Heat Conduction

Mathematical Models and Analytical Solutions

By: Liqiu Wang, Xuesheng Zhou, Xiaohao Wei

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Hardcover | 2 January 2008

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532 Pages

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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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"The book under review may stimulate interest among the researchers to explore the true nature of the heat. ... A good number of references at the end of the book may help the readers to further explore the subject. ... The book contains many solutions of the problems of technological importance in heat conduction of macro and micro scales and will be useful for researchers in mechanical engineering, chemical engineering, material sciences and applied mathematics." (K. N. Shukla, Zentralblatt MATH, Vol. 1237, 2012)

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You Can Find This Book In

Non-Fiction Science Physics Thermodynamics & Heat Engineering & Technology Mechanical Engineering & Materials Mechanical Engineering Energy Technology & Engineering Heat Transfer Processes Chemistry Physical Chemistry Mathematics

Calculus & Mathematical Analysis Differential Calculus & Equations Materials Science Engineering Thermodynamics Applied Mathematics

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Paperback

Published: 19th October 2010

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

Heat Conduction

Mathematical Models and Analytical Solutions

At a Glance

Hardcover

Industry Reviews

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How to return your order

Defective items

You Can Find This Book In

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