A New Mixed Nonpolynomial Spline Method for the Numerical Solutions of Time Fractional Bioheat Equation

Abstract

In this paper, a numerical approximation for a time fractional one-dimensional bioheat equation (transfer paradigm) of temperature distribution in tissues is introduced. It deals with the Caputo fractional derivative with order for time fractional derivative and new mixed nonpolynomial spline for second order of space derivative. We also analyzed the convergence and stability by employing Von Neumann method for the present scheme.

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

“…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.…”