The L2-convergence of the Legendre spectral Tau matrix formulation for nonlinear fractional integro differential equations

Abstract

The operational Tau method, a well known method for solving functional equations is employed to approximate the solution of nonlinear fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. The unique solvability of the linear Tau algebraic system is discussed. In addition, we provide a rigorous convergence analysis for the Legendre Tau method which indicate that the proposed method converges exponentially provided that the data in the given FIDE are smooth. To do so, Sobolev inequality with some properties of Banach algebras are considered. Some numerical results are given to clarify the efficiency of the method.

Fractional pseudospectral integration matrices for solving fractional differential, integral, and integro-differential equations

“…Consider the following FIDE [64] C 0 D 0.5 t g(t) = f 1 (t)g(t) + f 2 (t) + √ t · 0 I 1 t g 2 (t), t ∈ [0, 1]…”

Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations

“…Since K(x, t) is smooth then s 4 is a sufficiently large number. Finally, for sufficiently large N , desired estimate can be concluded by inserting the relations (21), (22), (25) and (27) into the relation (18), ignoring some unnecessary terms and using the Fractional Poincare-Friedreich inequality, i.e., Lemma 2.5.…”

“…In [10], authors applied the collocation method to solve the nonlinear FIDE's. In [22], Mokhtary and Ghoreishi, proved the L 2 convergence of Legendre Tau method for numerical solution of nonlinear FIDE's. But many of the techniques mentioned above or have not proper convergence analysis or if any, very restrictive conditions including smoothness of the exact solution are considered in their analysis.…”

The Müntz-Legendre Tau method for fractional differential equations

“…To recover the spectral rate of convergence the authors changed the original equation into a new equation which possesses smooth solution by applying a variable transformation method and then a collocation discretization scheme with spectral rate of convergence for the new equation was presented. An extension of the spectral Tau method for numerical solutions of fractional differential equations and fractional integro-differential equations based upon classical orthogonal polynomials was presented in [19,28]. Fakhr Kazemi and Ghoreishi [16] presented a RBF(Radial basis functions) collocation method for numerical solution of multi-order fractional differential equations by scattered interpolation.…”

“…Second, in the Tau method, generally the derivative of the trial functions may be expressed according to the same trial bases. But due to the fact that the fractional derivatives of classical orthogonal polynomials are not polynomials, another polynomial approximation is needed to display them in terms of trial functions [19,28]. Consequently, we have to consider suitable basis functions in the proposed scheme.…”