General one-dimensional model of the time-fractional diffusion-wave equation in various geometries

Abstract

This paper deals with the analysis of the time-fractional diffusion-wave equation as one-dimensional problem in a large plane wall, long cylinder, and sphere. The result of the analysis is the proposal of one general mathematical model that describes various geometries and different processes. Finite difference method for solving the time-fractional diffusion-wave equation using Grünwald-Letnikov definition for homogeneous or inhomogeneous material and for homogeneous or inhomogeneous boundary conditions is described. Dirichlet, Neumann and Robin boundary conditions are considered. Implementation of numerical methods for explicit, implicit, and Crank-Nicolson scheme were realised in MATLAB. Finally, illustrative examples of simulations using the developed toolbox are presented.

Transient Heat Conduction in a Semi-Infinite Domain with a Memory Effect: Analytical Solutions with a Robin Boundary Condition

“…Transport processes in non-local media are mainly modeled by the time-fractional diffusion equations [1][2][3][4] which macroscopically perform the fractal natures of the materials as non-localities in time [5,6]. Especially in the case of heat conduction and anomalous diffusion, the transient heat conduction via the application of the time-fractional Caputo derivative has been thermodynamically formulated in [7,8] applying the fading memory formalism.…”

“…Especially in the case of heat conduction and anomalous diffusion, the transient heat conduction via the application of the time-fractional Caputo derivative has been thermodynamically formulated in [7,8] applying the fading memory formalism. Recently, various techniques upon various initial and boundary conditions have been applied to solve time-fractional heat transfer equations [1][2][3][4][5][6][7][8][9][10][11][12][13][14] but a review of these studies is beyond the scope of this work.…”

“…The present analysis, and solutions developed, address the time-fractional heat conduction (which can also be considered as an anomalous diffusion equation) with the Robin boundary condition. The Robin boundary condition appears in many applied diffusion problems such as solute rejection [9,14,15], solidification of alloys [16,17], and heat transfer at the boundary by convection (as in this article) [18], and the dominating solutions are numerical [6,[9][10][11][12][15][16][17] considering finite domains [11,[14][15][16][17], while solutions in the semiinfinite domain are rare [9]. Moreover, estimates concerning the existence and uniqueness of the problem solution have been developed in [9,11,19].…”