A computational wavelet method for variable-order fractional model of dual phase lag bioheat equation

Abstract

Abstract In this study, we focus on the mathematical model of hyperthermia treatment as one the most constructive and effective procedures. Considering the sophisticated nature of involving phenomena in bioheat transfer inside a living tissue, several models with different levels of simplifications have been proposed. One of the general forms of the bioheat transfer equation which is introduced and studied in this paper for the first time, is the 2D-transient, dual phase lag (DPL), variable-order fractional energy equation. For finding the numerical solution of this general case, we propose an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs). Precisely, the variable-order fractional derivatives of the model are discretized in the first stage, and then the response of the model is expanded by the 2D LWs. Consequently, the main problem is transformed into an equivalent system of algebraic equations, which can be simply tackled. The stability of the proposed method is examined theoretically and experimentally. Also, the procedure is described for one example to examine the computational efficiency of method. The experimental results show the stability and spectral accuracy of the proposed method. According to the achieved results, increasing the fractional order from 0.1 to 1.0, leads to increment of maximum tissue temperatures by about 29% near the center of the targeted region.