On the solution of nonlinear generalized Caputo-Riesz fractional EFK and KS equations

This paper is concerned with a space-time fractional partial differential equation (FPDE) which gives a generalization of a class of fourth-order partial differential equation. In the newly proposed FPDE, the spatial derivative is in Riesz-Feller fractional derivative type and the derivative of time in Caputo sense. The studied equation represents the Swift-Hohenberg (SH), the extended Fisher Kolmogorov equations (EFK) and Kuramoto Sivashinsky (KS) equations in a generalized form. We describe the application of the optimal homotopy analysis method (OHAM) to obtain an approximate solution for the suggested fractional initial value problem. An averaged-squared residual error function is defined and used to determine an optimal convergence control parameter. Two different numerical examples are considered, the EFK equation and the KS equation, to justify the efficiency and the accuracy of the adopted approximation technique. The dependence of the solution on the order of the fractional derivative in space and time and model parameters is investigated via various graphs of the obtained optimal homotopy series.