A Pseudo-spectral Method with Least Squares Approach and Monomial Basis Functions for Fractional Ordinary Differential Equations

Abstract

The present work combines the pseudo-spectral method with least square technique to approximate non-linear and linear differential equations that are bounded and it makes use of monomials as trial functions. The pseudo-spectral method involves the approximation of solutions to differential equations by taking the sum of truncated basis functions, and this feature differentiates this method from difference-based methods. The idea of the least square technique is to optimally minimize the sum of squares of the residual function of the differential equation. The Caputo fractional differential operator is generally used given that it allows for the inclusion of the conditions at the initial state and the boundaries in formulating the equation(s). Non-linear fractional differential equations are linearized using the quasilinearization approach, which is based on the Newton-Raphson method. The resulting linear algebraic equations are regularized using the Tikhonov regularization technique due to the density of the equations. Some problems are solved to demonstrate the efficiency of the proposed method and display its convergence and accuracy, respectively, of approximate solutions obtained. Comparison of the numerical solutions and exact solutions (where obtainable) is conducted, and the numerical results are further presented in tabular form and graphically. The results obtained show that the proposed method is computationally efficient. The process of regularizing the numerical schemes of non-linear fractional differential equations was also shown to give more accurate solutions.