Operational-matrix-based algorithm for differential equations of fractional order with Dirichlet boundary conditions

Abstract

The fractional differential equations (FDEs) are ground-breaking tools to demonstrate the complex-nature scientific systems in the form of non-linear behavior endorsed by the scientific community to develop some new and accurate mathematical methods. The main objective of this paper is the development of an extended mathematical algorithm based on the Gegenbauer wavelet method for the fractional-order problem. The Gegenbauer wavelet operational matrix with their derivative is proposed in our study. Some new operational matrices for the derivative of fractional order with Dirichlet boundary condition is purposed by introducing the piecewise function. Furthermore, a successful use to analyze the solution for the set of algebraic equations governed through the extended Gegenbauer wavelets technique is performed. Analytical solutions of the mentioned problem are effectively obtained, and a comparative study is presented. The outcomes are obtained via the modified Gegenbauer wavelet method by endorsing the accuracy and effectiveness of the mentioned technique. The convergence and error bound analysis are enclosed in our investigation. It is further verified that the algorithm is quite accurate, and an efficient mathematical tool is used to tackle the nonlinear fractional-order complex-nature problems.