On Coupled Time-Fractional Heat-Equations

Abstract

In this paper, we consider a coupled time-fractional heat equations, one homogeneous and the other a non-homogeneous. Analytic solutions of the involved variables, the main and auxiliary, are presented with a special attention on the fractional order derivative α on the range of (0, 1]. The main variable may represent some density function while the auxiliary function plays the role of a source function. We consider the well-posedness of solution of the equation in the sense of Lipschitz and energy method. Scaling becomes important tools for the study of the solution near the singularity and larger numerical resolutions. We present the analytic solutions of the system using the method of Fourier and Laplace transforms, where one finds the solution in terms of a Green’s function, which is a scaled Mittag-Leffler’s function. The behaviour of the main and auxiliary variables is depicted using numerical approximation.