Time Fractional Zakharov-Kuznetsov Research Articles

The $$(2 + 1)$$-dimensional generalized time-fractional Zakharov Kuznetsov Benjamin Bona Mahony equation: its classical and nonclassical symmetries, exact solutions, and conservation laws

The $$(2 + 1)$$-dimensional generalized time-fractional Zakharov Kuznetsov Benjamin Bona Mahony equation: its classical and nonclassical symmetries, exact solutions, and conservation laws

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.

An analytical technique to obtain traveling wave solutions to nonlinear models of fractional order

The (3, 3, 3) time-fractional Zakharov–Kuznetsov (TFZK) equation demonstrates the characteristic of traffic flows, the viscoelasticity waves, the material science, sound waves, signal processing, the financial mathematics, optical fibers, the electromagnetic field, water wave mechanics, ion acoustic waves in plasmas, the chemical physics and so many. The (3, 3, 3) TFZK equation is the specific event of the general (a,b,c) TFZK equation, where a,b denote the space coordinates and c denotes the temporal coordinate. The object of this investigation is to consider determining the (3, 3, 3) TFZK equation by the modified (G′/G)-expansion scheme (MGD/GES). The derivatives are in the beta-derivative sense. The fractional transformation equation is utilized to convert the proposed nonlinear models of fractional order (NLMFO) into non-linear models of integer order. Additionally, numerous new wave solutions (WSs) are formed. The attractive and easy technique of the MGD/GES recommends that this technique can be applied to WSs of other NLMFOs straightforwardly.

A Fractional Analysis of Zakharov–Kuznetsov Equations with the Liouville–Caputo Operator

In this study, we used two unique approaches, namely the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM), to derive approximate analytical solutions for nonlinear time-fractional Zakharov–Kuznetsov equations (ZKEs). This framework demonstrated the behavior of weakly nonlinear ion-acoustic waves in plasma containing cold ions and hot isothermal electrons in the presence of a uniform magnetic flux. The density fraction and obliqueness of two compressive and rarefactive potentials are depicted. In the Liouville–Caputo sense, the fractional derivative is described. In these procedures, we first used the Yang transform to simplify the problems and then applied the decomposition and perturbation methods to obtain comprehensive results for the problems. The results of these methods also made clear the connections between the precise solutions to the issues under study. Illustrations of the reliability of the proposed techniques are provided. The results are clarified through graphs and tables. The reliability of the proposed procedures is demonstrated by illustrative examples. The proposed approaches are attractive, though these easy approaches may be time-consuming for solving diverse nonlinear fractional-order partial differential equations.

On group of Lie symmetry analysis, explicit series solutions and conservation laws for the time-fractional (2 + 1)-dimensional Zakharov-Kuznetsov (q,p,r) equation

This research article intends to utilize results on Lie symmetry analysis, explicit series solutions and conservation laws for the time-fractional Zakharov-Kuznetsov (q,p,r) equation in three-dimensional space. Such a fractional equation yields the mathematical model which describes an occurrence of stationary spatial stripe modalities in a three-dimensional system in the framework of the theory of conservation laws. The governing equation is solved analytically by the power series method, where the total derivative in the sense of Riemann-Liouville type. Simulation results are systematically validated through a series of test cases. Strong evidence shows that the model and the method are conservative and robust.

The singular manifold method for a class of fractional-order diffusion equations

The present article presents the technique for finding exact wave solutions of nonlinear fractional-order partial differential equations (NLFPDEs). The singular manifold method (SMM) is applied by starting with the fractional complex transformation (FCT) to reduce the equations to ordinary differential equations (ODEs). Then using the Bäcklund transformation (BT) and the resulting Schwarzian derivatives of the Eigen functions, a new variety of exact solutions is obtained. The method is applied to nonlinear (2 + 1) time-fractional Zoomeron equation, nonlinear (3 + 1) conformable time-fractional Zakharov–Kuznetsov equation, and nonlinear fractional Fokas equation. Some resulting solutions are graphically presented, revealing different solitary wave solutions.

The Numerical Investigation of Fractional-Order Zakharov–Kuznetsov Equations

In this article, a modified method called the Elzaki decomposition method has been applied to analyze time-fractional Zakharov–Kuznetsov equations. In this method, the Adomian decomposition technique and Elzaki transformation are combined. Two problems are investigated to show and validate the efficiency of the suggested method. It is also shown that the solutions achieved from the current producer are in good contact with the exact solutions to show with the help of graphs and table. It is observed that the suggested technique is to be reliable, efficient, and straightforward to implement for many related models of engineering and science.

Efficient techniques for traveling wave solutions of time-fractional Zakharov–Kuznetsov equation

This article manages the development of the traveling wave solutions of time-fractional Zakharov–Kuznetsov (ZK) equation by using the modified auxiliary equation method (MAEM) and analytical approximate solution that is constructed through the residual power series method (RPSM). The MAEM with time fractional beta-derivative is utilized to observe the effects of fractional parameter on the governing model. The RPSM is more useful to retrieve a particular form of wave solution when the initial behavior of the system is known. This method is observed by considering Caputo’s time fractional derivative. Some of the obtained results are graphically illustrated using 3D surface graphs and 2D line plots. The proposed methods are shown to be useful for construction of traveling wave solutions which are hoped to be beneficial in exploring different dynamical behaviors expressed by the equation.

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.

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.

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.

Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations

To model physical phenomena more accurately, fractional order differential equations have been widely used. Investigating exact solutions of the fractional differential equations have become more important because of the applications in applied mathematics, mathematical physics, and other areas. In this work, by means of the trial solution method and complete discrimination system, exact traveling wave solutions of the conformable time-fractional Zakharov–Kuznetsov equation and conformable time-fractional Zoomeron equation have been obtained and also solutions have been illustrated. Finding exact solutions of these equations that are encountered in plasma physics, nonlinear optics, fluid mechanics, and laser physics can help to understand nature of the complex phenomena.

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.

New exact solitary wave solutions to the space-time fractional differential equations with conformable derivative

The exact wave solutions to the space-time fractional modified Benjamin-Bona-Mahony (mBBM) and space time fractional Zakharov-Kuznetsov Benjamin-Bona-Mahony (ZKBBM) equations are studied in the sense of conformable derivative. The existence of chain rule and the derivative of composite functions permit the nonlinear fractional differential equations (NLFDEs) to convert into the ordinary differential equation using wave transformation. The wave solutions of these equations are examined by means of the expanding and effective two variable ( G' / G ,1/ G )-expansion method. The solutions are obtained in the form of hyperbolic, trigonometric and rational functions containing parameters. The method is efficient, convenient, accessible and is the generalization of the original ( G' / G )-expansion method.

A new iterative technique for a fractional model of nonlinear Zakharov–Kuznetsov equations via Sumudu transform

The main objective of this paper is to suggest a new computational technique namely new iterative Sumudu transform method (NISTM) to solve numerically nonlinear time-fractional Zakharov–Kuznetsov (FZK) equation in two dimensions. We implemented the proposed technique on two test examples, plotted the solution and compared the absolute error with the variational iterative technique (VIM) and homotopy perturbation transform method (HPTM).

Invariant analysis and conservation laws of ([formula omitted]) dimensional time-fractional ZK–BBM equation in gravity water waves

This paper intends to make an in-depth study on the symmetry properties and conservation laws of the (2+1) dimensional time fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZK–BBM) equation with Riemann–Liouville fractional derivative. Symmetry properties have been investigated here via Lie symmetry analysis method. In view of Erdélyi-Kober fractional differential operator, the reduction of (2+1) dimensional time fractional ZK–BBM equation has been done into fractional ordinary differential equation. To analyse the conservation laws, new theorem of conservation law has been proposed here for constructing the new conserved vectors for (2+1) dimensional time fractional ZK–BBM equation with the help of formal Lagrangian.

Exponential rational function method for space–time fractional differential equations

In this paper, exponential rational function method is applied to obtain analytical solutions of the space–time fractional Fokas equation, the space–time fractional Zakharov Kuznetsov Benjamin Bona Mahony, and the space–time fractional coupled Burgers’ equations. As a result, some exact solutions for them are successfully established. These solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.

Approximate analytical solution to a time-fractional Zakharov-Kuznetsov equation

In this paper we present the approximate analytical solution of a time-fractional Zakharov-Kuznetsov equation via the fractional iteration method. The fractional derivatives are described in the Caputo sense. The approximate results show that the fractional iteration method is a very ecient technique to handle fractional partial differential equations.