The development of numerical methods for fractional calculus is an extensive area of contemporary mathematics with various applications in semiconductors, nanophysics, plasma and thermodynamics, acoustics, and quantum optics. These lead to solving problems with fractional derivatives for conservation of mass, advection dispersion equations, time-space diffusion equations, and more.

The authors of this paper studied the numerical solution of the time fractional diffusion problem with boundary and initial conditions in a bounded domain for one to three spatial dimensions. The differential operator comprises an alpha Caputo fractional derivative. The existence of a unique smooth solution of the problem is assumed.

At first, for the Caputo derivative, a second-order finite difference scheme on a special monotonically decreasing non-uniform mesh with respect to time is constructed and a corresponding error analysis is performed. The obtained results are applied to construct a semi-discrete difference scheme for solving the considered fractional diffusion problem. Unconditional stability and convergence of the semi-discrete scheme are proved in the H¹ norm. Subsequently, a fully discrete difference scheme based on the fourth-order compact difference method is derived and analyzed. Some numerical experiments are also presented.

It should be noted that along with the formal standard discretization, a felicitous moving local refinement technique is introduced to improve the temporal accuracy of the numerical solution. In fact, it could be highlighted in a more complex example to show how the method solves specific real physical and engineering problems, along with the considered mathematical problem.