Approximate Solutions for Solving Time-Space Fractional Bioheat Equation Based on Fractional Shifted Legendre Polynomials

The aim of this article was employed a fractional-shifted Legendre polynomials (F-SLPs) in a matrix form to approximate the temporal and spatial derivatives of fractional orders for derived an approximate solutions for bioheat problem of a space-time fractional. The spatial-temporal fractional derivatives are described in the formula by the Riesz-Feller and the Caputo fractional derivatives of orders v (1,2] and γ (0,1], respectively. The proposed methodology applied for two examples for demonstrating its usefulness and effectiveness. The numerical results confirmed that the utilized technique is immensely effective, provides high accuracy and good convergence.


Introduction
Medical treatments like cryosurgery, cryopreservation, cancer hyperthermia, skin burns and thermal malady diagnostics, require an understanding of thermal phenomena and temperature behavior in living tissues. Therefore, bioheat transport in human tissues is a topic of high theoretical and applied benefit. Biothermal studies can assist the design of clinical thermal treatment equipment, evaluation of thermal treatment's effects on skin, and establishment of thermal protections for various purposes [1, 6,18].
The physical phenomenon that explain heat transport in human tissue that includes the influence of blood flux on tissue temperature in a continuum was presented by Pennes [14], Furthermore it suggested a mathematical model to describe heat flux in biological tissue. The model known as the bioheat equation which that is still widely utilize [3].
In this article, will introducing the approximate algorithm for solving one dimensional time-space fractional bioheat equation (T-SFBHE) based on F-SLPs.

Time-Space Fractional Bioheat Equation
The time-space fractional version of the one-dimensional unsteady state Pennes bioheat equation can be obtain by replacing the first order time derivative by Caputo fractional derivative of order ∈ 0,1 and second order space derivative by Riesz-Feller derivative of fractional arbitrary positive real order ∈ 1, 2 . The T-SFBHE is given according to [17] , * , , , 0, 0 , where , , , , , , , , and symbolizes density, specific heat, thermal conductivity, temperature, time, distance, artillery temperature ,blood perfusion rate, metabolic heat generation in skin tissue and external heat exporter in skin tissue respectively. The units and values of the symbols expressed in this equation are mentioned in Table1[5]. where, is the heat flux on the skin surface.

Preliminaries and Notations
In this section, remind the principles essentials of the fractional calculus theory that will be used in this article.
Definition 3. The Caputo definition of fractional differential operator is defined as [10]: , .

Fractional Shifted Legendre Polynomials
Define the F-SLPs by introducing the change of variable and 1 on shifted Legendre polynomials. The F-SLPs is symbolized by . The F-SLPs are a particular solution of the normalized eigenfunctions of the Sturm-Liouville problem. This is generate 1 algebraic equations by multiplying for 0,1,2, … , ; 0,1,2, … , , integrating from 0 to 1 and using the orthogonal property, to get * ˊ , with the initial condition from equation (2)  should note that in order to construct the approximate ̃ , to the error function , , only equation (58) needs to be recomputed in the same procedure as doing before for the solution of equation (1).

Numerical Examples
In this section, apply the algorithm, which presented in section 6 for solving the T-SFBHE in the two examples based on F-SLPs. In order to showing a capability of the collocation method for achieving the

Conclusions
In this work, the approximate algorithm structured on the F-SLPs in the matrix form to estimate the fractional derivatives to found the numerical solutions of the T-SFBHE. The Caputo formula utilized into approximate the fractional derivatives. Figs. 1-4 and Tables 2-3 indicated that the numerical results for Example 1 and 2 of a present technicality has a higher accuracy, good convergence, reasonable stability as well as a minimal computational effort by utilizing a few mesh grid. Concluded that the target numerical approach can be solve a various kinds of models of any fractional orders. In addition expected that the present methodology may present a more exact estimate by employing some other families based on orthogonal polynomials.