Abstract
In this article, a comparative study between optimal homotopy asymptotic method and multistage optimal homotopy asymptotic method is presented. These methods will be applied to obtain an approximate solution to the seventh-order Sawada-Kotera Ito equation. The results of optimal homotopy asymptotic method are compared with those of multistage optimal homotopy asymptotic method as well as with the exact solutions. The multistage optimal homotopy asymptotic method relies on optimal homotopy asymptotic method to obtain an analytic approximate solution. It actually applies optimal homotopy asymptotic method in each subinterval, and we show that it achieves better results than optimal homotopy asymptotic method over a large interval; this is one of the advantages of this method that can be used for long intervals and leads to more accurate results. As far as the authors are aware that multistage optimal homotopy asymptotic method has not been yet used to solve fractional partial differential equations of high order, we have shown that this method can be used to solve these problems. The convergence of the method is also addressed. The fractional derivatives are described in the Caputo sense.
Highlights
Some phenomena in many disciplines are usually modeled by fractional differential equations or fractional integrodifferential equations
Fractional differential equations have been solved by some series methods, homotopy analysis transform [1], Adomian decomposition [2,3,4], fractional Jacobi collocation [5], variational iteration [6, 7], differential transform [8, 9], homotopy perturbation (HP) [10,11,12], homotopy analysis (HA) [13,14,15], least squares [16], and others [17,18,19,20,21]
OHAM results in to satisfactory solutions on short domains, but when the interval becomes longer, the accuracy of the method decreases, so a new approach was proposed by Anakira et al, which is called multistage optimal homotopy asymptotic method (MOHAM) that is suitable for analytic approximate solutions for any long interval [23, 24]
Summary
Some phenomena in many disciplines are usually modeled by fractional differential equations or fractional integrodifferential equations. An advantage of OHAM, in comparison with HAM, is that there is no need of h-curves study This method provides us with a convenient way to control the convergence of the solution series and allows the adjustment of the convergence region, wherever it is needed. Several authors have demonstrated the effectiveness, generalizability, and reliability of this method Another advantage of OHAM is built in convergence a criterion that is controllable. OHAM results in to satisfactory solutions on short domains, but when the interval becomes longer, the accuracy of the method decreases, so a new approach was proposed by Anakira et al, which is called multistage optimal homotopy asymptotic method (MOHAM) that is suitable for analytic approximate solutions for any long interval [23, 24]. The approximate solutions obtained from both methods are compared with the exact solution
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