Meshless Technique for the Solution of Time-Fractional Partial Differential Equations Having Real-World Applications

Abstract

In this article, radial basis function collocation scheme is adopted for the numerical solution of fractional partial differential equations. This method is highly demanding because of its meshless nature and ease of implementation in high dimensions and complex geometries. Time derivative is approximated by Caputo derivative for the values of $\alpha \in \left(0,1\right)$ and $\alpha \in \left(1,2\right)$ . Forward difference scheme is applied to approximate the 1st order derivative appearing in the definition of Caputo derivative for $\alpha \in \left(0,1\right)$ , whereas central difference scheme is used for the 2nd order derivative in the definition of Caputo derivative for $\alpha \in \left(1,2\right)$ . Numerical problems are given to judge the behaviour of the proposed method for both the cases of $\alpha$ . Error norms are used to asses the accuracy of the method. Both uniform and nonuniform nodes are considered. Numerical simulation is carried out for irregular domain as well. Results are also compared with the existing methods in the literature.