A semi-analytical method for 1D, 2D and 3D time fractional second order dual-phase-lag model of the heat transfer

Abstract

In this paper we present a new numerical technique for 1D, 2D, and 3D time-fractional second order dual-phase-lag model of heat transfer. The problem is formulated as a solution of a time fractional partial differential equation (TFPDE) subjected to specific initial and boundary conditions. In the framework of the presented technique we first find an analytical function which satisfies the boundary condition of the original problem at each time moment. Then, using this function we transform the original problem into the problem with zero boundary conditions. This problem admits of the approximate solution in the form of the Fourier series expansion. Applying the Fourier expansion, the original TFPDE is reduced into a sequence of independent multi-term fractional ordinary differential equations (FODEs). These FODEs are solved separately, using the efficient backward substitution method and the Mu¨ntz polynomials in the approximate solution of each FODE. The accuracy and convergence of the proposed method are examined experimentally with several examples. In particular, the equations with coefficients that correspond to the real thermophysical parameters of the biological tissue have been considered. In the last example we demonstrate the ability of the method presented to solve 3D FPDEs with time-dependent coefficients.

Keywords

• Heat transfer
• Dual-phase-lag model
• Time fractional
• Fourier transform
• Numerical method