Numerical Methods for Fractional Percolation Equation with Riesz Space Fractional Derivative

Abstract

The aims of this paper are to propose mixed fractional derivative, clarify the problems suggested by the tacit approach of finite difference, and the study of consistency, stability, and convergence methods. An effective computational approach for solving Factional Percolation Equations Riesz Space Mixed Fractional Derivative (FPERSMFD) using Implicit Finite Difference Methods (IFDMs) is proposed. The moved Grunwald estimate is analyzed for the mixed fractional derivatives. However, the given method is successfully applied to the mixed fractional derivative classes with Riesz space in order to solve various Fractional Percolation Equations (FPEs). The numerical method of fractional order is defined as consistent, stable, and convergent. Four illustrative examples are given to illustrate the efficiency and the validity of the algorithm proposed and to compare the results with the exact solution. For these four examples and from the Tables illustrated in this work, a high-precision approximation, the approximate solution values of the various grid points provided by the implicit finite difference methods are similar to the exact solution values. Different and random values for fractional derivative have been used to prove the efficiency of this proposed method where the error equals to zero between the proposed method and the exact method. It can also be seen that the accuracy improves with the approximation order. The data have been presented in Tables by using the software package MathCAD 12 and MATLAB when necessary. In solving Factional percolation equations Riesz space mixed fractional derivative, the implicit finite difference methods have appeared to be effective and reliable.