Eur. Phys. J. Plus (2018) 133: 412
https://doi.org/10.1140/epjp/i2018-12204-x

Regular Article

A comprehensive numerical study of space-time fractional bioheat equation using fractional-order Legendre functions

¹ Department of Mechanical Engineering, Fasa University, Fasa, Iran
² Department of Mathematics, Shiraz University of Technology, Shiraz, Iran
³ Department of Civil Engineering, Fasa University, Fasa, Iran
⁴ Department of Statistics, Fasa University, Fasa, Iran

^* e-mail: heydari@sutech.ac.ir

Received: 25 July 2018
Accepted: 10 August 2018
Published online: 12 October 2018

Abstract

The determination of a temperature field within the living tissues and especially during the thermal therapies requires a comprehensive modeling due to the involved complex heat transfer mechanisms. Anomalous structure of the blood vessels, history-dependent heat transfer, temperature-dependent metabolic heat generation and various types of available thermal therapy procedures are some of the difficulties arising for a realistic modeling. To tackle the mentioned problem, in the present investigation the general form of the space-time fractional heat conduction equation with locally variable initial condition and time-dependent boundary conditions is solved. Moreover, the heat generation source term is assumed to be a function of both time and space. A computational method based on the fractional-order Legendre functions (F-OLFs) and Galerkin method is proposed to solve the problem. The main advantage of the proposed method is that it obtains a global solution for the problem. In addition, the method reduces the problem under consideration to a simpler problem that consists of solving a system of nonlinear algebraic equations. The developed mathematical method is applied to three common clinical thermal therapies: instantaneous and gradual internal magnetic heat generation and skin laser exposure. The effect of various physiological and clinical parameters is investigated. According to the obtained results, the temperature field is a strong function of the time and space fractional order. Additionally, it is shown that the instantaneous heat source which is commonly utilized in the literature leads to substantial different results in comparison to the more realistic case of gradually increasing heat generation.