Solution of fractional bioheat equation in terms of Fox’s H-function

Present paper deals with the solution of time and space fractional Pennes bioheat equation. We consider time fractional derivative and space fractional derivative in the form of Caputo fractional derivative of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in \left( 0,1\right]$$\end{document}α∈0,1 and Riesz–Feller fractional derivative of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in \left( 1,2\right]$$\end{document}β∈1,2 respectively. We obtain solution in terms of Fox’s H-function with some special cases, by using Fourier–Laplace transforms.

partial differential equation for Cauchy problem in a whole-space domain and signalling problem in a half-space domain. Shang (2015) gave analytic solution of viral infection dynamics in vivo through a time-inhomogeneous Markov chain characterization by Lie algebraic approach.
In present study, we consider fractional form of Pennes bioheat equation by replacing first order time derivative by Caputo fractional derivative of order α ∈ (0, 1] and second order space derivative by Riesz-Feller fractional derivative of order β ∈ (1, 2] respectively. We make an attempt to solve the fractional model by dividing it into two sections. In section one, time fractional derivative is considered while in section two, space fractional derivative is taken into account. We apply Laplace-Fourier transform and obtain the solution in term of Fox H-function.

Preliminaries and notations
Fractional derivative of order α is denoted as a D α t f (t), the subscripts a and t denote the two limits related to the operation of fractional differentiation, which are called the terminal of fractional differentiation. If α is negative then it denotes the fractional integrals of arbitrary order.
Definition 1 (Kilbas et al. 2006) The Riemann-Liouville fractional derivative of order α > 0 for Real(α) > 0 and m ∈ N , t > a is defined as Definition 2 (Kilbas et al. 2006) The Caputo fractional derivative of order α > 0, for Real(α) > 0 and m ∈ N , t > a is defined as  Definition 7 (Podlubny 1999) Mittag-Leffler function for one parameter is denoted by E α (z) and defined as Mittag-Leffler function for two parameter is denoted by E α , β (z) and defined as Podlubny (1999) reported the Laplace transform of a derivative of Mittag Leffler function as and inverse Laplace transform of (16) is also existing as Definition 8 (Mathai et al. 2010) The H-function is defined by means of a Mellin-Barners type integral in the following manner where, and an empty product is interpreted as unity, m, n, p, q (0, ∞) and C being the complex number field. The counter is infinite contour which separates all the poles of Ŵ(1 − a j + sA l ), j = 1, . . . , n.
The relation between derivative of Mittag-Leffler function and H-Function (Langlands 2006) is given as

Fractional bioheat equation
The Pennes bioheat model (Pennes 1948) is widely used for study of the heat transfer in skin tissue due to its simplicity, ease application and effectiveness. Pennes (1948) suggested that the rate of heat transfer between blood and tissue is proportional to the product of the volumetric perfusion rate and difference between the arterial blood temperature and the local tissue temperature. Pennes equation is employed to describe the heat transfer process as where ρ, c and k represent density, specific heat and thermal conductivity respectively. T, t and x represent, temperature, time and distance respectively; the subscript b denotes for blood. T a and W b are artillery temperature and blood perfusion rate respectively. Q met and Q ext are metabolic heat generation and external heat source in skin tissue respectively. Considering Eq. (20) and replacing first order time derivative by Caputo fractional derivative of order α ∈ (0, 1] and second order space derivative by Riesz-Feller fractional derivative of order β ∈ (1, 2]. The fractional form of Pennes bioheat equation is given as Initial and boundary conditions of Eq. (21) are given below as To find solution of Eq. (21) to (23). We divide it into two parts, in first part, the time fractional is considered while in second part, the space fractional is considered.