A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations

Abstract

The time-fractional coupled Korteweg---de Vries (KdV) system is a generalization of the classical coupled KdV system and obtained by replacing the first order time derivatives by fractional derivatives of orders $$\nu _1$$?1 and $$\nu _2$$?2, $$(0<\nu _1,\nu _2\le 1).$$(0<?1,?2≤1). In this paper, an accurate and robust numerical technique is proposed for solving the time-fractional coupled KdV equations. The shifted Legendre polynomials are introduced as basis functions of the collocation spectral method together with the operational matrix of fractional derivatives (described in the Caputo sense) in order to reduce the time-fractional coupled KdV equations into a problem consisting of a system of algebraic equations that greatly simplifies the problem. In order to test the efficiency and validity of the proposed numerical technique, we apply it to solve two numerical examples.