value Problems in Unbounded Domains     397
Potential Theory     403
Potentials     403
Generalized Integrals with Parameters     406
Solid Angle and Russin Surface     409
Properties of Surface Potentials     411
Transformation of Boundary-Value Problems of Laplace Equations to Integral Equations     414
Integral Equations     414
Transformation of Boundary-Value Problems into Integral Equations     416
Boundary-Value Problems of Poisson Equations     419
Two-Dimensional Potential Equations     420
Special Functions     425
Bessel and Legendre Equations     425
Bessel Functions     427
Properties of Bessel Functions     432
Legendre Polynomials     436
Properties of Legendre Polynomials     438
Associated Legendre Polynomials      440
Integral Transformations     447
Fourier Integral Transformation     447
Fourier Integral     447
Fourier Transformation     450
Generalized Functions and the [delta]-function     455
Generalized Fourier Transformation     458
The Multiple Fourier Transformation     462
Laplace Transformation     464
Laplace Transformation     464
Properties of Laplace Transformation     466
Determine Inverse Image Functions by Calculating Residues     469
Convolution Theorem     471
Tables of Integral Transformations     475
Eigenvalue Problems     483
Regular Sturm-Liouville Problems     483
The Lagrange Equality and Self-Conjugate Boundary-Value Problems     484
Properties of S-L Problems     485
Singular S-L Problems     487
References     491
Index     511

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