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

We propose a new modification of homotopy perturbation method (HPM) called the delta-homotopy perturbation transform method (delta-HPTM). This modification consists of the Laplace transform method, HPM, and a control parameter delta. This control convergence parameter delta in this new modification helps in adjusting and controlling the convergence region of the series solution and overcome some limitations of HPM and HPTM. The delta-HPTM and q-homotopy analysis transform method (q-HATM) are considered to study the generalized time-fractional perturbed (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 of delta in delta-HPTM and the auxiliary parameters n and h in q-HATM gives a guaranteed convergence of series solution, but the difference between the two techniques is that the embedding parameter p in delta-HPTM varies from zero to nonzero delta, whereas the embedding parameter q in q-HATM varies from zero to 1/n, 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.

Automated summary

The study focuses on the application of the new modified homotopy perturbation method delta-HPTM and q-HATM to the generalized time-fractional perturbed equation, demonstrating the reliability and efficiency of these methods by adjusting and controlling convergence parameters to improve the convergence of series solutions.