29. A novel Pennesí bioheat transfer equation with memory-dependent derivative
Department of Applied Mathematics, University of Calcutta, Kolkata-700 009, India
Received 15 November 2016; received in revised form 19 December 2016; accepted 20 December 2016
Abstract. In this paper, a new mathematical model for Pennesí bioheat equation using the new memory-dependent derivative is established. The one-dimensional thermal behavior in living tissue subject to instantaneous surface heating is investigated. Numerical calculations are performed to study the temperature transients in the skin exposed to instantaneous surface heating. Numerical results are plotted in the form of two-dimensional graphs and discussed. In this novel model, the time delay parameter is a new indicator of bio-heat efficiency in living tissues.
Keywords: Pennesí bioheat transfer equation, Fourierís law of heat conduction, memory‑dependent derivative, time-delay parameter, Kernel function.
 Wang J. L., Li H. F. Surpassing the fractional derivative: Concept of the memory-dependent derivative. Computers and Mathematics with Applications, Vol. 62, 2011, p. 1562‑1567.
 Diethelm K. Analysis of Fractional Differential Equation: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer-Verlag, Berlin, Heidelberg, 2010.
 Caputo M. Linear models of dissipation whose Q is almost frequency independent II. Geophysical Journal of the Royal Astronomical Society, Vol. 13, 1967, p. 529‑539.
 Mishura Y. S. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer‑Verlag, Berlin, Heidelberg, 2008.
 Ezzat M. A., El-Karamany A. S., El-Bary A. A. Generalized thermoelasticity with memory‑dependent derivatives involving two temperatures. Mechanics of Advanced Materials and Structures, Vol. 23, 2016, p. 545‑553.
 Yu Y.-J., Hu W., Tian X.-G. A novel generalized thermoelasticity model based on memory-dependent derivative. International Journal of Engineering Science, Vol. 811, 2014, p. 123‑134.
 Lord H. W., Shulman Y. A. Generalized dynamical theory of thermoelasticity. Journal of the Mechanics and Physics of Solids, Vol. 15, 1967, p. 299‑309.
 Pennes H. H. Analysis of tissue and arterial blood temperatures in the resting human forearm. Journal of Applied Physiology, Vol. 1, 93, p. 122‑1948.
 Ozisik M. N., Tzou D. Y. On the wave theory in heat conduction. Journal of Heat Transfer, Vol. 116, 1994, p. 526‑535.
 Zakian V. Numerical inversions of Laplace transforms. Electronics Letters, Vol. 5, 1969, p. 120‑121.
 Bachher M., Sarkar N., Lahiri A. Fractional order thermoelastic interactions in an infinite voids material due to distributed time-dependent heat sources. Meccanica, Vol. 50, 2015, p. 2167‑2178.
 Liu J., Zhang X., Wang C. Generalized time delay bioheat equation and preliminary analysis on its wave nature. Chinese Science Bulletin, Vol. 42, 1997, p. 289‑292.
Cite this article
Sarkar N. A novel Pennesí bioheat transfer equation with memory‑dependent derivative. Mathematical Models in Engineering, Vol. 2, Issue 2, 2016, p. 151‑157.
Mathematical Models in Engineering. December 2016, Volume 2, Issue 2
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