Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsov equation in dusty plasma

Abstract

AbstractWe propose a new modification of homotopy perturbation method (HPM) called theδ-homotopy perturbation transform method (δ-HPTM). This modification consists of the Laplace transform method, HPM, and a control parameterδ. This control convergence parameterδin this new modification helps in adjusting and controlling the convergence region of the series solution and overcome some limitations of HPM and HPTM. Theδ-HPTM and q-homotopy analysis transform method (q-HATM) are considered to study the generalized time-fractional perturbed$(3+1)$(3+1)-dimensional Zakharov–Kuznetsov equation with Caputo fractional time derivative. This equation describes nonlinear dust-ion-acoustic waves in the magnetized two-ion-temperature dusty plasmas. The selection of an appropriate value ofδinδ-HPTM and the auxiliary parametersnandħin q-HATM gives a guaranteed convergence of series solution, but the difference between the two techniques is that the embedding parameterpinδ-HPTM varies from zero to nonzeroδ, whereas the embedding parameterqin q-HATM varies from zero to$\frac{1}{n}, n\geq{1}$1n,n≥1. We examine the effect of fractional order on the considered problem and present the error estimate when compared with exact solution. The outcomes reveal complete reliability and efficiency of the proposed algorithm for solving various types of physical models arising in sciences and engineering. Furthermore, we present the convergence and error analysis of the two methods.