Optimal collocation method for time-fractional advection-dispersion equation using modified generalized Laguerre polynomials and particle swarm optimaization algorithm

Abstract

In this article, time-fractional advection-dispersion equation is considered. Fractional derivative is in the Caputo sense and for approximating the first and second derivatives, the modified generalized Laguerre polynomials (MGLP s) have been used. The MGLP s (L α,β n (x)) have two parameter α > −1 and β > 0. These polynomials, orthogonal polynomial on the [0,∞) by weight function ω α,β(x) = x αe −βx. For solving these time-fractional advection-dispersion equations, we introduce optimal MGLP s collocation method. The error of the proposed algorithm depends on the parameters of α and β. The (P SO) algorithm is used to find the optimal parameters so that the error is minimized. In other words, we try to find the optimal value for parameters α and β so that the error of the method is minimized. In practice, since the exact solution is not available, we will face a problem to measure the error. To overcome this problem, the best parameters for approximating the function that describes the initial condition with the help of P SO algorithm are found and these parameters are used to better approximate the exact solution of the problem. A few numerical experiments are carried out to support the theoretical claims. The presented examples confirm that the optimal parameters for the initial condition can reduce the error of the method.