An accurate approximation technique based on Schröder operational matrices for numerical treatments of time-fractional nonlinear generalized Kawahara equation

Abstract

This study presents a novel numerical method for solving the nonlinear time-fractional generalized Kawahara equations (NTFGKE) under uniform initial boundary conditions (IBCs). To satisfy the aforementioned IBCs, we construct a class of modified Schröder polynomials (MSPs). The suggested approach makes use of operational matrices (OMs) to compute MSPs’ fractional derivatives (FDs) as well as the ordinary derivatives (ODs). These OMs are integrated alongside the spectral collocation method (SCM) to achieve the computational framework. The proposed approach’s convergence and error analysis are provided. We show three numerical test cases to illustrate the accuracy and usefulness of our methodology. The efficacy of the method is demonstrated by benchmarking the obtained numerical outcomes against established solutions. Tables and graphs demonstrate that the suggested method yields very accurate approximate solutions (ASols). Our method demonstrates significant improvements in accuracy and computational efficiency, making it particularly valuable for complex nonlinear problems. This research contributes to the field by providing a robust tool for accurately solving time-fractional equations, which are crucial in modeling various physical phenomena.