Approximate solution of integro-differential equation of fractional (arbitrary) order

Abstract

In the present paper, we study the integro-differential equations which are combination of differential and Fredholm-Volterra equations that having the fractional order with constant coefficients by the homotopy perturbation and the variational iteration. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented.

Approximate solutions of Volterra-Fredholm integro-differential equations of fractional order

“…Therefore, finding more accurate solutions using numerical schemes can be helpful. Some numerical algorithm for solving integrodifferential equation of fractional order can be summarized as follows: but not limited to; Adomian decomposition method [16,18,19], Laplace decomposition method [32], Taylor expansion method [9], least squares method [17] differential transform method [5,21], Spectral collocation method [14], Legendre wavelets method [24,26], Haar wavelets method [7], Chebyshev wavelets method [29,33,37], piecewise collocation methods [23,36], Chebyshev pseudo-spectral method [10,31], homotopy analysis method [1,35,38], homotopy perturbation method [6,20,25] and variational iteration method [6,20].…”

On the local transformed based method for partial integro-differential equations of fractional order

“…Therefore researchers have solved these problems numerically. For the solution of fractional integro-differential equations (FIDEs) various numerical method have been used over the years, some of these may be found in the references [9,25,26,32,33]. Brunner and co-authors have developed very efficient collocation methods for the solution of Volterra integral equations, some of these can be found in the references [2,3,7], and some other accurate numerical methods are referred in [8,27,31,42].…”

A method for fractional Volterra integro-differential equations by Laguerre polynomials

“…Since the fractional calculus has attracted much more interest among mathematicians and other scientists, the solutions of the fractional differential and integro-differential equations have been studied frequently in recent years [2][3][4][5][6][7][8][9][10]. The methods that are used to find the solutions of the fractional Volterra integro-differential equations are given as Adomian decomposition [11], Bessel collocation [12,13], CAS wavelets [14], Chebyshev pseudo-spectral [15], cubic B-spline wavelets [16], Euler wavelet [17], fractional differential transform [18], homotopy analysis [19], homotopy perturbation [20][21][22][23], Jacobi spectral-collocation [24,25], Legendre collocation [26], Legendre wavelet [27], linear and quadratic interpolating polynomials [28], modification of hat functions [29], multi-domain pseudospectral [30], normalized systems functions [31], novel Legendre wavelet Petrov-Galerkin method [32], operational Tau [33], piecewise polynomial collocation [34], quadrature rules [35], reproducing kernel [36], second Chebyshev wavelet [37], second kind Chebyshev polynomials [38], sinccollocation [39,40], spline collocation [41], Taylor expansion…”