The Implicit Numerical Method for the Radial Anomalous Subdiffusion Equation

Abstract

This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation’s solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional one. The proposed method is an extension of the fractional Crank–Nicolson method, based on the discretization of the equivalent integral-differential equation. To validate the method, the obtained results were compared with a solution obtained through the Laplace transform. The analytical solution in the image of the Laplace transform was inverted using the Gaver–Wynn–Rho algorithm implemented in the specialized mathematical computing environment, Wolfram Mathematica. The results clearly show the mutual convergence of the solutions obtained via the two methods.

Numerical Algorithms for Approximation of Fractional Integrals and Derivatives Based on Quintic Spline Interpolation

“…In general, the values of 6N coefficients c k,i (for k = 0, ..., 5 and i = 0, ..., N − 1) need to be determined. In accordance with the above-mentioned considerations, here, a system of 6N linear equations is constructed that satisfies both dependencies defined by Equation ( 9), written out using (7) and (8) as…”

“…In order to calculate, for example, the time or space fractional integrals and/or derivatives at a given time or a given point, then knowledge of the function at all previous times or positions is required. There are many applications of fractional calculus in the fields of science and engineering (see, e.g., recent works [5][6][7][8][9][10][11]), and it is impossible to list all their applications.…”