Finite integral transform techniques are applied to solve the one-dimensional (1D) dual-phase heat conduction problem, and a comprehensive analysis is provided for general time-dependent heat generation and arbitrary combinations of various boundary conditions (Dirichlet, Neumann, and Robin). Through the dependence on the relative differences in heat flux and temperature relaxation times, this analytical solution effectively models both parabolic and hyperbolic heat conduction. In order to demonstrate several exemplary physical phenomena, four distinct cases that illustrate the wavelike heat conduction behavior are presented. In the first model, following an initial temperature spike in a slab, the thermal evolution portrays immediate dissipation in parabolic systems, whereas the dual-phase solution depicts wavelike temperature propagation—the intensity of which depends on the relaxation times. Next, the analysis of periodic surface heat flux at the slab boundaries provides evidence of interference patterns formed by temperature waves. In following, the study of Joule heating driven periodic generation inside the slab demonstrates that the steady-periodic parabolic temperature response depends on the ratio of pulsatile electrical excitation and the electrical resistivity of the slab. As for the dual-phase model, thermal resonance conditions are observed at distinct excitation frequencies. Building on findings of the other models, the case of moving constant-amplitude heat generation is considered, and the occurrences of thermal shock and thermal expansion waves are demonstrated at particular conditions.

References

References
1.
Maxwell
,
J. C.
,
1873
,
A Treatise on Electricity and Magnetism
,
Dover Publications
,
Mineola, NY
.
2.
Ölçer
,
N. Y.
,
1964
, “
On the Theory of Conductive Heat Transfer in Finite Regions
,”
Int. J. Heat Mass Transfer
,
7
(
3
), pp.
307
314
.
3.
Özışık
,
M.
,
1989
,
Boundary Value Problems of Heat Conduction
,
Courier Corporation
, North Chelmsford, MA.
4.
Vernotte
,
P.
,
1958
, “
Les Paradoxes de la Théorie Continue de Léquation de la Chaleur
,”
C. R. Hebd. Seances Acad. Sci.
,
246
(
22
), pp.
3154
3155
.
5.
Cattaneo
,
C.
,
1958
, “
Sur une Forme de Lequation de la Chaleur Eliminant le Paradoxe Dune Propagation Instantanee
,”
C. R. Hebd. Seances Acad. Sci.
,
247
(
4
), pp.
431
433
.
6.
Chester
,
M.
,
1963
, “
Second Sound in Solids
,”
Phys. Rev.
,
131
(
5
), pp.
2013
2015
.
7.
Morse
,
P. M.
, and
Feshbach
,
H.
,
1953
,
Methods of Theoretical Physics
, McGraw-Hill, New York.
8.
Ordonez-Miranda
,
J.
, and
Alvarado-Gil
,
J.
,
2009
, “
Thermal Wave Oscillations and Thermal Relaxation Time Determination in a Hyperbolic Heat Transport Model
,”
Int. J. Therm. Sci.
,
48
(
11
), pp.
2053
2062
.
9.
Peshkov
,
V.
,
1946
, “
Determination of the Velocity of Propagation of the Second Sound in Helium II
,”
J. Phys. USSR
,
10
, pp.
389
398
.
10.
Peshkov
,
V.
,
2013
, “
The Second Sound in Helium II
,” Helium 4: The Commonwealth and International Library: Selected Readings in Physics, Elsevier, Amsterdam, The Netherlands, p.
166
.
11.
Pellam
,
J. R.
,
1949
, “
Investigations of Pulsed Second Sound in Liquid Helium II
,”
Phys. Rev.
,
75
(
8
), pp.
1183
1194
.
12.
Wang
,
M.
,
Yang
,
N.
, and
Guo
,
Z.-Y.
,
2011
, “
Non-Fourier Heat Conductions in Nanomaterials
,”
J. Appl. Phys.
,
110
(
6
), p.
064310
.
13.
Wang
,
H.-D.
,
2014
,
Theoretical and Experimental Studies on Non-Fourier Heat Conduction Based on Thermomass Theory
,
Springer Science and Business Media
, Berlin.
14.
Tang
,
D.
, and
Araki
,
N.
,
1999
, “
Wavy, Wavelike, Diffusive Thermal Responses of Finite Rigid Slabs to High-Speed Heating of Laser-Pulses
,”
Int. J. Heat Mass Transfer
,
42
(
5
), pp.
855
860
.
15.
Eckert
,
E. R. G.
, and
Drake
,
R. M.
,
1972
,
Analysis of Heat Transfer and Mass Transfer
,
McGraw-Hill
, New York.
16.
Chester
,
M.
,
1966
, “
High-Frequency Thermometry
,”
Phys. Rev.
,
145
(
1
), pp.
76
80
.
17.
Nettleton
,
R.
,
1960
, “
Relaxation Theory of Thermal Conduction in Liquids
,”
Phys. Fluids
,
3
(
2
), pp.
216
225
.
18.
Maurer
,
M. J.
,
1969
, “
Relaxation Model for Heat Conduction in Metals
,”
J. Appl. Phys.
,
40
(
13
), pp.
5123
5130
.
19.
Francis
,
P.
,
1972
, “
Thermo-Mechanical Effects in Elastic Wave Propagation: A Survey
,”
J. Sound Vib.
,
21
(
2
), pp.
181
192
.
20.
Luikov
,
A.
,
1965
, “
Application of the Methods of Thermodynamics of Irreversible Processes to the Investigation of Heat and Mass Transfer
,”
J. Eng. Phys.
,
9
(
3
), pp.
189
202
.
21.
Mitra
,
K.
,
Kumar
,
S.
,
Vedevarz
,
A.
, and
Moallemi
,
M. K.
,
1995
, “
Experimental Evidence of Hyperbolic Heat Conduction in Processed Meat
,”
ASME J. Heat Transfer
,
117
(
3
), pp.
568
573
.
22.
Kaminski
,
W.
,
1990
, “
Hyperbolic Heat Conduction Equation for Materials With a Nonhomogeneous Inner Structure
,”
ASME J. Heat Transfer
,
112
(
3
), pp.
555
560
.
23.
Wang
,
L.
,
2000
, “
Solution Structure of Hyperbolic Heat-Conduction Equation
,”
Int. J. Heat Mass Transfer
,
43
(
3
), pp.
365
373
.
24.
Wang
,
L.
,
Zhou
,
X.
, and
Wei
,
X.
,
2007
,
Heat Conduction: Mathematical Models and Analytical Solutions
,
Springer Science and Business Media
, Berlin.
25.
Kronberg
,
A.
,
Benneker
,
A.
, and
Westerterp
,
K.
,
1998
, “
Notes on Wave Theory in Heat Conduction: A New Boundary Condition
,”
Int. J. Heat Mass Transfer
,
41
(
1
), pp.
127
137
.
26.
Ali
,
Y.
, and
Zhang
,
L.
,
2005
, “
Relativistic Heat Conduction
,”
Int. J. Heat Mass Transfer
,
48
(
12
), pp.
2397
2406
.
27.
Bai
,
C.
, and
Lavine
,
A.
,
1995
, “
On Hyperbolic Heat Conduction and the Second Law of Thermodynamics
,”
ASME J. Heat Transfer
,
117
(
2
), pp.
256
263
.
28.
Barletta
,
A.
, and
Zanchini
,
E.
,
1997
, “
Hyperbolic Heat Conduction and Local Equilibrium: A Second Law Analysis
,”
Int. J. Heat Mass Transfer
,
40
(
5
), pp.
1007
1016
.
29.
Glass
,
D.
,
Özişik
,
M.
, and
Vick
,
B.
,
1987
, “
Non-Fourier Effects on Transient Temperature Resulting From Periodic On-Off Heat Flux
,”
Int. J. Heat Mass Transfer
,
30
(
8
), pp.
1623
1631
.
30.
Al-Khairy
,
R. T.
, and
Al-Ofey
,
Z. M.
,
2009
, “
Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium Under the Effect of Time-Dependent Laser Heat Source
,”
J. Appl. Math.
,
2009
, p.
604695
.
31.
Lewandowska
,
M.
, and
Malinowski
,
L.
,
2006
, “
An Analytical Solution of the Hyperbolic Heat Conduction Equation for the Case of a Finite Medium Symmetrically Heated on Both Sides
,”
Int. Commun. Heat Mass Transfer
,
33
(
1
), pp.
61
69
.
32.
Taitel
,
Y.
,
1972
, “
On the Parabolic, Hyperbolic and Discrete Formulation of the Heat Conduction Equation
,”
Int. J. Heat Mass Transfer
,
15
(
2
), pp.
369
371
.
33.
Özişik
,
M.
, and
Vick
,
B.
,
1984
, “
Propagation and Reflection of Thermal Waves in a Finite Medium
,”
Int. J. Heat Mass Transfer
,
27
(
10
), pp.
1845
1854
.
34.
Gembarovič
,
J.
, and
Majernik
,
V.
,
1988
, “
Non-Fourier Propagation of Heat Pulses in Finite Medium
,”
Int. J. Heat Mass Transfer
,
31
(
5
), pp.
1073
1080
.
35.
Kar
,
A.
,
Chan
,
C.
, and
Mazumder
,
J.
,
1992
, “
Comparative Studies on Nonlinear Hyperbolic and Parabolic Heat Conduction for Various Boundary Conditions: Analytic and Numerical Solutions
,”
ASME J. Heat Transfer
,
114
(
1
), pp.
14
20
.
36.
Tang
,
D.
, and
Araki
,
N.
,
1996
, “
Analytical Solution of Non-Fourier Temperature Response in a Finite Medium Under Laser-Pulse Heating
,”
Heat Mass Transfer
,
31
(
5
), pp.
359
363
.
37.
Yuen
,
W.
, and
Lee
,
S.
,
1989
, “
Non-Fourier Heat Conduction in a Semi-Infinite Solid Subjected to Oscillatory Surface Thermal Disturbances
,”
ASME J. Heat Transfer
,
111
(
1
), pp.
178
181
.
38.
Barletta
,
A.
, and
Zanchini
,
E.
,
1996
, “
Hyperbolic Heat Conduction and Thermal Resonances in a Cylindrical Solid Carrying a Steady-Periodic Electric Field
,”
Int. J. Heat Mass Transfer
,
39
(
6
), pp.
1307
1315
.
39.
Barletta
,
A.
,
1996
, “
Hyperbolic Propagation of an Axisymmetric Thermal Signal in an Infinite Solid Medium
,”
Int. J. Heat Mass Transfer
,
39
(
15
), pp.
3261
3271
.
40.
Hector
,
L. G.
,
Woo-Seung
,
K.
, and
Özisik
,
M. N.
,
1992
, “
Propagation and Reflection of Thermal Waves in a Finite Medium Due to Axisymmetric Surface Sources
,”
Int. J. Heat Mass Transfer
,
35
(
4
), pp.
897
912
.
41.
Chan
,
S.
,
Low
,
M.
, and
Mueller
,
W.
,
1971
, “
Hyperbolic Heat Conduction in Catalytic Supported Crystallites
,”
AIChE J.
,
17
(
6
), pp.
1499
1501
.
42.
Jiang
,
F.
,
2006
, “
Solution and Analysis of Hyperbolic Heat Propagation in Hollow Spherical Objects
,”
Heat Mass Transfer
,
42
(
12
), pp.
1083
1091
.
43.
Zaіtsev
,
V.
, and
Polyanin
,
A.
,
1995
,
Handbook of Linear Partial Differential Equations
,
Nauka
,
Fizmatlit, Moscow
.
44.
Han-Taw
,
C.
, and
Jae-Yuh
,
L.
,
1994
, “
Analysis of Two-Dimensional Hyperbolic Heat Conduction Problems
,”
Int. J. Heat Mass Transfer
,
37
(
1
), pp.
153
164
.
45.
Wiggert
,
D.
,
1977
, “
Analysis of Early-Time Transient Heat Conduction by Method of Characteristics
,”
ASME J. Heat Transfer
,
99
(
1
), pp.
35
40
.
46.
Glass
,
D. E.
,
Özişik
,
M. N.
,
McRae
,
D. S.
, and
Vick
,
B.
,
1985
, “
On the Numerical Solution of Hyperbolic Heat Conduction
,”
Numer. Heat Transfer
,
8
(
4
), pp.
497
504
.
47.
Han-Taw
,
C.
, and
Jae-Yuh
,
L.
,
1993
, “
Numerical Analysis for Hyperbolic Heat Conduction
,”
Int. J. Heat Mass Transfer
,
36
(
11
), pp.
2891
2898
.
48.
Torii
,
S.
, and
Yang
,
W.-J.
,
2005
, “
Heat Transfer Mechanisms in Thin Film With Laser Heat Source
,”
Int. J. Heat Mass Transfer
,
48
(
3
), pp.
537
544
.
49.
Blackwell
,
B.
,
1990
, “
Temperature Profile in Semi-Infinite Body With Exponential Source and Convective Boundary Condition
,”
ASME J. Heat Transfer
,
112
(
3
), pp.
567
571
.
50.
Zubair
,
S.
, and
Chaudhry
,
M. A.
,
1996
, “
Heat Conduction in a Semi-Infinite Solid Due to Time-Dependent Laser Source
,”
Int. J. Heat Mass Transfer
,
39
(
14
), pp.
3067
3074
.
51.
Anderson
,
D. A.
,
Tannehill
,
J. C.
, and
Pletcher
,
R. H.
,
1984
,
Computational Fluid Mechanics and Heat Transfer
, Hemisphere,
New York
, p.
166
.
52.
Lewandowska
,
M.
,
2001
, “
Hyperbolic Heat Conduction in the Semi-Infinite Body With a Time-Dependent Laser Heat Source
,”
Heat and Mass Transfer
,
37
(
4–5
), pp.
333
342
.
53.
Yang
,
H.
,
1992
, “
Solution of Two-Dimensional Hyperbolic Heat Conduction by High-Resolution Numerical Methods
,”
Numer. Heat Transfer
,
21
(
3
), pp.
333
349
.
54.
Yen
,
C.-C.
, and
Wu
,
C.-Y.
,
2003
, “
Modelling Hyperbolic Heat Conduction in a Finite Medium With Periodic Thermal Disturbance and Surface Radiation
,”
Appl. Math. Modell.
,
27
(
5
), pp.
397
408
.
55.
Tang
,
D.
, and
Araki
,
N.
,
1996
, “
The Wave Characteristics of Thermal Conduction in Metallic Films Irradiated by Ultra-Short Laser Pulses
,”
J. Phys. D: Appl. Phys.
,
29
(
10
), pp.
2527
2533
.
56.
Chih-Yang
,
W.
,
1989
, “
Hyperbolic Heat Conduction With Surface Radiation and Reflection
,”
Int. J. Heat Mass Transfer
,
32
(
8
), pp.
1585
1587
.
57.
Herwig
,
H.
, and
Beckert
,
K.
,
2000
, “
Fourier Versus Non-Fourier Heat Conduction in Materials With a Nonhomogeneous Inner Structure
,”
ASME J. Heat Transfer
,
122
(
2
), pp.
363
364
.
58.
Guyer
,
R. A.
, and
Krumhansl
,
J.
,
1996
, “
Solution of the Linearized Phonon Boltzmann Equation
,”
Phys. Rev.
,
148
(
2
), pp.
766
778
.
59.
Tzou
,
D.
,
1995
, “
A Unified Field Approach for Heat Conduction From Macro-to Micro-Scales
,”
ASME J. Heat Transfer
,
117
(
1
), pp.
8
16
.
60.
Wang
,
L.
,
Xu
,
M.
, and
Zhou
,
X.
,
2001
, “
Well-Posedness and Solution Structure of Dual-Phase-Lagging Heat Conduction
,”
Int. J. Heat Mass Transfer
,
44
(
9
), pp.
1659
1669
.
61.
Al-Nimr
,
M.
,
Naji
,
M.
, and
Abdallah
,
R.
,
2004
, “
Thermal Behavior of a Multi-Layered Thin Slab Carrying Periodic Signals Under the Effect of the Dual-Phase-Lag Heat Conduction Model
,”
Int. J. Thermophys.
,
25
(
3
), pp.
949
966
.
62.
Xu
,
M.
, and
Wang
,
L.
,
2002
, “
Thermal Oscillation and Resonance in Dual-Phase-Lagging Heat Conduction
,”
Int. J. Heat Mass Transfer
,
45
(
5
), pp.
1055
1061
.
63.
Xu
,
M.
, and
Wang
,
L.
,
2005
, “
Dual-Phase-Lagging Heat Conduction Based on Boltzmann Transport Equation
,”
Int. J. Heat Mass Transfer
,
48
(
25
), pp.
5616
5624
.
64.
Quintanilla
,
R.
,
2008
, “
A Well-Posed Problem for the Dual-Phase-Lag Heat Conduction
,”
J. Therm. Stresses
,
31
(
3
), pp.
260
269
.
65.
Ordonez-Miranda
,
J.
, and
Alvarado-Gil
,
J. J.
,
2011
, “
On the Stability of the Exact Solutions of the Dual-Phase Lagging Model of Heat Conduction
,”
Nanoscale Res. Lett.
,
6
(
1
), pp.
1
6
.
66.
Lam
,
T. T.
,
2014
, “
A Generalized Heat Conduction Solution for Ultrafast Laser Heating in Metallic Films
,”
Int. J. Heat Mass Transfer
,
73
, pp.
330
339
.
67.
Fabrizio
,
M.
, and
Lazzari
,
B.
,
2014
, “
Stability and Second Law of Thermodynamics in Dual-Phase-Lag Heat Conduction
,”
Int. J. Heat Mass Transfer
,
74
, pp.
484
489
.
68.
Körner
,
C.
, and
Bergmann
,
H.
,
1998
, “
The Physical Defects of the Hyperbolic Heat Conduction Equation
,”
Appl. Phys. A
,
67
(
4
), pp.
397
401
.
69.
Shen
,
B.
, and
Zhang
,
P.
,
2008
, “
Notable Physical Anomalies Manifested in Non-Fourier Heat Conduction Under the Dual-Phase-Lag Model
,”
Int. J. Heat Mass Transfer
,
51
(
7–8
), pp.
1713
1727
.
70.
Antaki
,
P. J.
,
2005
, “
New Interpretation of Non-Fourier Heat Conduction in Processed Meat
,”
ASME J. Heat Transfer
,
127
(
2
), pp.
189
193
.
71.
Tzou
,
D.
, and
Chiu
,
K.
,
2001
, “
Temperature-Dependent Thermal Lagging in Ultrafast Laser Heating
,”
Int. J. Heat Mass Transfer
,
44
(
9
), pp.
1725
1734
.
72.
Liu
,
K.-C.
,
2005
, “
Analysis of Dual-Phase-Lag Thermal Behaviour in Layered Films With Temperature-Dependent Interface Thermal Resistance
,”
J. Phys. D: Appl. Phys.
,
38
(
19
), pp.
3722
3732
.
73.
Shiomi
,
J.
, and
Maruyama
,
S.
,
2006
, “
Non-Fourier Heat Conduction in a Single-Walled Carbon Nanotube: Classical Molecular Dynamics Simulations
,”
Phys. Rev. B
,
73
(
20
), p.
205420
.
74.
Tzou
,
D. Y.
,
1995
, “
The Generalized Lagging Response in Small-Scale and High-Rate Heating
,”
Int. J. Heat Mass Transfer
,
38
(
17
), pp.
3231
3240
.
75.
Kulish
,
V. V.
, and
Novozhilov
,
V. B.
,
2004
, “
An Integral Equation for the Dual-Lag Model of Heat Transfer
,”
ASME J. Heat Transfer
,
126
(
5
), pp.
805
808
.
76.
Smith
,
A. N.
,
Hostetler
,
J. L.
, and
Norris
,
P. M.
,
1999
, “
Nonequilibrium Heating in Metal Films: An Analytical and Numerical Analysis
,”
Numer. Heat Transfer, Part A
,
35
(
8
), pp.
859
873
.
77.
Chen
,
J.
,
Beraun
,
J.
, and
Tzou
,
D.
,
2000
, “
A Dual-Phase-Lag Diffusion Model for Predicting Thin Film Growth
,”
Semicond. Sci. Technol.
,
15
(
3
), pp.
235
241
.
78.
Al-Nimr
,
M.
,
Naji
,
M.
, and
Arbaci
,
V.
,
2000
, “
Nonequilibrium Entropy Production Under the Effect of the Dual-Phase-Lag Heat Conduction Model
,”
ASME J. Heat Transfer
,
122
(
2
), pp.
217
223
.
79.
Lin
,
C.-K.
,
Hwang
,
C.-C.
, and
Chuag
,
Y.-P.
,
1997
, “
The Unsteady Solutions of a Unified Heat Conduction Equation
,”
Int. J. Heat Mass Transfer
,
40
(
7
), pp.
1716
1719
.
80.
Lee
,
Y.-M.
,
Lin
,
P.-C.
, and
Tsai
,
T.-W.
,
2009
, “
Green's Function Solution of Dual-Phase-Lag Model
,”
ASME
Paper No. MNHMT2009-18425.
81.
Alkhairy
,
R.
,
2012
, “
Green's Function Solution for the Dual-Phase-Lag Heat Equation
,”
Appl. Math.
,
3
(
10
), pp.
1170
1178
.
82.
Chen
,
J.
,
Beraun
,
J.
, and
Tzou
,
D.
,
1999
, “
A Dual-Phase-Lag Diffusion Model for Interfacial Layer Growth in Metal Matrix Composites
,”
J. Mater. Sci.
,
34
(
24
), pp.
6183
6187
.
83.
Dai
,
W.
, and
Nassar
,
R.
,
1999
, “
A Finite Difference Scheme for Solving the Heat Transport Equation at the Microscale
,”
Numer. Methods Partial Differ. Equations
,
15
(
6
), pp.
697
708
.
84.
Dai
,
W.
,
Shen
,
L.
,
Nassar
,
R.
, and
Zhu
,
T.
,
2004
, “
A Stable and Convergent Three-Level Finite Difference Scheme for Solving a Dual-Phase-Lagging Heat Transport Equation in Spherical Coordinates
,”
Int. J. Heat Mass Transfer
,
47
(
8
), pp.
1817
1825
.
85.
Dai
,
W.
, and
Nassar
,
R.
,
2001
, “
A Finite Difference Scheme for Solving a Three-Dimensional Heat Transport Equation in a Thin Film With Microscale Thickness
,”
Int. J. Numer. Methods Eng.
,
50
(
7
), pp.
1665
1680
.
86.
Zhang
,
J.
, and
Zhao
,
J. J.
,
2001
, “
High Accuracy Stable Numerical Solution of 1D Microscale Heat Transport Equation
,”
Commun. Numer. Methods Eng.
,
17
(
11
), pp.
821
832
.
87.
Kunadian
,
I.
,
McDonough
,
J.
, and
Kumar
,
R. R.
,
2005
, “
An Efficient Numerical Procedure for Solving Microscale Heat Transport Equation During Femtosecond Laser Heating of Nanoscale Metal Films
,”
ASME
Paper No. IPACK2005-73376.
88.
Prakash
,
G. S.
,
Reddy
,
S. S.
,
Das
,
S. K.
,
Sundararajan
,
T.
, and
Seetharamu
,
K. N.
,
2000
, “
Numerical Modelling of Microscale Effects in Conduction for Different Thermal Boundary Conditions
,”
Numer. Heat Transfer, Part A
,
38
(
5
), pp.
513
532
.
89.
Ho
,
J.-R.
,
Kuo
,
C.-P.
, and
Jiaung
,
W.-S.
,
2003
, “
Study of Heat Transfer in Multilayered Structure Within the Framework of Dual-Phase-Lag Heat Conduction Model Using Lattice Boltzmann Method
,”
Int. J. Heat Mass Transfer
,
46
(
1
), pp.
55
69
.
90.
Liu
,
K.-C.
, and
Cheng
,
P.-J.
,
2006
, “
Numerical Analysis for Dual-Phase-Lag Heat Conduction in Layered Films
,”
Numer. Heat Transfer, Part A
,
49
(
6
), pp.
589
606
.
91.
Chou
,
Y.
, and
Yang
,
R.-J.
,
2008
, “
Application of CESE Method to Simulate Non-Fourier Heat Conduction in Finite Medium With Pulse Surface Heating
,”
Int. J. Heat Mass Transfer
,
51
(
13
), pp.
3525
3534
.
92.
Chou
,
Y.
, and
Yang
,
R.-J.
,
2009
, “
Two-Dimensional Dual-Phase-Lag Thermal Behavior in Single-/Multi-Layer Structures Using CESE Method
,”
Int. J. Heat Mass Transfer
,
52
(
1
), pp.
239
249
.
93.
Ghazanfarian
,
J.
, and
Abbassi
,
A.
,
2009
, “
Effect of Boundary Phonon Scattering on Dual-Phase-Lag Model to Simulate Micro-and Nano-Scale Heat Conduction
,”
Int. J. Heat Mass Transfer
,
52
(
15
), pp.
3706
3711
.
94.
Ghazanfarian
,
J.
, and
Abbassi
,
A.
,
2012
, “
Investigation of 2D Transient Heat Transfer Under the Effect of Dual-Phase-Lag Model in a Nanoscale Geometry
,”
Int. J. Thermophys.
,
33
(
3
), pp.
552
566
.
95.
Ghazanfarian
,
J.
, and
Shomali
,
Z.
,
2012
, “
Investigation of Dual-Phase-Lag Heat Conduction Model in a Nanoscale Metal-Oxide-Semiconductor Field-Effect Transistor
,”
Int. J. Heat Mass Transfer
,
55
(
21
), pp.
6231
6237
.
96.
Basirat
,
H.
,
Ghazanfarian
,
J.
, and
Forooghi
,
P.
,
2006
, “
Implementation of Dual-Phase-Lag Model at Different Knudsen Numbers Within Slab Heat Transfer
,”
International Conference on Modeling and Simulation (MS)
, Montreal, QC, Canada, May 24–26, pp. 895–899.
97.
Dai
,
W.
,
Han
,
F.
, and
Sun
,
Z.
,
2013
, “
Accurate Numerical Method for Solving Dual-Phase-Lagging Equation With Temperature Jump Boundary Condition in Nano Heat Conduction
,”
Int. J. Heat Mass Transfer
,
64
, pp.
966
975
.
98.
Titchmarsh
,
E. C.
,
1948
,
Introduction to the Theory of Fourier Integrals
, Vol.
2
,
Clarendon Press
,
Oxford, UK
.
99.
Sneddon
,
I. N.
,
1955
,
Fourier Transforms
,
Courier Corporation
, North Chelmsford, MA.
100.
Barletta
,
A.
, and
Zanchini
,
E.
,
1995
, “
Steady Periodic Heat Transfer in a Flat Plate Conductor Carrying an Alternating Electric Current
,”
Int. Commun. Heat Mass Transfer
,
22
(
2
), pp.
241
250
.
101.
Landau
,
L. D.
, and
Lifshitz
,
E. M.
,
1960
,
Electrodynamics of Continuous Media
,
Pergamon Press
,
Oxford, UK
.
You do not currently have access to this content.