The Numerical Technique Based on Shifted Jacobi-Gauss-Lobatto Polynomials for Solving Two Dimensional Multi-Space Fractional Bioheat Equations

: This article deals with the approximate algorithm for two dimensional multi-space fractional bioheat equations (M-SFBHE). The application of the collection method will be expanding for presenting a numerical technique for solving M-SFBHE based on “shifted Jacobi-Gauss-Labatto polynomials” (SJ-GL-Ps) in the matrix form. The Caputo formula has been utilized to approximate the fractional derivative and to demonstrate its usefulness and accuracy, the proposed methodology was applied in two examples. The numerical results revealed that the used approach is very effective and gives high accuracy and good convergence.


Introduction
Fractional calculus has been utilized to ameliorate the modeling fineness of many phenomena naturalistic in science and engineering. It was applied in the assorted fields such as diffusion problems, viscoelasticity, mechanics of solids, biomedical engineering, control theory, and economics, etc. (1).
Pennes' suggested in (1948) the essential structure of the mathematical designing that describes temperature propagation in human tissues, the model known as the bioheat equation remains extensively used in the hyperthermal and freezing treatments (2). The fractional bioheat model which extracted the focus of the researchers and these contributed to a significant amount of the researches based on approximate and analytic methodology, for example (Singh et al. in (3), finite difference and homotopy perturbation method, Jiang and Qi in (4), Taylor's series expansion, Damor et al. in (5), implicit finite difference method, Ezzat et al. in (6), Laplace transform mode, Ferrás et al. and Kumar et al. in (7)(8), implicit finite difference method, "backward finite difference method" and "Legendre wavelet Galerkin scheme", Qin and Wu and Damor et al. in (9)(10), quadratic spline collocation method and Fourier-Laplace transforms, Kumar and Rai in (11), finite element based on Legendre wavelet Galerkin method, Roohi et al. in (12), Galerkin scheme, Hosseininia et al. and Al-Saadawi and Al-Humedi in (13)(14), "Legendre wavelet method" and "Collocation method").
In this paper, the SJ-GL-Ps in the matrix form is employed for the present numerical approach in order to solving the following twodimensional M-SFBHE. Therefore, the spacefractional version of the two-dimensional unsteady state Pennes bioheat equation can be obtained by replacing the space derivatives with the derivatives of arbitrary positive real orders 1 , 2 ∈ (1,2] as: where , , , , , , , , 0 , = , and symbolizes density, specific heat, thermal conductivity, temperature, time, distances with , , artillery temperature, heat flux on the skin surface, blood exudation rate, metabolic heat obstetrics in lacing tissue and external heat exporter in skin tissue respectively. The units and value of the symbolizations that expressed in this equation are tabulating in Table1. The sections of this article are structured as follows: In the next section, some definitions of essentials principles of the fractional calculus will be showing. Followed by the shifted Jacobi polynomials operational matrix for ordinary derivatives and their fractional derivatives, the approximate approach for 1D, 2D and 3D temperature function in matrix form depending on shifted Jacobi polynomials for fractional differentiation are given, to establish a numerical solution for M-SFBHE, so a method for solution is explained, after that to determine an error bound ( , , ) is called for, an efficient error estimation for the SJ-GL-Ps will be given. The final section deals with the numerical results for the M-SFBHE.

Preliminaries and Notations
The essentials principles of the fractional calculus theory that utilized in this article will be explain.

Shifted Jacobi Polynomials for Ordinary Derivatives and Fractional Derivatives
The Jacobi polynomials which are orthogonal in the interval [−1, 1] are defined as the following formula: For transform Jacobi polynomials on a region 0 ≤ ≤ , one can procedure the replace of variables = 2 − 1 in the above formula.
The analytic form of the SJPs , ( , ) ( ) of degree is given as following: From the SJPs, the formulas that most utilized can be obtain which are the "shifted Legendre polynomials" (SLPs) ( ); the "shifted Chebyshev polynomials" (SCPs) of the first kind , ( ); the "shifted Chebyshev polynomials" of the second kind , ( ); the nonsymmetric SJPs, the two important special cases of "shifted Chebychev polynomials" of third(fourth) kinds , ( ) and , ( ); and also, the symmetric SJPs that called "Gegenbauer where (1) is the ( + 1) × ( + 1) shifted Jacobi operational matrix of derivative introduced by (19): ( − + 1, + + + + 2, + + + + 2, + + 1 + + 2, 2 + + + 2 ; 1) For example, for even we have To generalize the shifted Jacobi operational matrix of ordinary derivatives into the fractional derivative. By utilizing Eq. (18), it is obvious that where ∈ and the superscript in (1) , symbolizes matrix powers. Thus , it is clear that the SJPs for derivatives in the matrix form for integer calculus is in complete agreement with the SCPs for derivatives in the matrix form for integer calculus.  where ( ) is the ( + 1) × ( + 1) shifted Jacobi operational matrix of derivatives of fractional order in the Caputo formula and is defined by: and is given by Proof. By apply Eq. (11) … (24) Now, approximate − by ( + 1) terms of shifted Jacobi series, the obtained result: where the coefficients can be obtain as following One can notes that if = ∈ , then above theorem gives the same formula as in Eq. (18). Corollary (1):-If = = 0 and = 1, then is given as follows: By the aid of properties of the SJPs with simplification, the obtained result is: Then one can easily demonstrated that where is given as in (1). It is clear that the SJPs for derivatives in the matrix form for fractional calculus with = = 0, is in complete agreement with the SLPs for derivatives in the matrix form for fractional calculus as in (1) Then one can easily elucidate that where and , are given as in (20). It is clear that the SJPs for derivatives in the matrix form for any arbitrary fractional order with = = − 1 2 , is complete accord with the SCPs for derivatives in the matrix form for fractional calculus obtained by (20).

Shifted Jacobi Operational Matrix of Fractional Differentiation
A temperature function ( ) define for 0 ≤ ≤ 1 may be expressed in terms of the SJPs as where the coefficients are given by In practice, consider the ( 1 + 1), ( 2 + 1) and ( + 1)-terms triple SJPs with respect to , , so that where

Method for Solution
The selection of collocation points is playing significant role in the efficiency and convergence of the "collocation method". For boundary value problems, the "Gauss-Lobatto" points represent one of the principal keys utilized for approximation. It should be renowned that for a differential equation with the singularity at = 0 in the region [0, ] one is unable to apply the "collocation method" with "Jacobi-Gauss-Lobatto" points because the two assigned abscissas 0 and are necessary to use as two points from the collocation nodes. Use the "collocation method" with "Jacobi-Gauss-Lobatto" nodes to treat the two dimensional M-FSBHE; i.e., collocate this equation only at the × ( 1 − 1) × ( 2 − 1) "Jacobi-Gauss-Lobatto" points (0, ), (0, 1 ) and (0, 2 ) respectively. These equations and with initial, boundary conditions generate ( + 1) × ( 1 + 1) × ( 2 + 1) nonlinear algebraic equations by using one of the iteration methods can be solved. For any ∅( ) ∈ 1 (0, 1 ), we have This generates ( + 1) × ( 1 + 1) × ( 2 + 1) nonlinear algebraic equations, which can be solved by using a Levenberg-Marquardt MATLAB code algorithm effective and be more robust than other methods (because this algorithm combines the advantages of gradient-descent and Gauss-Newton methods) (21,22), taking ̈ as its variable, with an initial guess of all zeros, to reduce Eqs. (47-48), consequently, the approximate solution , 1 , 2 ( , , ) at the point ( 1 , 1 , , 2 , 2 , , , , ) given in Eq. (38) can be calculated. Hence, an upper bound of the maximum absolute errors achieved for the approximate solution. The convergence of the recommended method depends fundamentally on the above error bound. Moreover, the speed of convergence of "Jacobi collocation methods" was proved be fast for any choice of shifted Jacobi parameters (25,26).

Estimation of the Error Function
In this section, an efficient error estimation will present for the SJ-GL-Ps and also a technique for obtaining the corrected solution of the M-SFBHE as in equation (1)   . . . (68) The above error estimation depends on the convergence rates of expansion in Jacobi polynomial (22). Therefore, it provided reasonable convergence rates in spatial and temporal discretization.

Numerical Examples
In this section, the approach presented in section (method for solution), has been applied for solving the two dimensional M-SFBHE in the two examples based on SJ-GL-Ps. The two dimensional M-SFBHE were transformed into non-linear algebraic Eqs. (47-48) respectively. The Levenberg-Marquardt MATLAB code technique, taking ̈ as its variable, was used to minimize these equations as a set of least squares problems. This ̈ is then used in Eq. (38) to acquire our approximate surface of ( , , ).

Conclusions
In this article, an approximate approach for solving two-dimensional M-SFBHE has been introduced. The fractional derivatives are described in the Caputo form. The proposed technique depends on the collocation method of operational matrix formula for the shifted Jacobi-Gauss-Lobatto polynomials. The error of the approximate solution is estimated theoretically and the convergence average of the suggested approach in both spatial and temporal nodes graphically is investigated analyzed. The approximate calculations show that the present technique has higher accurate, good convergence (depending on Figures 2 and 4) by using few grid points.