On the Solution of Time-Fractional Diffusion Models

In many situations, the analysis of viscoelastic materials, like polymers, takes benefit from the introduction of fractional operators in the mathematical formalization. In addition, fractional differential models have been applied in a wide variety of fields, from biology to thermodynamics, from diffusion of information to dehydration/rehydration of food. Thus, a great interest is paid both in the analytical and in the numerical solution of fractional differential problems. The present paper considers a class of time-fractional diffusion problems with Dirichlet boundary conditions. Using Duhamel's principle, the analytical solution is found. As usual in this context, the solution is given in series form and depends on the Mittag-Leffler function. We suggest a computational procedure to evaluate the solution with high accuracy, in a computing environment. Some test examples are presented both in the subdiffusion and in the superdiffusion case, to illustrate the behavior of the solution for different values of the fractional index. Test cases have been carried out in Matlab.