Fractional Zakharov-Kuznetsov Equation Research Articles

Discovering novel soliton solutions for (3+1)-modified fractional Zakharov–Kuznetsov equation in electrical engineering through an analytical approach

Discovering novel soliton solutions for (3+1)-modified fractional Zakharov–Kuznetsov equation in electrical engineering through an analytical approach

A novel approach for numerical treatment of traveling wave solution of ion acoustic waves as a fractional nonlinear evolution equation on Shehu transform environment

Abstract In this paper, we develop and employ an efficient numerical technique for traveling wave solution of the Time Fractional Zakharov-Kuznetsov (TFZK) equation, also known as the nonlinear evolution equation, using the Modified Adomian Decomposition Approach (MADA) in collaboration with the cubic order convergence of the Newton-Raphson method (also known as the improvised Newton-Raphson method) on the Shehu Transform environment (STE). In the current study, the time fractional Caputo-Fabrizio Derivative (CFD) is used in singular and non-singular kernel derivatives to address the influence of fractional parameters. Some of the current numerical and analytical results are displayed utilizing 3D plots, while others are depicted in the form of a legend 2D plots for comparison. To validate the robustness of the current approach, the uniqueness, stability, and convergence analyses are described. The current result is compared to the analytical solution as well as previous solutions in order to demonstrate the efficiency of our suggested technique.

MODIFIED COMPUTATIONAL METHOD BASED ON INTEGRAL TRANSFORM FOR SOLVING FRACTIONAL ZAKHAROV-KUZNETSOV EQUATIONS

This work proposes a new computational method, namely the He-Elzaki transform method (HETM) formulated by He’s variation iteration method and modified Laplace transform called Elzaki integral transform to solve nonlinear fractional Zakharov-Kuznetsov equations. The fractional derivatives are described by Caputo sense. The beauty of this technique is that one has no need to evaluate the Lagrange multiplier by integration or taking the convolution theorem. The suggested method is implemented on two examples and the results obtained are compared with those of the Variation iteration method (VIM), homotopy perturbation transform method (HPTM), and new iteration Sumudu transform method (NISTM), and optimum homotopy analysis method (OHAM). The innovative computational technique is an efficient high accurate method and facilitates solving fractional differential equations.

NON-DIFFERENTIABLE EXACT SOLUTIONS OF THE LOCAL FRACTIONAL ZAKHAROV–KUZNETSOV EQUATION ON THE CANTOR SETS

In this study, a new fractional Zakharov–Kuznetsov equation (ZKE) within the local fractional derivative (LFD) is derived. Yang’s non-differentiable (ND) traveling wave transform is introduced, then two novel techniques namely the Mittag-Leffler function-based method (MLFBM) and Yang’s special function method (Y-SFM) are adopted to seek for the ND exact solutions for the first time. With the aid of the Mathematica software, the dynamic behaviors of the different solutions on the Cantor sets are illustrated via the 3D plots by assigning the appropriate parameters. The attained results confirm that the mentioned methods are effective and straightforward, which can be used to study the ND exact solutions of the local fractional partial differential equations (PDEs).

A robust computational approach for Zakharov-Kuznetsov equations of ion-acoustic waves in a magnetized plasma via the Shehu transform

The application of the q-homotopy analysis Shehu transform method (q-HAShTM) to discover the estimated solution of fractional Zakharov-Kuznetsov equations is investigated in this study. In the presence of a uniform magnetic field, the Zakharov-Kuznetsov equations regulate the behaviour of nonlinear acoustic waves in a plasma containing cold ions and hot isothermal electrons. The q-HAShTM is a stable analytical method that combines homotopy analysis and the Shehu transform. This q-homotopy investigation Shehu transform is a constructive method that leads to the Zakharov-Kuznetsov equations, which regulate the propagation of nonlinear ion-acoustic waves in a plasma. It is a more semi-analytical method for adjusting and controlling the convergence region of the series solution and overcoming some of the homotopy analysis method’s limitations.

Construction of an Approximate Analytical Solution for Multi-Dimensional Fractional Zakharov–Kuznetsov Equation via Aboodh Adomian Decomposition Method

In this paper, the Aboodh transform is utilized to construct an approximate analytical solution for the time-fractional Zakharov–Kuznetsov equation (ZKE) via the Adomian decomposition method. In the context of a uniform magnetic flux, this framework illustrates the action of weakly nonlinear ion acoustic waves in plasma carrying cold ions and hot isothermal electrons. Two compressive and rarefactive potentials (density fraction and obliqueness) are illustrated. With the aid of the Caputo derivative, the essential concepts of fractional derivatives are mentioned. A powerful research method, known as the Aboodh Adomian decomposition method, is employed to construct the solution of ZKEs with success. The Aboodh transform is a refinement of the Laplace transform. This scheme also includes uniqueness and convergence analysis. The solution of the projected method is demonstrated in a series of Adomian components that converge to the actual solution of the assigned task. In addition, the findings of this procedure have established strong ties to the exact solutions to the problems under investigation. The reliability of the present procedure is demonstrated by illustrative examples. The present method is appealing, and the simplistic methodology indicates that it could be straightforwardly protracted to solve various nonlinear fractional-order partial differential equations.

Numerical Solutions of Time Fractional Zakharov-Kuznetsov Equation via Natural Transform Decomposition Method with Nonsingular Kernel Derivatives

In this paper, we have studied the time-fractional Zakharov-Kuznetsov equation (TFZKE) via natural transform decomposition method (NTDM) with nonsingular kernel derivatives. The fractional derivative considered in Caputo-Fabrizio (CF) and Atangana-Baleanu derivative in Caputo sense (ABC). We employed natural transform (NT) on TFZKE followed by inverse natural transform, to obtain the solution of the equation. To validate the method, we have considered a few examples and compared with the actual results. Numerical results are in accordance with the existing results.

Adequate soliton solutions to the time fractional Zakharov-Kuznetsov equation and the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation

The time fractional (2, 2, 2) Zakharov-Kuznetsov (ZK) equation and the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation demonstrate the characteristic of shallow water waves, turbulent motion, waves of electro-hydro-dynamics in the local electric field, sound wave, waves of driving flow of fluid, ion acoustic waves in plasmas, traffic flow, financial mathematics, etc. The time-fractional (2, 2, 2) ZK equation is the particular case of the general time-fractional ZK equation, where represent the space coordinate and represents the temporal coordinate. Hereinto to evade the complexity and to ascertain soliton solutions of this model, we accept and in this case, the general ZK equation is called the time-fractional (2, 2, 2) ZK equation. In this article by making use of the concept of fractional complex transformation, the auxiliary equation method is put in use to search the closed form soliton solutions to the above indicated fractional nonlinear equations (FNLEs).The ascertained solutions are in the form of exponential, rational, hyperbolic and trigonometry functions with significant precision. We illustrate the soliton solutions relating to physical concern by setting the definite values of the free parameters through depicting diagram and interpreted the physical phenomena. The developed solutions assert that the method is effective, able to measure NLEEs, influential, powerful and offer vast amount of travelling wave solutions of nonlinear evolution equations in the area of mathematical sciences and engineering.

Numerical Simulation of Fractional Zakharov–Kuznetsov Equation for Description of Temporal Discontinuity Using Projected Differential Transform Method

The core aim of this study is to propose a novel computational procedure, namely, Elzaki transform iterative method to work out two‐dimensional nonlinear time‐fractional Zakharov–Kuznetsov equation numerically. We execute the suggested iterative procedure on two models and results are presented graphically in the form of surface plot and absolute error is compared with the VIM and HPM to show that the method is more powerful than VIM and HPM and deduce that the offered numerical pattern is more efficient in simulating linear and nonlinear fractional order models.

Improved Fractional Subequation Method and Exact Solutions to Fractional Partial Differential Equations

In this paper, the improved fractional subequation method is applied to establish the exact solutions for some nonlinear fractional partial differential equations. Solutions to the generalized time fractional biological population model, the generalized time fractional compound KdV-Burgers equation, the space-time fractional regularized long-wave equation, and the (3+1)-space-time fractional Zakharov-Kuznetsov equation are obtained, respectively.

On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations

A family of dispersive equations is considered which links a higher dimensional Benjamin-Ono equation and the Zakharov-Kuznetsov equation. For these fractional Zakharov-Kuznetsov equations new well-posedness results are proved using transversality and localization of time to small frequency dependent time intervals.

Constructions of the optical solitons and other solitons to the conformable fractional Zakharov–Kuznetsov equation with power law nonlinearity

The current research manifests kink wave answers, mixed singular optical solitons, the mixed dark-bright lump answer, the mixed dark-bright periodic wave answer, and periodic wave answers to the conformable fractional ZK model, including power law nonlinearity by plugging the revised -expansion process. The constraint requirements for the occurrence of substantial solitons are provided. Under the selection of proper values of a, b, n, t, λ, μ and α, the 2D and 3D pictures to a few of the recorded answers are sketched. From our obtained solutions, we might decide that the investigated procedure is hugely muscular, sincere, and essential in rendering various new soliton solutions of distinct nonlinear conformable fractional evolution equations and accordingly, we shall bring it up in our future investigations.

Numerical solution of nonlinear fractional Zakharov–Kuznetsov equation arising in ion-acoustic waves

The main purpose of this work is to suggest an efficient hybrid computational technique, namely the q-homotopy analysis transform method (q-HATM) to find the solution of the nonlinear time-fractional Zakharov–Kuznetsov (FZK) equation in two dimensions. The uniqueness and convergence analysis of the nonlinear time-FZK equation is presented. The Laplace decomposition method (LDM) is also employed to get the approximate solution of the nonlinear FZK equation. We implemented these techniques on two numerical examples, plotted the solution and compared the absolute error with the variational iteration technique and homotopy perturbation transform technique to show the efficiency of these techniques.

Closed form wave solutions to the time fractional Boussinesq-type equation and the Zakharov-Kuznetsov equation

In this article, we have considered the time fractional Boussinesq-type equation and the (3, 3,.3) time fractional Zakharov-Kuznetsov (FZK) equation, as the model of fluid flow and heat transfer, temperature changes from one place to another, waves on a free-moving fluid surface, periodic and shallow water waves, conservation of mass, ion-acoustic waves in plasma, electromagnetic waves, traffic flow and signal processing waves of optical fibers, etc. Applying the generalised (G'/G)-expansion method and by means of fractional complex transformation, we have examined fresh, useful and further generalised exact travelling wave solutions to the above mentioned equations. We have shown that the method used in this article is a more generalised, straightforward and efficient technique that can be used to establish a large number of fresh solutions of the fractional nonlinear differential equations involved in mathematical physics. We have also discussed the physical explanation for its definite values of emerging parameters of the solutions, which are examined from these equations. Finally we have depicted the 3D and 2D figures to the sense of physical phenomena of the obtained solutions.

Solution for fractional Zakharov–Kuznetsov equations by using two reliable techniques

In this paper, the approximated analytical solutions for nonlinear dispersive fractional Zakharov-Kuznetsov (FZK(a, b, c)) equations are obtained with the help of two novel techniques, called fractional natural decomposition method (FNDM) and q-homotopy analysis transform method (q-HATM). In order to validate and illustrate the efficiency of the proposed techniques, we consider two special cases FZK(2, 2, 2) and FZK(3, 3, 3) in FZK(a, b, c). The numerical simulations have been conducted in order to verify the proposed techniques are reliable and accurate. The outcomes are revealed through the plots and tables. The comparison between the obtained solutions with the exact solutions exhibits that, both the featured schemes are efficient and effective in solving nonlinear complex problems.

Lie Symmetry Analysis and Exact Solutions of Generalized Fractional Zakharov-Kuznetsov Equations

This paper considers the Lie symmetry analysis of a class of fractional Zakharov-Kuznetsov equations. We systematically show the procedure to obtain the Lie point symmetries for the equation. Accordingly, we study the vector fields of this equation. Meantime, the symmetry reductions of this equation are performed. Finally, by employing the obtained symmetry properties, we can get some new exact solutions to this fractional Zakharov-Kuznetsov equation.

All single travelling wave patterns to fractional Jimbo–Miwa equation and Zakharov–Kuznetsov equation

By the complete discrimination system of polynomial method, we obtain the classification and representation of all single travelling wave solutions to \((3+1)\)-dimensional conformal fractional Jimbo–Miwa equation and fractional Zakharov–Kuznetsov equation. These solutions show rich evolution patterns of models described by these two equations.

Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity

Abstract In this work, we consider the (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity. Solitary wave solutions, soliton wave solutions, elliptic wave solutions, and periodic (hyperbolic) wave rational solutions are obtained by means of the unified method. The solutions showed that this method provides us with a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences.

Numerical method for fractional Zakharov-Kuznetsov equations with He’s fractional derivative

In this paper, a fractional Zakharov-Kuznetsov equation with He's fractional derivative is studied by the fractional complex transform and He's homotopy perturbation method. The solution process is elucidated step by step to show its simplicity and effectiveness of the proposed method.

Approximate solutions to nonlinear factional order partial differential equations arising in ion-acoustic waves

An approximative procedure named asymptotic homotopy perturbation method (AHPM) is introduced to obtain solutions of the non-linear fractional order models. The two special cases, FZK$(3, 3, 3)$ and FZK$(2, 2, 2)$ of fractional Zakharov-Kuznetsov equations are chosen for the illustrative purpose of our method. AHPM is a very recent new procedure as compare with other existing homotopy perturbation procedures. A new auxiliary function has been introduced in AHPM. The AHPM solutions are compared with solutions of fractional complex transform FCT using variational iteration method VIM and exact solutions. Further, the surface graph of AHPM solutions are compared with surface graph of solutions of homotopy perturbation transform method (HPTM) solutions. In comparison, the solutions computed by AHPM are in agreement with exact solutions of the problems. The simulation section reveals that our new developed procedure is effective and explicit.