本文主要探討應用混合拉氏轉換法求解一般傅立葉熱傳問題和非傅立葉熱傳問題，而非傅立葉問題包括了有熱波模式(CV wave model)以及雙相差模式(dual-phase-lag model )，此兩種模式皆屬於雙曲線方程式。熱波模式由於有一項溫度對時間的二次微分，此項會造成有類似波動傳遞的波前現象，而雙相差模式又比熱波模式多了一項溫度對空間二次微分和對時時間一次微分項，這些都會導致在數值分析上很容易有數值震盪或不穩定之狀況。
在解析一般傅立葉熱傳問題時，主要應用混合拉氏轉換法搭配有限差分法求解，並逐漸更換邊界條件和增加熱源項，最後將計算結果與正確解作比較，結果兩者都很吻合。
在求解非傅立葉熱傳問題由於容易產生數值震盪的情況，故使用和一般傅立葉問題不同之控制體積法，再搭配逆拉氏轉換法求解，以克服數值震盪的情況發生。在解析非傅立葉熱傳問題時，先求解熱波模式，再求解雙相差模式。在求解此兩種模式時也是逐漸更換邊界條件和增加熱源項，最後將計算結果與正確解作比較，結果兩者也都非常吻合。

The present study employs the hybrid Laplace transform method to investigate Fourier and non-Fourier heat transfer problems. They belong to the hyperbolic equations. The CV wave model produces the wave-like transfer with a front due to its second-order temperature-time derivative term. In addition to this term, the dual-phase-lag model has another second-order temperature-space and first-order temperature-time derivative term. These terms easily make the solutions oscillated and unstable.
When solving the traditional Fourier heat transfer problems, the hybrid Laplace transform method is primarily used to handle the time-dependent terms in the governing equation and boundary conditions, and then the transformed equations are discretized using the finite difference scheme to yield solutions in s domain. The solutions then inversely transform to the real transient values. Comparisons of the present numerical and analytical solutions are performed. Results show that the present numerical results agree well with the analytical ones.
Unlike the numerical solution of the traditional Fourier heat transfer problems, the solution of non-Fourier heat transfer problems encounters the oscillations in the vicinity of discontinuity. Accordingly, in additional to deal with the time-dependent terms with Laplace transform method, the control volume scheme is applied to discretize the transformed equations. In the numerical analyses of CV wave model and dual-phase-lag model, the simulated results agree fairly well with those from the analytical solutions.

1. Ceylan, H. T., “Long-Time Solutions to Heat Conduction Transient with Time-Dependent Inputs,” Thesis, Mechanical Engineering Department, University of Wisconsin-Madison, WS., 1979.
2. Mikhailov, M. D., and Ozisik, M. N., “General Solution of One-Dimensional, Time Dependent Coupled Heat Flow Systems,” Int. COMM. Heat Mass Transfer, vol. 12, pp. 393-404, 1985.
3. Rizzo, F. J., and Shippy, D. J., “A method of solution for certain problems of transient heat conduction,” AIAA J, vol. 8, pp. 2004-2009, 1970.
4. Holzlohner, U., “A finite element analysis for time-dependent problems,” Int. J. Numer. Meth. Engng, Vol. 8, pp. 55-69, 1974.
5. Aral, M. M., and Gulcat, U., “A finite element Laplace transform solution technique for the wave equation,” Int. J. Numer. Meth. Engng., vol. 11, pp. 1719-1732, 1977.
6. Liggett, J. A., and Liu, P. L. F., “Unsteady flow in confined aquifers : a comparison of two boundary integral methods,” Water Resources Research, vol. 15, pp. 861-866, 1979.
7. Tamma, K. K. and Railkar, S. B., “Hybrid transfmite element methodology for nonlinear transient, thermal problems,” Numer. Heat Transfer, vol. 11, pp. 443-459, 1987.
8. Tamma, K. K., and Railkar, S. B., “Transfinite element methodology for nonlinear/linear transient thermal modeling/analysis : process and recent advances,” Int. J. Numer. Meth. Engng., vol. 25, pp. 475-494, 1988.
9. Tamma, K. K. and Railkar, S. B., “A new hybrid transfinite element computational methodology for applicability to conduction/convection/radiation heat transfer,” Int. J. Numer. Meth. Engng., vol. 26, pp. 1087-1100, 1988.
10. Tamma, K. K. and Railkar, S. B., “Specially tailored transfinite-element formulation for hyperbolic heat conduction involving non-Fourier effects,” Numer. Heat Transfer, vol. l5 (Part B)., pp. 211-226, 1989.
11. Chen, H. T., “Application of hybrid numerical method to transient heat conduction problems,” Ph. D. Thesis, Mechanical Engineering Department, National Cheng Kung University, R.O.C, 1987.
12. Chen, H. T. and Lin, J. Y., “Application of the Laplace transform to nonlinear transient problems,” Appl. Math. Modelling, vol. 15, pp. 144-151, 1991.
13. Chen, H. T. and Lin, J. Y., “Hybrid Laplace transform technique for non-linear transient thermal problems,” Int. J. Heat Mass Transfer, vol. 34, pp. 1301-1308, 1991.

14. Chen, H. T. and Lin, J. Y., “Application of the hybrid method to transient heat conduction in one-dimensional composite layers,” Comp. Struct., vol. 39, pp. 451-458, 1991.
15. Chen, H. T. and Lin, J. Y., “Application of the Laplace transform to one-dimensional non-linear transient heat conduction in hollow cylinders,” Comm. Appl. Numer. Meth., vol. 7, pp. 241-252, 1991.
16. Lin, J. Y. and Chen, H. T., “Radiar'axrsymmetric transient heat conduction in composite hollow cylinders with variable thermal conductivity,” Engng. Anal. Boundary Elements, vol. 10, pp. 27-33, 1992.
17. 林傑毓, “解析暫態熱傳問題之新數值方法,” 國立成功大學機械工程研究所博士論文, 1994
18. Fourier, J. B., Theorie Analytique de la Chaleur, Paris, English translation by Freeman, A., The Analytical Theory of Heat, Dover Publication, Inc., New York, 1995.
19. Joseph, D. D., and Preziosi, L., “Heat wave, ” Rev. mod Phys. 61, pp. 41-73, 1989
20. Caviglia, G., Morro, A., and Straughan, B., ”Thermoelasticity at cryogenic temperatures, ” Int. J. Non-Linear Mech.27, pp. 251-263, 1992.
21. Antaki, P. J., “Hotter than you think,Mach.,” Des. 13, pp. 116-118, 1995.
22. Cattaneo, C., “A Form of Heat Conduction Equation Which Eliminates the Paradox of Instantaneous Propagation,” Compte Rendus, vol. 47, pp. 431-433, 1958
23. Vernotte, P., “Les Paradoxes de la Theorie Continue del’equation De La Chaleur, ” Compte Rendus, vol. 246, pp. 3154-3155, 1958.
24. Vernotte, P., “Some Possible Complications in the Phenomena of Thermal Conducction,” Compte Rendus, vol. 252, pp. 2190-2191, 1961.
25. Carey, G. F. and Tsai, M., “Hyperbolic heat transfer with reflection,” Heat Transfer, vol. 5, pp. 309-327, 1982.
26. Glass, D. E., Ozisik, M. N., McRae, D. S. and Vick, B., “On the numerical solution of hyperbolic heat conduction, ” Numer. Heat Transfer, vol. 8, pp. 497-504, 1985
27. Tzou, D. Y., “A Unifide Field Approch for Heat Conduction form Micro-to Macro-Scales, ” ASME Journal of Heat Transfer, vol. 117, pp. 8-16, 1995.
28. Tzou, D. Y., “The Generalized Lagging Response in Small-Scale and High-Rate Heating,” International Journal of Heat and Mass Transfer, vol. 38, pp. 3231-3240, 1995.

29. Tzou, D. Y., “Experimental Support for the Lagging Response in Heat Propagation,” AIAA Journal of Thermophysics and Heat Transfer, vol. 9, pp. 686-693, 1995.
30. Tzou, D. Y., Ozisik, M. N. and Chiffelle, R. J., “The lattice temperature in the microscopic two-step model,” ASME J. Heat Transfer, vol. 116, pp. 1034-1038, 1994.
31. Anyaki, P. J. , “Solution for Non-Fourier Dual Phase Lag Heat Conduction in a Semi-infinite Slab with Surface Heat Flux, ” International journal of Heat and Mass transfer, vol. 41, pp. 2253-2258, 1998.
32. Tang, D. W., Araki, N., Wavy, wavelike, “Diffusiiv thermal responses of finite rigid slab to high-speed heating of laser-pulses, ” Int. J. Heat Mass Transfer, vol. 42, pp. 855-860, 1999.
33. 陳良彰, “熱遲滯現象之數值分析,” 國立成功大學工程科學系碩士論文, 2007
34. Honig, G., and Hirdes, U., “A Method for the Numerical Inversion of Laplace Transforms,” J. of Computational and Applied Mathematics, vol. 10, pp. 113-133, 1984.
35. Durnin, F., “Numerical inversion of Laplace transforms：an efficient improvement to Dubner and Abate’s method,” Comput. J., vol. 17, pp. 371-376, 1973.
36. Anderson, D. A., Tannehill, J. C. and Pletcher, R. H., “Computational Fluid Mechanics and Heat Trans,” Hemisphere, 1990.
37. Tzou, D. Y., Chiu and Kevin K.S., “Depth of Thermal Penetration ：Effect of Relaxation and Thermalization,” AIAA of Thermophysics & Heat Transfer, vol. 13, pp. 166-269, 1999.
38. Liu, K. C. and Chen, H. T., “Numerical analysis for the hyperbolic heat conduction problem under a pulsed surface disturbance,” Appl. Math. Comput., vol. 159, pp. 887-901, 2004.
39. Taitel, Y., “On the parabolic, hyperbolic and discrete formulation of the heat conduction equation, ” Int. J. Heat Transfer, vol. 15, pp.369-371, 1972.
40. Wu, C. Y., “Integral equation solution for hyperbolic heat conduction with surface radiation,” Int. Comm. Heat Mass Transfer, vol. 15, pp. 365-374, 1988
41. Qiu, T. Q. and Tien, C. L., “Short-Pulse Laser Heating on Matals, ”International Journal of Heat and Mass Transfer, vol. 35, pp. 719-726, 1992.
42. Qiu, T. Q. and Tien, C. L., “Heat Transfer Mechanisms During Short-Pulse Laser Heating of Metals,” ASME Journal of Heat Transfer, vol. 115, pp. 835-841, 1992.
43. Qiu, T. Q. and Tien, C. L., “Femtosecond Laser Heating of Multi-Layered Metals-I. Analysis,” International Journal of Heat and Mass Transfer, vol. 37, pp. 2789-2797, 1994.