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)-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}, ngeq{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.
Highlights
It is worth noting that the Laplace transform method alone in some cases is insufficient in handling nonlinear problems because of the difficulties that may arise by the nonlinear terms
We propose a new modification of homotopy perturbation method (HPM), called the δ-homotopy perturbation transform method (δ-HPTM), which consists of HPM, the Laplace transform method, and a control parameter δ
6 Conclusion In this paper, we proposed a new modification of the homotopy perturbation method (HPM), called the δ-homotopy perturbation transform method (δ-HPTM), which consists of HPM, the Laplace transform method, and a control parameter δ for solving integer- and
Summary
The study of fractional partial differential equations (FPDEs) has enticed the interest of many researchers in the field of applied sciences and engineering by virtue of its enormous applications in electrodynamics, random walk, biotechnology, viscoelasticity, chaos theory, signal and image processing, nanotechnology, and many other areas [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. 1–6, we present the response of the obtained solutions by the proposed methods with regard to the real and imaginary parts in terms of 2D and 3D plots. The selection of the auxiliary parameters δ in δ-HPTM and in q-HATM are very crucial to guarantee fast convergence of the series solutions. The comparative study for the case γ = 1 of the real and imaginary parts of the results obtained by δ-HPTM, q-HATM, and the exact solution as the benchmark are considered in Tables 1– 6.
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