SOLVING THE FRACTIONAL FORNBERG-WHITHAM EQUATION BY MEANS OF THE OPTIMAL q-HOMOTOPY ANALYSIS METHOD (Oq-HAM)

Abstract

The main aim of this paper is to propose a new and simple algorithm namely optimal q-homotopy analysis method (Oq-HAM), to obtain the approximate analytical solutions to solve the nonlinear Fornberg-Whitham equation with fractional time derivative.  The fractional derivatives are taken in the Caputo sense.  Comparison of Oq-HAM with the homotopy analysis method (HAM) and the
homotopy perturbation method (HPM) are made.  The results reveal that the Oq-HAM has more accuracy than the others.  Finally, numerical example is given to illustrate the accuracy and stability of this method.  Comparison of the approximate solution with the exact solutions also we show that the proposed method is very efficient and computationally attractive.  A new efficient approach is proposed to obtain the optimal value of convergence controller parameter ħ to guarantee the convergence of the obtained series solution.