Analytical solutions for time-fractional diffusion equation with heat absorption in spherical domains

Abstract

Under the time-variable Dirichlet condition, the time-fractional diffusion equation with heat absorption in a sphere is taken into consideration. The time-fractional derivative with the power-law kernel is used in the generalized Cattaneo constitutive equation of the thermal flux. The Laplace transform and a suitable transformation of the independent variable and function are used to determine the analytical solution of the problem in the Laplace domain. To obtain the temperature distribution in the real domain, the inverse Laplace transforms of two functions of exponential type are obtained. These formulae are new in the literature. The particular cases of the classical Cattaneo law of heat conduction and of the classical Fourier's law are obtained from the solutions corresponding to the time-fractional generalized Cattaneo law.

Keywords

• Fractional derivative
• Generalized Cattaneo’s law, Analytical solutions
• Dirichlet conditions