Document Type : Research Paper

**Authors**

Mechanical Engineering Department, Tarbiat Modarres University, Tehran, Iran

**Abstract**

Based on the recently introduced fractional Taylor’s formula, a fractional heat conduction constitutive equation is formulated by expanding the single-phase lag model using the fractional Taylor’s formula. Combining with the energy balance equation, the derived fractional heat conduction equation has been shown to be capable of modeling diffusion-to-Thermal wave behavior of heat propagation by changing the order of fractional differentiation. Finally, finite sine-Fourier and Laplace transforms are employed to find the exact solution of a signaling problem.

**Keywords**

- Fractional calculus
- Non-Fourier heat conduction
- Generalized Taylor’s Formula
- Diffusion-to-Thermal wave propagation
- Exact solution

**Main Subjects**

[1] Hilfer, R., "*Applications of Fractional Calculus in Physics*", World Scientific, Singapore, (2000).

[2] Magin, R.L., "*Fractional Calculus in Bioengineering*", Begell House, Inc, (2006).

[3] Podlubny, I., "*Fractional Differential Equations*", Academic Press, New York, (1999).

[4] Flik, M. I., Choi, B. I., and Goodson, K. E., "Heat Transfer Regimes in Microstructures", J. Heat Transfer, Vol. 114, pp. 666-674, (1992).

[5] Ozisik, M. N., and Tzou, D. Y., "On the Wave Theory in Heat Conduction", J. Heat Transfer, Vol. 116, pp. 526-535, (1994).

[6] Hoashi, E., Yokomine, T., Shimizu, A., and Kunugi, T., "Numerical Analysis of Wave-type Heat Transfer Propagating in a Thin Foil Irradiated By Short-pulsed Laser", Int. J. Heat Mass Transfer, Vol. 46, pp. 4083–4095, (2003).

[7] Ai, X., and Li, B.Q., "Numerical Simulation of Thermal Wave Propagation During Laser Processing of Thin Films", Journal of Electronic Materials, Vol. 34, No. 5, pp. 583-591, (2005).

[8] Dai, W., Wang H., Pedro M. J., Ronald E. M., and Bejan, A., "A Mathematical Model for Skin Burn Injury Induced By Radiation Heating", Int. J. Heat Mass Transfer, Vol. 51, pp. 5497–5510, (2008).

[9] Jaunich, M., Raje, S., Kim, K., Mitra, K., and Guo, Z., "Bio-heat Transfer Analysis During Short Pulse Laser Irradiation of Tissues", Int. J. Heat Mass Transfer, Vol. 51(23-24), pp. 5511-5521, (2008).

[10] Mitra, K., Kumar, A., Vedavarz, A., and Moallemi, M. K., "Experimental Evidence of Hyperbolic Heat Conduction in Processed Meat", J. Heat Transfer, Trans. ASME, Vol. 117(3), pp. 568-573, (1995).

[11] Ozen S., Hehel S., and Cerezci, O., "Heat Analysis of Biological Tissue Exposed to Microwave by using Thermal Wave Model of Bio-heat Transfer (TWMBT) ", Burns, Vol. 34, pp. 45-49, (2008).

[12] Kaminski, W., "Hyperbolic Heat Conduction Equation for Materials with a Nonhomogeneous Inner Structure", ASME J. Heat Transfer, Vol. 112, pp. 555-560, (1990).

[13] Straughan, B., "Thermal Convection with the Cattaneo–Christov Model", Int. J. Heat Mass Transfer, Vol. 53, pp. 95-98, (2010).

[14] Antaki, P. J., "Importance of Nonfourier Heat Conduction in Solid-phase Reactions", Combustion and Flame, Vol. 112, pp. 329-341, (1998).

[15] Agwu Nnanna, A. G., Haji-Sheikh, A., and Harris, K. T., "Experimental Study of Non-Fourier Thermal Response in Porous Media", J. Porous Media, Vol. 8(1), pp. 31-44, (2005).

[16] Antaki, P. J., "New Interpretation of Non-Fourier Heat Conduction in Processed Meat", J. Heat Transfer, Trans. ASME, Vol. 127(2), pp. 189-193, (2005).

[17] Xu, F., Seffen. K.A., and Lu, T. J., "Non-Fourier Analysis of Skin Biothermomechanics", Int. J. Heat Mass Transfer, Vol. 51, pp. 2237-2259, (2008).

[18] Odibat, Z. M., and Shawagfeh, N. T., "Generalized Taylor's Formula", Applied Mathematics and Computation, Vol. 186, pp. 286-293, (2007).

[19] Stephen, W., Wheatcraft, and Meerschaert, M. M., "Fractional Conservation of Mass", Advances in Water Resources, Vol. 31, pp. 1377–1381, (2008).

September 2010

Pages 66-80