Operational Tau approximation for a general class of fractional integro-differential equations

Abstract

Abstract. In this work, an extension of the algebraic formulation of the operational Tau method (OTM) for the numerical solution of the linear and nonlinear fractional integro-differential equations (FIDEs) is proposed. The main idea behind the OTM is to convert the fractional differential and integral parts of the desired FIDE to some operational matrices. Then the FIDE reduces to a set of algebraic equations. We demonstrate the Tau matrix representation for solving FIDEs based on arbitrary orthogonal polynomials. Some advantages of using the method, error estimation and computer algorithm are also presented. Illustrative linear and nonlinear experiments are included to show the validity and applicability of the presented method.Mathematical subject classification: 65M70, 34A25, 26A33, 47Gxx.

Iterative refinement for a system of linear integro-differential equations of fractional type

“…One of the most popular methods which attracted the attention of many mathematicians is the Adomian decomposition method [1,2]. Among the other methods are the differential transform method [3], Taylor expansion methods [4,5], spectral methods [6], as well as the variational iteration and homotopy perturbation methods [7,8] -only to name a few. The widespread of these methods enabled the subject of fractional calculus to become a well established discipline in applied mathematics, and thus finding its way into the toolbox of many commercial packages like MATLAB and MATHEMATICA.…”

A numerical method for solving systems of higher order linear functional differential equations

“…In physics, chemistry, biology and engineering, a lot of problems are modelled by di erential equations, delay differential equations [10][11][12][13] and their systems [1][2][3][4][5][6][7][8][9][14][15][16][17].…”

Solving Nonlinear Fractional Integro-Differential Equations of Volterra Type Using Novel Mathematical Matrices

“…In fact, such equations appear in signal processing [1], strength of disorder materials [2], heat conduction [3], cosine transform [4], unmixed mechanics [5], econometrics [6], fluid dynamics [7], nuclear reactor dynamics, acoustic waves [8], and glass forming process (more details about the sources, where these equations arise, can be found in physics, biology, and engineering applications books). There are different methods for approximating the solutions of Fredholm integro-differential equations such as the Galerkin method, wavelet Galerkin method [9,10], Chebyshev wavelet method [11], Taylor method [12], cosine and sine (CAS) wavelet method [13], Legendre method, Legendre wavelet method [14,15], the Adomian decomposition method [16], differential transform method, generalized differential transform method [17,18], Tau approximation method [19], Chebyshev pseudospectral method [20], Jacobi operational method, Jacobi spectral-collocation method [21,22], and the hybrid function method [23].…”