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

Main Article Content

Firas Amer Al-Saadawi
Hameeda Al-Humedi1

Abstract

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.

Downloads

Download data is not yet available.

Article Details

How to Cite
1.
Al-Saadawi FA, Al-Humedi1 H. The Numerical Technique Based on Shifted Jacobi-Gauss-Lobatto Polynomials for Solving Two Dimensional Multi-Space Fractional Bioheat Equations. Baghdad Sci.J [Internet]. 2020Dec.1 [cited 2021Mar.7];17(4):1271. Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/3747
Section
article

References

Saadatmandi A. Dehghan M. A new operational matrix for solving fractional-order differential equations. Comput Math Appl. 2010 Feb; 59(3): 1326-1336.

Bhrawy A. A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations. Numer. Algor. 2015 Dec; 73(1): 91-113.

Singh J, Gupta P, Rai K. Solution of fractional bioheat equations by finite difference method and HPM. Math. Comput. Model. 2011; 54: 2316-2325.

Jiang X. Qi H. Thermal wave model of bioheat transfer with modified Riemann-Liouville fractional derivative. J Phys. A Math. Theor. 2012 Nov; 45: 1-11.

Damor R, Kumar S, Shukla A. Numerical solution of fractional bioheat equation with constant and sinusoidal heat flux condition on skin tissue. Amer. J Math. Anal. 2013; 1(2): 20-24.

Ezzat M, AlSowayan N, Al-Muhiameed Z, Ezzat S. Fractional modeling of Pennes’ bioheat transfer equation. Heat Mas Trans. 2014 Jul; 50(7): 907-914.

Ferrás L, Ford N, Nobrega J, Rebelo M. Fractional Penns’ bioheat equation: Theoretical and numerical studies. Fract. Calc. Appl. Anal. 2015 Aug; 18(4): 1080-1106.

Kumar P, Kumar D, Rai K. A mathematical model for hyperbolic space-fractional bioheat transfer during thermal therapy. Procedia Eng. 2015 Dec; 127: 56-62.

Qin Y, Wu K. Numerical solution of fractional bioheat equation by quadratic spline collocation method. J Nonlinear Sci Appl. 2016 Jul; 9: 5061-5072.

Damor R, Kumar S, Shukla A. Solution of fractional bioheat equation in term of Fox’s H-function. Springer Plus. 2016; 111(5):1-10.

Kumar D, Rai K. Numerical simulation of time fractional dual-phase-lag model of heat transfer within skin tissue during thermal therapy. J Therm Biol. 2017 Jul; 67: 49-58.

Roohi R, Heydari M, Aslami M. A comprehensive numerical study of space-time fractional bioheat equation using fractiona-order Legendre functions. Eur Phys J Plus. 2018 Oct; 133(412): 1-15.

Hosseninia M, Heydari M, Roohi R. , Avazzadeh Z. A computational wavelet method for variable-order fractional model of dual phase lag bioheat equation. J Comput Phys. 2019 Oct; 395(15): 1-18.

Al-Saadawi F, Al-Humedi H, Fractional shifted Legendre polynomials for solving time-fractional Bioheat equation, J Basrah Research sci, 2019; 45(2): 108-118.

Dehghan M, Sabouri M. A spectral element method for solving the Pennes bioheat transfer equation by using triangular and quadrilateral elements. Appl Math Model. 2012 Dec; 36(12):6031-6049.

Chen Y, Sun Y, Liu L. Numerical solution of fractional partial differential equations with variable coefficients using generalized fractional-order Legendre functions. Appl Math Comput. 2014 Oct; 244(1): 847-858.

Huang Q, Zhao F, Xie J, Ma L, Wang J, Li Y. Numerical approach based on two-dimensional fractional-order Legendre functions for solving fractional differential equations. Disc Dynam Natu Soc. 2017; 1-12.

Doha E, Bhrawy A, Ezz-Eldien S. A new Jacobi operational matrix: An application for solving fractional differential equations. Appl Math Model. 2012 Dec; 36(10): 4931-4943.

Doha E, Bhrawy A, Ezz-Eldien S. Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equation. Appl Math Model. 2011 Dec; 35(12): 5662-5672.

Doha E, Bhrawy A, Ezz-Eldien S. A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput and Math Appl. 2011 Sep; 62(5): 2364-2373.

Moré J. The Levenberg-Marquardt algorithm implementation and theory. Numer Anal.1978; 105-116.

Ghasemi A, Gitizadeh M. Detection of illegal consumers using pattern classification approach combined with Levenberg-Marquardt method in smart grid. Inter J Elec Pow Ener Sys. 2018 Jul; 99,363-375.

Main M, Deves L. The convergence rates of expansions in Jacobi polynomials. Numer. Math. 1977; 27(2): 219-255.

Bavinck H. On absolute convergence of Jacobi series. J Appr Theo. 1971; 4(4): 387-400.

Eslahchi M, Dehghan M, Parvizi M. Application of the collocation method for solving nonlinear fractional integro-differential equations. J Comput Appl Math. 2014 Jul; 257: 105-128.

Ma X, Hang C. Spectral collocation method for linear fractional integro-differential equations. Appl. Math. Model. 2014 Feb; 38(4): 1434-1448.

Yüzbaşı Ş. Shifted Legendre method with residual error for delay linear Fredholm integro-differential equations. J Taibah Univ Sci. 2017 Mar;11(2),344-352.