Adomian Legendre Method for Fractional Differential Equations

Abstract

Fractional differential equations have gained significant attention in various scientific and engineering disciplines due to their ability to model complex systems. The goal of this dissertation is to create a novel numerical method utilising the fractional Legendre polynomials in Adomian decomposition method for finding numerical solutions to fractional differential equations. Firstly, a generic extension and review of the modified Adomian decomposition method for fractional differential equations with variable coefficients is worked on and several examples having constant coefficients are also included to determine how accurate the existing method is for constant coefficients. We aslo predict that the method could further be extended for ψ-fractional differential equations with variable or constant coefficients. Later, the focus shifted to our primary interest, which involved the development of a numerical method based on the Adomian decomposition method using generalized Legendre polynomials for solving the initial value problem for single and multi derivative term fractional differential equations, where we employ both the Adomian polynomials firstly to reduce the non-linear term in a given FDE and later approximate integration of it in terms of Legendre polynomials to obtain the accurate solutions. Both polynomials are employed in our methods for function approximation and linearization of non-linear term help us to achieve higher accuracy, allowing for flexibility in approximation. The integral approximation of function in terms of Legendre polynomials makes ease in accomodating a variety of forcing functions. Subsequently we conducted a comparative analysis of our method with the Reproducing Kernel Hilbert Space method employing the Caputo fractional derivative to check accuracy of approximate solution obtained by our method for the initial values problems. We took into account examples for comparsion and found results with the help of MATLAB. This comparison was made to highlight how effective and accurate our findings are as compared to existing numerical methods.