Numerical Solution of Nonlinear Volterra Integro-Differential Equations of Fractional Order by Using Adomian Decomposition Method

Abstract

In this work we present a computational method for for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on Adomian Decom-position Method. Convergence analysis is dependable enough to estimate the maximum absolute truncated error of the Adomian series solution. Numerical example is included to demonstrate the validity and applicability of the method.

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].…”

Usage of the Modified Variational Iteration Technique for Solving Fredholm Integro-Differential Equations

“…Recently, many authors focus on the development of numerical and analytical techniques for integrodifferential equations. For instance, we can remember the following works: Abbasbandy and Elyas [2] studied some applications on variational iteration method for solving system of nonlinear Volterra integro-differential equations, Hamoud and Ghadle [6] applied the hybrid methods for solving nonlinear Volterra-Fredholm integrodifferential equations, Alao et al [4] used Adomian decomposition and variational iteration methods for solving integro-differential equations, Yang and Hou [21] applied the Laplace decomposition method to solve the…”

Existence and Convergence Results for Caputo Fractional Volterra Integro-Differential Equations

“…The fractional integro-differential equations have attracted much more interest of mathematicians and physicists which provides an efficiency for the description of many practical dynamical arising in engineering and scientific disciplines such as, physics, biology, electrochemistry, chemistry, economy, electromagnetic, control theory and viscoelasticity [2,5,8,7,9,10,17,18,20]. In recent years, many authors focus on the development of numerical and analytical techniques for fractional integrodifferential equations.…”

“…For instance, we can remember the following works. An application of fractional derivatives was first given in 1823 by Abel [1] who applied the fractional calculus in the solution of an integral equation that arises in the formulation of the Tautochrone problem, Al-Samadi and Gumah [3] applied the homotopy analysis method for fractional SEIR epidemic model, Zurigat et al [23] applied HAM for system of fractional integro-differential equations, Yang and Hou [20] applied the Laplace decomposition method to solve the fractional integro-differential equations, Mittal and Nigam [18] applied the Adomian decomposition method to approximate solutions for fractional integro-differential equations, and Ma and Huang [17] applied hybrid collocation method to study integro-differential equations of fractional order. Moreover, properties of the fractional integro-differential equations have been studied by several authors [11,12,21,23].…”