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 \(\frac{1}{n}, n\geq{1}\). 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.

我们提出了一种新的同态扰动方法（HPM），称为δ-同伦扰动变换方法（δ- HPTM）。该修改包括拉普拉斯变换方法HPM和控制参数δ。此新修改中的控制收敛参数δ有助于调整和控制串联解决方案的收敛区域，并克服了HPM和HPTM的某些限制。考虑采用δ -HPTM和q同伦分析变换方法（q-HATM）研究广义时间分数扰动\（（3 + 1）\）Caputo分数时间导数的三维Zakharov–Kuznetsov方程。该方程式描述了在磁化的两个离子温度的粉尘等离子体中的非线性粉尘离子声波。的一个适当的值的选择δ在δ -HPTM和辅助参数Ñ和ħ在Q-HATM给出的系列溶液保证收敛，但是这两种技术之间的区别在于，嵌入参数p在δ -HPTM变化从零到非零δ，而q-HATM中的嵌入参数q从零到\（\ frac {1} {n}，n \ geq {1} \）。我们研究了分数阶对所考虑问题的影响，并提出了与精确解相比时的误差估计。结果揭示了所提出算法用于解决科学和工程学中出现的各种物理模型的完全可靠性和效率。此外，我们介绍了两种方法的收敛性和误差分析。