Derivation and Implementation of the Collocation Methods for Solving Single and Multi Time-Fractional Order Differential Equations

Standard collocation (SCM) and perturbed collocation (PCM) are utilized as effective numerical techniques for solving fractional-order differential equations (FODEs) which focus on constructing orthogonal polynomials to serve as basis functions for approximating the solutions to these equations. The approach began by assuming an approximate solution, expressed in the constructed orthogonal polynomials. These assumed solutions were then substituted into the original FODEs. Following this, the problem was converted into a system of algebraic linear equations by collocating the equations at evenly spaced interior points. Numerical examples and the results indicated that the SCM and PCM are easy, efficient, and in good agreement compared with some existing methods and the results presented in the tables and graphs unequivocally demonstrate the efficacy of the proposed methods in solving fractional-order differential equations, yielding solutions of remarkable accuracy. However, the SCM and PCM exhibit comparable accuracy, making it difficult to identify a single superior approach, we conclude that both the proposed methods are effective and viable options for solving fractional order differential equations.