Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsov equation in dusty plasma
Lanre Akinyemi, Mehmet Şenol, Shaheed N. Huseen
Abstract
Abstract We propose a new modification of homotopy perturbation method (HPM) called the δ -homotopy perturbation transform method ( δ -HPTM). This modification consists of the Laplace transform method, HPM, and a control parameter δ . This control convergence parameter δ 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 δ -HPTM and q-homotopy analysis transform method (q-HATM) are considered to study the generalized time-fractional perturbed $(3+1)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:math> -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 δ in δ -HPTM and the auxiliary parameters n and ħ in q-HATM gives a guaranteed convergence of series solution, but the difference between the two techniques is that the embedding parameter p in δ -HPTM varies from zero to nonzero δ , whereas the embedding parameter q in q-HATM varies from zero to $\frac{1}{n}, n\geq{1}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>n</mml:mi> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:math> . 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.