Green–CAS Wavelet Schemes with Fast Algorithms for Generalized Nonlinear Time–Space Fractional Partial Differential Equations

Abstract

This study aims to acquire numerical schemes to detect the numerical solutions of fractional partial differential equations of arbitrary order, subject to prescribed initial and boundary conditions. This novel approach, referred to as the Green-CAS technique, integrates Green’s function with CAS wavelets to construct an efficient and systematic computational framework. The present approach is not only simple and easy to implement due to the Green function, but it also eliminates the need for operational matrices for boundary conditions. To further enhance computational efficiency, a fast algorithm is coupled with the Green-CAS wavelets, enabling effective handling of fractional partial differential equations. While tackling the nonlinear fractional partial differential equation of arbitrary order, the Picard iterative method is employed to transform the equation into a sequence of linear problems, which are then solved using the proposed techniques. Moreover, the order of convergence for two parameters has also been demonstrated in the  convergence analysis, which further strengthens the effectiveness of the proposed technique. To show the validity and accuracy of the recommended techniques, the acquired outcomes are compared with the conventional CAS wavelets and various other renowned techniques. In addition, the results of various applications are presented in the form of graphics and tables, which elaborate on the effectiveness and correctness of the discussed method.