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 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 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 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.

The homotopy perturbation method (HPM) was developed by He [45][46][47][48][49][50] by combining the perturbation and standard homotopy for solving numerous physical problems. We refer the reader to He's works for a clear understanding of HPM, where further insights can be found. Recently, an improved modification of HPM, called the parameterized homotopy perturbation method (PHPM), was proposed in [51,52]. Another formulation, called the He-Laplace method, was proposed to obtain an exact closed approximate solution of nonlinear models [53,54]. The HPM and well-known Laplace transformation method were combined to produce a highly effective technique, called the homotopy perturbation transform method (HPTM), for solving many nonlinear problems [55,56]. 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. In this present study, we propose a new modification of HPM, called the δ-homotopy perturbation transform method (δ-HPTM), which consists of HPM, the Laplace transform method, and a control parameter δ. Similarly to the control parameters n and in q-HATM, the control parameter δ in δ-HPTM also helps in adjusting and controlling the convergence region of the series solutions and can overcome some limitations of HPM, HPTM, and He-Laplace method. It is worth mentioning that the present modification (δ-HPTM) requires neither polynomials like ADM nor Lagrange multipliers like VIM and overcomes the limitations of these methods.
To elucidate the reliability and effectiveness of the proposed modification, we consider the generalized time-fractional perturbed (3 + 1)-dimensional Zakharov-Kuznetsov (gpZK) equation given by where W represents the electrostatic potential, k is a positive number, γ is the fractional order, ξ represents a smallness parameter, and the physical quantities β 1 , β 2 , and β 3 are constants. Zhen et al. [57] and Seadawy et al. [58,59] have outlined these physical quantities. This equation is used to describe the nonlinear dust-ion-acoustic waves in the magnetized two-ion-temperature dusty plasmas [60,61]. The study of ion-acoustic waves and structures in dense quantum plasmas has attracted a lot of consideration in recent years. The ZK equation comprises the nonlinear term W ∂W ∂x and third-order dispersion term Equation (2) is limited to the waves of small amplitudes only. The width of the soliton and its velocity deviate from the predictions of this equation as the amplitude of the wave increases. The pZK equation (1) with fractional order γ = 1 and k = 1 includes an of extra fifth-order dispersion term ξ ∂ 5 W ∂x 5 was proposed to overcome this problem (see [57][58][59]62], for more detail). The proposed δ-HPTM and q-HATM are employed to compute numerical solutions of Eq. (1). The two algorithms provide the solutions in a rapid convergent series, which can lead the solutions to a closed form. To the author's knowledge, the approximate solutions of the gpZK (1) was not addressed in the literature before.
The rest of the paper is structured as follows. Useful notations and definitions are provided in Sect. 2. The essential idea of the two methods with convergence and error analysis are presented in Sect. 3. The applications of δ-HPTM and q-HATM on the generalized time-fractional pZK equation are detailed in Sect. 4. Numerical comparison and discussion are provided in Sect. 5. Lastly, Sect. 6 concludes the paper.

Preliminaries
This section contains some helpful notations and definitions. [63].

Definition 3
The fractional derivative of W (t) (denoted by D γ W (t)) in the Caputo sense for m -1 < γ < m, m ∈ N, is defined as [23,65] where with the following properties:

Definition 4
The Laplace transform (denoted by L ) of a Riemann-Liouville fractional integral (J γ t W (t)) and Caputo fractional derivative (D γ t W (t)) of a function W ∈ C ω (ω ≥ where s is a parameter.

Analysis of the proposed methods
Here we give the general idea of the δ-HPTM and q-HATM. We also present some convergence and error analysis of the two methods. Consider the general nonlinear FPDE of the form with initial conditions where D γ t represents the Caputo fractional derivative, M and N denote, respectively, the linear and nonlinear differential operators, W = W (x, y, z, t) specifies the unknown function, and = (x, y, z, t) is the provided source term. Applying the Laplace transform (denoted by L ) to both sides of Eq. (7), we have Using the differentiation property of the Laplace transform with the initial conditions (8), upon simplification and the inverse Laplace transform (denoted by L -1 ), we obtain
The solution of Eq. (7) is given as Remark 1 The particular case where δ = 1 is the standard HPTM [55,56].

Convergence and error analysis
Theorem 1 Let W = W (x, y, z, t) be defined in a Banach space B [67]. Then the series solution is convergent for a prescribed value of δ if where 0 < < |δ|.
We need to show that {S r } ∞ r=0 is a Cauchy sequence in the Banach space B. For δ = 0, we have For all r, k ∈ N with r ≥ k, applying the triangle inequality, we obtain Since 0 < < |δ| and δ = 0, we have 1 -( |δ| ) r-k < 1. Then Since Therefore {S r } ∞ r=0 is a Cauchy sequence in the Banach space B, so the series solution Eq. (16) converges.

Theorem 2
If the truncated series K r=0 W r (x, y, z, t; δ) = K r=0 W r δ r is employed as an approximate solution of Eq. (7), then the maximum absolute truncation error is estimated as Proof It follows from inequality (21) in Theorem 1. For M ≥ K , we have For a prescribed value of δ = 0, S M → W as M → ∞, and 1 -( |δ| ) M-K < 1 (since 0 < |δ| < 1). Thus where W 0 = W 0 .

The q-homotopy analysis transform method (q-HATM)
To exemplify the idea of q-HATM [68][69][70][71][72][73][74][75], we construct the zeroth-order deformation equation for 0 ≤ q ≤ 1 n , n ≥ 1, as where φ = φ(x, y, z, t; q), and N [φ] from Eq. (9) is defined as where q indicates the embedded parameter, the nonzero represents an auxiliary parameter, and H = 0 is an auxiliary function. From Eq. (27) with q = 0, 1 n we get As q rises from 0 to 1 n , the solutions φ ranges from the initial guess W 0 to the solution W . In case that W 0 , , and H are all selected appropriately the solutions φ in Eq. (27) hold for 0 ≤ q ≤ 1 n . Hence application of Taylor series expansion [76] to φ gives where If we choose W 0 , , and H adequately, then Eq. (30) converges at q = 1 n . From Eq. (29) we obtain Differentiating Eq. (27) r times with respect to q, setting q = 0, and multiplying by 1 The vector W r is expressed as Taking the inverse LT of Eq. (33), we obtain where and In Eq. (36), H r denotes the homotopy polynomial defined as

Convergence and error analysis
Here we present some helpful theorems with detailed proofs in [74,75] for the purpose of completeness.
Theorem 3 (Convergence theorem [74,75]) Let B be a Banach space, and let F : B → B be a nonlinear mapping. Suppose that where 0 < < 1. Then has a fixed point in light of Banach's fixed point theory [77]. Furthermore, for arbitrary choice of W 0 , W 0 ∈ B, the sequence generated by the q-HATM converges to a fixed point of , and Theorem 4 ([75]) Suppose that the series solution defined in Eq. (32) converges to the solution W for prescribed values of n and and that there is a real number 0 < < 1 satisfying If the truncated series is utilized as an approximation to the solution of problem (7), then the maximum absolute truncated error is evaluated as

Application of the proposed methods
We have carefully chosen the generalized time-fractional perturbed (3 + 1)-dimensional Zakharov-Kuznetsov (gpZK) equation and apply δ-HPTM and q-HATM to obtain analytical approximate solutions in the form of convergent series. Consider with initial condition Example 1 Consider Eq. (44) with k = 1 given as with initial condition where e 0 , p, q, and φ are arbitrary constants. The exact solution for γ = 1 is given by δ-HPTM Solution: By equating the identical power terms of p in Eq. (49) we generate the sequence of δ-HPTM as . . .
Accordingly, we can derive the remaining terms.
Example 2 Consider Eq. (44) with k = 2 given as with initial condition where e 0 , p, q, and φ are arbitrary constants. The exact solution for γ = 1 is given by
Example 3 Consider Eq. (44) with k = 4 given as with initial condition where p and q are arbitrary constants. The exact solution for γ = 1 is given by
Hence, using initial condition Eq. (66), we derive: By following this procedure we can obtain other terms.

q-HATM Solution:
Implementing LT on Eq. (65) with Eq. (66), we obtain The nonlinear operator N (φ), φ = φ(x, y, z, t; q), is given as Figure 1 The plots of the real part of δ-HPTM, q-HATM, and exact solution for Example 1 Respectively, we can derive the remaining terms.

Numerical comparison
In this section, the δ-HPTM and q-HATM formulations are tested upon the generalized perturbed (3 + 1)-dimensional Zakharov-Kuznetsov (gpZK) equation with Caputo fractional derivative. The δ-HPTM solution is presented as and the q-HATM solution is presented as We observe that setting δ = 1 n in Eq. (74) yields which is the solution of q-HATM. Thus we can conclude that this present modification (δ-HPTM) is more reliable and general. In Figs. 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 2D and 3D plots show the graphical comparison of the fourterm approximation solutions obtain by δ-HPTM and q-HATM and their exact solutions. The 2D plots also present the effect and behavior of the distinct fractional orders on the solution profile. In addition, Figs. 1-4 exhibit different shapes of the exact and approximate 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. For this reason, in Figs. 7-9, we have provided the so-called δ-curves and -curves of the two proposed methods, which serve as a guide in our optimal selection of values in the present analysis. The horizontal line test is employed to attain the intervals containing optimal values. 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. From these tables and plots we can observe that the solutions obtained by the proposed methods are very accurate and in agreement with their respective exact solutions.    Table 1 The comparative study of Re [W (3) ] solutions of δ-HPTM, q-HATM, and exact solution for Example 1 at β 1 = 1, β 2 = 2, β 3 = 0.1, y = 2, z = 2, ξ = 0.1, e 0 = 3, p = 0.5, q = 0.5, φ = 1, and t = 0.01  Table 2 The comparative study of Im [W (3) ] solutions of δ-HPTM, q-HATM, and exact solution for Example 1 at β 1 = 1, β 2 = 2, β 3 = 0.1, y = 2, z = 2, ξ = 0.1, e 0 = 3, p = 0.5, q = 0.5, φ = 1, and t = 0.01   Table 4 The comparative study of Im [W (3) ] solutions of δ-HPTM, q-HATM, and exact solution for Example 2 at β 1 = 1, β 2 = 2, β 3 = 0.1, y = 2, z = 2, ξ = 0.1, e 0 = 1, p = 0.8, q = 0.8, φ = 1, and t = 0.01