本文以拉氏轉換法與控制體積法的混合應用，並配合雙曲線形狀函數來探討各種不同的雙曲線型擴散問題，如具有位勢場的非費克(non-Fickian)質量擴散問題、雙層複合材料的非傅立葉(non-Fourier)熱傳導問題和具熱波效應的生物熱傳問題。本文數值方法乃先以拉氏轉換法移除統制方程式和邊界條件中的時間項，而後以控制體積法聯合雙曲線形狀函數離散拉氏轉換後的統制方程式。由於雙曲線型擴散問題可能發生陡峻變化的現象，以致數值解在陡峻變化處可能會有數值振盪出現，是求取本文中各種問題之數值解過程的主要障礙。為了驗證本文數值方法的精確性，本文數值結果將與解析解比較。結果顯示，本文數值結果與解析解甚為吻合，且於波前不連續處附近也未見明顯的數值振盪現象發生。
　　非傅立葉或非費克的波傳效應僅發生於極短時程內，該效應會隨時間增長而消失，而且傳遞介質的幾何形狀對溫度分佈或質量濃度分佈具有相當的影響。由於複合兩材料間的物理性質差異，導致熱波或質量波在交界面處會發生傳遞-反射的現象。並且，位勢場於非費克質量擴散問題中扮演極重要的角色。位勢場梯度循質量波傳遞方向若為負時，會增強質量傳播能力；若為正時，質量傳播將會受到阻礙。但位勢場梯度的值和質量波的傳遞速度無關。
　　複合材料於交界面的接觸熱阻(contact resistance)，對於熱傳導具有阻礙。本文以對流邊界條件(convection boundary condition)模式界定複合材料之交界面熱阻，進而探討熱阻對熱傳遞的影響情形。由本文數值結果觀察得到，熱波在複合材料的傳遞行為也受到材料熱性質比的影響，進而改變了傳遞波和反射波的強度。
　　探討二維生物熱傳的熱波問題時，即使生物組織之非傅立葉熱傳問題的統制方程式帶有非齊性項，本文數值方法也僅以五個節點來離散統制方程式，並能有效壓制數值振盪現象得到穩定可靠之數值分析結果。數值結果顯示血流灌注率在生物組織的溫度調節上扮演極重要角色，其對生物組織之溫度分佈的影響是不容忽視的。

　　The present study employs a hybrid application of the Laplace transform method and the control-volume method in conjunction with the hyperbolic shape functions to investigate the hyperbolic diffusion problems, such as non-Fickian mass diffusion problem in a potential field, non-Fourier heat conduction problem in a two-layer composite, and bio-heat transfer problem under thermal wave effect. The Laplace transform method is used to remove the time-dependent terms in the governing equation and the boundary conditions, and then the transformed equations are discretized by the control volume scheme. The primary difficulty encountered in the numerical solutions is numerical oscillations in the vicinity of sharp discontinuities. To evidence the accuracy of the present numerical method, a comparison between the present numerical results and analytical solution is made. Results show that the present numerical results agree well with the analytic solution and do not exhibit numerical oscillations in the vicinity of the jump discontinuity.
　　The wave nature of the hyperbolic diffusion problems is significant only for short times and quickly dissipates with time, and the geometry of the transmission medium strongly affects the temperature distribution or the mass concentration distribution. Due to the difference of the physical properties of two adjacent materials, the thermal wave or the mass wave creates the transmission-reflection phenomenon at the interface of the composite. In addition, the potential field plays an import role in the non-Fickian mass diffusion problem. If the potential gradient is negative in the direction of mass transfer, it will increase the mass transmission ability; if the potential gradient is positive, it will obstruct the transmission of mass. However, the potential field does not affect the propagation speed.
　　The contact resistance at the interface of composite is obstructive for heat conduction. This study uses the convection boundary condition model to define the interface thermal resistance, and then investigating the effect of the interface thermal resistance on the heat transfer. The thermal property ratios of two dissimilar materials also affect the propagation behavior of the thermal wave in the composite and alter the strength of the reflected and transmitted waves.
　　While investigating the heat wave problem of the 2-D bio-heat transfer, the present numerical method can discretize the governing equation that has non-homogenous term with five nodes and efficiently suppress the numerical oscillations for a stable and reliable result of analysis. The present numerical results show that the blood perfusion rate can adjust the temperature of biological tissue and its effect on the temperature distribution can not be ignored.

[1] Kelly, D.C., “Diffusion: a relativistic appraisal,” American J. Phys., vol. 36, pp. 585-591, 1968.

[2] Peshkov, V., “Second sound in helium II,” J. Phys. USSR, vol. 8, pp. 381, 1944.

[3] Cattaneo, C., “Sulla conduzione de calore,” Atti del Semin. Mat. E Fis. Univ. Modena, vol. 3, pp. 3, 1948.

[4] Vedavarz, A., Mitra, K., and Kumar, S., “Hyperbolic temperature profile for laser surface interactions,” J. Appl. Phys., vol. 76, pp. 5014-5021, 1994.

[5] Tang, D.W. and Araki, N., “Wavy, wavelike, diffusive thermal responses of finite rigid slabs to high-speed heating of laser-pulses,” Int. J. Heat Mass Transfer, vol. 42, pp. 855-860, 1999.

[6] Mitra, K., Kumar, S., Vedavarz, A., and Moallemi, M.K., “Experimental evidence of hyperbolic heat conduction in processed meat,” ASME. J. Heat Transfer, vol. 117, pp. 568-573, 1995.

[7] Liu, J. and Xu, L.X. “Boundary information based diagnostic on the thermal states of biological bodies,” Int. J. Heat Mass Transfer, vol. 43, pp. 2827-2839, 2000.

[8] Desocio, L.M. and Gualtieri, G., “A hyperbolic stefan problem,” Quar. Appl. Math., July, pp. 253-259, 1983.

[9] Solomon, A.D., Alexiades, V., Wilson, D.G., and Drake, J., “On the formulation of hyperbolic stefan problems,” Quar. Appl. Math., October, pp. 295-304, 1985.

[10] 徐雲生, 過增元, 羅曉迎, “半導體擊穿過程的熱分析,” 工程熱物理學報, 第十四卷, 第三期, pp. 298-302, 科學出版社, 北京, 1993.

[11] Guo, Z.Y. and Xu, Y.S., “Non-fourier heat conduction in IC chip,” IEEE InterSociety Conference on Thermal Phenomena, pp. 271-275, 1992.

[12] Bai, C. and Lavine, A.S., “Hyperbolic heat conduction in a superconducting film,” ASME. Thermal Engineering Proceedings, vol. 4, pp. 87-92, 1991.

[13] Jou, D., Camacho, J., and Grmela, M., “On the nonequilibrium thermodynamics of non-Fickian diffusion,” Macromolecules, vol. 24, pp. 3597-3602, 1991.

[14] Krapivnyi, N.G., Kleshnya, V.B., and Sobornitskii, V.I., “Allowing for the finite rate of propagation of the hydrogen concentration wave during hydrogen diffusion in metals,” Soviet Electrochemistry, vol. 28, pp. 369-373, 1992.

[15] Galenko, P. and Sobolev, S., “Local nonequilibrium effect on undercooling in rapid solidification of alloys” Physical Review E, vol. 55, pp. 343-355, 1997.

[16] 淮秀蘭, 姜任秋, 劉登瀛, 孟群, “快速瞬態傳質過程中非費克效應的實驗與理論研究,” 工程熱物理學報, 第二十一卷, 第五期, pp. 595-599, 科學出版社, 北京, 2000.

[17] Skryl, Y., “The effects of hyperbolic diffusion in liquid junctions,” Phys. Chem. Chem. Phys., vol. 2, pp. 2969-2976, 2000.

[18] Wong E.H., Chan, K.C., Lin, T.B., and Lam, T.F., “Non-Fickian moisture properties characterisation and diffusion modeling for electronic packages,” IEEE Electronic Components and Technology Conference, pp. 302-306, 1999.

[19] 蔣方明, 劉登瀛, “非傅立葉導熱的最新研究發展,” 力學進展, 第三十一卷, 第一期, pp. 128-139, 中科院力學研究所, 北京, 2002.

[20] Tzou, D.Y., Macro- to microscale heat transfer, Taylor & Francis, Washington, 1996.

[21] Honner, M. and Kunes, J., “On the wave diffusion and parallel nonequilibrium heat conduction,” ASME. J. Heat Transfer, vol. 121, pp. 702-707, 1999.

[22] 姜任秋, 熱傳導、質擴散與動量傳遞中的瞬態衝擊效應, 科學出版社, 北京, 1997.

[23] Weymann, H.D., “Finite speed of propagation in heat conduction, diffusion, and viscous shear motion,” American J. Phys., vol. 35, pp. 488-496, 1967.

[24] Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, second ed., Oxford University, New York, 1959.

[25] Maxwell, J.C., “On the dynamic theory of gases,” Philos. Trans. Soc. London, vol. 157, pp. 49-88, 1867.

[26] Tisza, L., “Sur la supraconductibilité thermique de l’helium II liquide et la statistique de Bose-Einstein,” C. R. Acad. Sci. vol. 207, pp. 1035, 1938.

[27] Landau, L., “The theory of superfluidity of helium II,” J. Phys. Vol. 5, pp. 71, 1941.

[28] Morse, P.M. and Feshbach, H., Methods of theoretical physics, vol. 1, McGraw-Hill, New York, 1953.

[29] Cattaneo, C., “A form of heat conduction equation which eliminates the paradox of instantaneous propagation,” Compute Rendus, vol. 247, pp. 431-433, 1958.

[30] Vernotte, P., “Les paradoxes de la théorie continue de l’equation de la chaleur,” Compute Rendus, vol. 246, pp. 3154-3155, 1958.

[31] Vernotte, P., “Some possible complications in the phenomena of thermal conduction,” Compute Rendus, vol. 252, pp. 2190-2191, 1961.

[32] Joseph, D.D. and Preziosi, L., “Heat waves,” Rev. Mod. Phys. Vol. 61, pp. 41-73, 1989.

[33] Joseph, D.D. and Preziosi, L., “Addendum to the paper ‘heat waves’,” Rev. Mod. Phys. Vol. 62, pp. 375-391, 1990.

[34] Özisik, M.N., and Tzou, D.Y., “On the wave theory in heat conduction,” ASME J. Heat Transfer, vol. 116, pp. 526-535, 1994.

[35] Chester, M., “Second sound in solids,” Physics Review, vol. 131, pp. 2013-2015, 1963.

[36] Baumeister, K.J. and Hamill, T.D., “Hyperbolic heat-conduction equation ¾ a solution for the semi-infinite body problem,” ASME. J. Heat Transfer, vol. 93, pp. 126-127, 1971.

[37] Chan, S.H., Low, J.D., and Mueller, W.K., “Hyperbolic heat conduction in catalytic supported crystallites,” AIChE J., vol. 17, pp. 1499-1501, 1971.

[38] Mourer, M.J. and Thompson, H.A., “Non-Fourier effects at high heat flux,” ASME. J. Heat Transfer, vol. 95, pp. 284-286, 1973.

[39] Tzou, D.Y., “On the thermal shock wave induced by a moving heat source,” ASME J. Heat Transfer, vol. 111, pp. 232-238, 1989.

[40] 張鵬, 王如竹, “低溫傳熱中的熱波現象及其對超流氦膜沸騰發生的影響,” 工程熱物理學報, 第十八卷, 第四期, pp. 459-463, 科學出版社, 北京, 1997.

[41] Nettleton, R.E., “ Relaxation theory of thermal conduction in liquids,” Phys. Fluids, vol. 3, pp. 216-225, 1960.

[42] Francis, H.P., “Thermo-mechanical effects in elastic wave propagation: a survey,” J. Sound Vibration, vol. 21, pp. 181-192, 1972.

[43] Sieniutycz, S., “The variational principles of classical type for non-coupled nonstationary irreversible transport processes with convective motion and relaxation,” Int. J. Heat Mass Transfer, vol. 20, pp. 1221-1231, 1977.

[44] Crank, J., The mathematics of diffusion, 2nd ed., Oxford University Press, New York, 1975.

[45] Shewmon, P., Diffusion in Solids, 2nd ed., the Minerals, Metals and materials Society, Pennsylvannia, 1989.

[46] Galenko, P., “Local-nonequilibrium phase transition model with relaxation of the diffusion flux,” Physics Letters A, vol. 190, pp. 292-294, 1994.

[47] Godoy, S. and García-Colín, L.S., “From the quantum random walk to classical mesoscopic diffusion in crystalline solids,” physical ReviewE, vol. 53, pp. 5779-5785, 1996.

[48] Overbergh, N., Berghmans, H., and Smets, G., “Crystallization of isotactic polystyrene induced by organic vapors,” Polymer, vol. 16, pp. 703, 1975.

[49] Vrentas, J.S., Duda, J.L., and Hou, A.C., “Anomalous sorption in poly(ethyl methacrylate),” J. Appl. Polym. Sci., vol. 29, pp. 399, 1984.

[50] Hadeler, K.P., “Reaction transport equations in biological modeling,” Mathematical and Computer Modelling, vol. 31, pp. 75-81, 2000.

[51] Das, A.K., “A non-Fikian diffusion equation,” J. Appl. Phys., vol. 70, pp. 1355-1358, 1991.

[52] Das, A.K., “Some non-Fickian diffusion equation: theory and applications,” Defect and Diffusion Forum, vols. 162-163, pp. 97-118, 1998.

[53] Frankel, J.I., Vick, B., and Özisik, M.N., “General formulation and analysis of hyperbolic heat conduction in composite media,” Int. J. Heat Mass Transfer, vol. 30, pp. 1293-1305, 1987.

[54] Wiggert, D.C., “Analysis of early-time heat conduction by method of characteristics,” J. Heat Transfer, vol. 99, pp. 35-40, 1977.

[55] Carey, G.F. and Tsai, M., “Hyperbolic heat transfer with reflection,” numer. Heat Transfer, vol. 5, pp. 309-327, 1982.

[56] Glass, D.E., zisik, 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.

[57] Chen, H.T. and Lin, J.Y., “Numerical analysis for hyperbolic heat conduction,” Int. J. Heat Mass Transfer, vol. 36, pp. 2891-2898, 1993.

[58] Dubner, W.M. and Abate, J., “Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform,” J. Assoc. Comput. Mach., vol. 15, pp. 115-123, 1968.

[59] Stechfest, H., “Numerical inversion of Laplace transforms,” Commun. ACM, vol. 13, pp. 47-50, 1970.

[60] Honig, G. and Hirdes, U., “A method for the numerical inversion of Laplace transforms,” J. Comp. Appl. Math., vol. 10, pp. 113-132, 1984.

[61] Lin, J.Y. and Chen, H.T., “Numerical solution of hyperbolic heat conduction in cylindrical and spherical systems,” Applied Mathematics Modelling, vol. 18, pp. 384-390, 1994.

[62] Camera-Roda, G. and Sarti, G., “Mass transport with relaxation in polymers,” AIChE J., vol. 36, pp. 851-860, 1990.

[63] Hayes, C.K. and Cohen, D., “The evolution of steep fronts in non-Fickian polymer-penetrant systems,” J. Polymer Sci.: Part B: Polymer Physics, vol. 30, pp. 145-161, 1992.

[64] Ramos-Barrado, J.R., Motenegro, P.G., and Cambón, C.C., “A generalized Warburg impedance for a nonvanishing relaxation process,” J. Chem. Phys., vol. 105, pp. 2813-2815, 1996.

[65] Edwards, D.A., “Non-Fickian diffusion in thin polymer films,” J. Polymer Sci.: Part B: Polymer Physics, vol. 34, pp. 981-997, 1996.

[66] Chen, H.T. and Lin, J.Y., “Study of Hyperbolic Heat Conduction with Temperature-Dependent Thermal Properties,” J. Heat Transfer, vol. 116, pp. 750-753, 1994.

[67] Guy, A.G., Introduction to Materials Science, McGraw-Hill, New York, 1972.

[68] Taitel, Y., “On the parabolic, hyperbolic and discrete formulation of the heat conduction equation,” Int. J. Heat Mass Transfer, vol. 15, pp. 369-371, 1971.

[69] Zhang, Z. and Liu, D.Y., “Non-Fourier effects in rapid transient heat conduction in a spherical medium,” J. Eng Thermophy. Vol. 19, PP. 601-605, 1998.

[70] Özisik, M.N., Heat Conduction, second ed., Oxford University, New York, 1980.

[71] Kao, T.T., “Non-Fourier heat conduction in thin surface layers,” ASME J. Heat Transfer, vol. 99, pp. 343-345, 1977.

[72] Thomson, W.T. and Dahleh, M.D., Theory of Vibration Applications, Prentice Hall, Upper Saddle River, N.J., 1998.

[73] Ding, S. and Petuskey, W.T., “Solution to Fick’s second law of diffusion with a sinusoidal excitation,” Solid State Ionics, vol. 109, pp. 101-110, 1998.

[74] Wang, M.Y., Ko, F.H., Wang, C.C., and Huang, T.Y., “Characterization and modeling of out-diffusion of manganese zinc impurities from deep ultraviolet photoresist,” J. Electrochemical Society, vol. 146, pp. 3455-3460, 1999.

[75] Frankel, J.I., Vick, B., and Özisik, M.N., “Hyperbolic heat conduction in composite regions,” presented at the Eighth International Heat Transfer Conference, San Francisco, CA, 1986.

[76] Bourdillon, A. and Tan Bourdillon, N.X., High temperature superconductors: processing and science. Academic Press Inc., San Diego, CA, 1993.

[77] Lor, W.B. and Chu, H.S., ‘Hyperbolic heat conduction in thin-film high Tc superconductors with interface thermal resistance,” Cryogenics, vol. 39, pp. 739-750, 1999.

[78] Kazimi, M.S. and Erdman, C.A., “On the interface temperature of two suddenly contacting materials,” ASME J. Heat Transfer, vol. 97, pp. 615-617, 1975.

[79] Yang, H.Q. “Characteristics-based high-order accurate and nonoscillatory numerical method for hyperbolic heat conduction,” Numerical Heat Transfer, Part B, Vol. 18, pp. 221-241, 1990

[80] Lor, W.B. and Chu, H.S., “Effect of interface thermal resistance on heat transfer in a composite medium using the thermal wave model,” Int. J. Heat Mass Transfer, vol. 43, pp. 653-663, 2000.

[81] Little, W.A., “The transport of heat between dissimilar solids at low temperatures,” Canadian of Journal of Physics, vol. 37, pp. 334-349, 1959.

[82] Phelan, P.E., “Thermal response of thin-film high Tc superconductors to modulated irradiation,” J Thermophysics Heat Transfer, vol 9, pp. 397-402, 1995.

[83] Phelan, P.E., “Application of diffuse mismatch theory to the prediction of thermal boundary resistance in thin-film high Tc superconductors,” ASME J Heat Transfer, vol. 120, pp. 37-43, 1998.

[84] Kelkar, M., Phelan, P.E. and Gu, B., “Thermal boundary resistance for thin-film high Tc superconductors at varying interfacial temperature drops,” Int. J. Heat Mass Transfer, vol. 40, pp. 2637-2645, 1997.

[85] Smith, A.N., Hosterler, J.L., and Norris, P.M., “Thermal boundary resistance measurements using a transient thermoreflectance technique,” Microscale Thermophysical Engineering, vol. 4, pp. 51-60, 2000.

[86] Honner, M., “Heat waves simulation,” Comput. Math. Appl., vol. 38, pp. 233-243, 1999.

[87] Sadd, M.H. and Cha, C.Y., “Axisymmetric non-Fourier temperature in cylindrically bounded domains,” Int. J. Nonlinear Mechanics, vol. 17, pp. 129-136, 1982.

[88] Barletta, A. and Zanchini, E., “Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady-periodic electric field,” Int. J. Heat Mass Transfer, vol. 39, pp. 1307-1315, 1996.

[89] Barletta, A., “Hyperbolic propagation of an axisymmetric thermal signal in an infinite solid medium,” Int. J. Heat Mass Transfer, vol. 39, PP. 3261-3271, 1996.

[90] Barletta, A. and Pulvirenti, B., “Hyperbolic thermal waves in a solid cylinder with a non-stationary boundary heat flux,” Int. J. Heat Mass Transfer, vol. 41, pp. 107-116, 1998.

[91] Abdul Azeez, M.F. and Vakakis, A.F. “Axisymmetric transient solutions of the heat diffusion problem in layered composite media,” Int. J. Heat Mass Transfer, vol. 43, pp. 3883-3895, 2000.

[92] Glass, D.E., zisik, M.N., and Vick, B., “Hyperbolic heat conduction with surface radiation,” Int. J. Heat Mass Transfer, vol. 28, pp. 1823-1830, 1985.

[93] Barletta, A. and Zanchini, E., “A thermal potential formulation of hyperbolic heat conduction,” ASME J. Heat Transfer, vol. 121, pp. 166-169, 1999.

[94] Liu, J. Chen X., and Xu, L.X. “New thermal wave aspects on burn evaluation of skin subjected to instantaneous heating,” IEEE Transactions on Biomedical Engineering, vol. 64, pp. 420-428, 1999.

[95] Deng, Z.S. and Liu, J., “Blood perfused-based model for characterizing the temperature fluctuation in living tissues,” Physica A, vol. 300, pp. 521-530, 2001.

[96] Arkin, H., Xu, L.X., and Holmes, K.R., “Recent developments in modeling heat transfer in blood perfused tissues,” IEEE. Biomedical Engineering, vol. 41, pp. 97-107, 1994.

[97] Kaminski, W., “Hyperbolic heat conduction equation for material with a nonhomogenous inner structure,” ASME J. Heat Transfer, vol. 112, pp. 555-560, 1990.

[98] Lu, W.Q., Liu, J., and Zeng, Y., “ Simulation of the thermal wave propagation in biological tissues by the dual reciprocity boundary element method,” Engineering Analysis with Boundary Elements, vol 22, pp. 167-174, 1998.

[99] Yang, H.Q., “Solution of two-dimensional hyperbolic heat conduction equation by high resolution numerical method,” Numerical Heat Transfer, Part A, vol. 21, pp. 333-349, 1992.

[100] Wu, J.P., Shu, Y.P., and Chu, H.S., “Transient heat-transfer phenomenon of two-dimensional hyperbolic heat conduction problem,” Numerical Heat Transfer, Part A, Vol 33, pp. 635-652, 1998.

[101] Chen, H.T. and Lin, J.Y., “Analysis of two-dimensional hyperbolic heat conduction problems,” Int. J. Heat Mass Transfer, vol. 37, pp. 153-164, 1994.

[102] Arkin, H., Holmes K.R., Chen M. M., and Bottje, “ Thermal pulse decay method for simultaneous measurement of local thermal conductivity and blood perfusion: a theoretical analysis,” ASME J. Biomechanical Engineering, vol. 108, pp. 208-214, 1986.