The Numerical Approximation of the Bioheat Equation of Space-Fractional Type Using Shifted Fractional Legendre Polynomials

Abstract

The aim of this paper is to employ the fractional shifted Legendre polynomials (FSLPs) in the matrix form to approximate the fractional derivatives and find the numerical solutions of the one-dimensional space-fractional bioheat equation (SFBHE). The Caputo formula was utilized to approximate the fractional derivative. The proposed methodology applied for two examples showed its usefulness and efficiency. The numerical results showed that the utilized technique is very efficacious with high accuracy and good convergence.

Numerical Approximations of a One-Dimensional Time-Fractional Semilinear Parabolic Equation

“…Suppose that 𝑢 ̃𝑖 𝑛 is the approximate solution of equation ( 1 Based on (15), the above error formula can be written in matrix form as follows:…”

“…Many effective methods, such as the finite difference method, spectral method and finite element method, have been used to solve time (space) fractional differential equations, see for instance [7][8][9]. In fact, many authors have focused on numerical solutions of linear types of time (space)-fractional differential equations, see for instance [9][10][11][12][13][14][15][16], whereas other semilinear or nonlinear types have been considered by only a few authors, see for instance [17][18][19][20][21]. However, they are still in the early stage of research.…”