Optimal homotopy asymptotic method for solving fractional relaxation-oscillation equation

Abstract

In this paper, an approximate analytical solution of linear fractional relaxation-oscillation equations in which the fractional derivatives are given in the Caputo sense, is obtained by the optimal homotopy asymptotic method (OHAM). The studied OHAM is based on minimizing the residual error. The results given by OHAM are compared with the exact solutions and the solutions obtained by generalized Taylor matrix method. The reliability and efficiency of the proposed approach are demonstrated in three examples with the aid of the symbolic algebra program Maple.

Φ-Haar Wavelet Operational Matrix Method for Fractional Relaxation-Oscillation Equations ContainingΦ-Caputo Fractional Derivative

“…This approach is based on a fractional version of Taylor's formula, which was first proposed in [17]. The numerical solution of problem ( 3) is achieved by the optimal homotopy asymptotic approach in [18], where the fractional derivative is given in the Caputo sense. In [19], a trapezoidal approximation of the fractional integral is used to get the numerical solution of problem (3) with Caputo fractional derivative.…”

A computational algorithm for simulating fractional order relaxation–oscillation equation

“…To the best of our knowledge, the following approximative and numerical schemes have been proposed for the model problem (1.1)- (1.3). These include the generelaized Taylor marix method [7], the cubic B-spline wavelet collocation method [3], the optimal homotopy asymptotic method [10], and the Aboodh transform method [26]. In this work, we are aiming to propose an approximation algorithm as extension of the above mentioned papers.…”

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“…Furthermore, they used the same procedure to obtain an approximate solution of nonlinear equations that arise in steady state flow of a fourth-grade fluid past a porous plate, and for the solution of a nonlinear equation arising in heat transfer. OHAM has been successfully employed to solve different kinds of differential equations in science and engineering [18][19][20][21][22][23][24].…”