Kawahara Equation Research Articles

Numerical approximation of modified Kawahara equation using Kernel smoothing method

In this article, a numerical approximation of modified Kawahara equation is investigated by Kernel smoothing method. The spatial derivatives involved in the modified Kawahara equation are approximated by smoothing Kernel method. Whereas, for the time integration, we employ Crank–Nicolson method. The conservative nature of the proposed scheme is demonstrated by the mass conservation constant (I1) and energy conservation constant (I2). To quantify the quality of the proposed scheme, we also have performed numerical testing on a collection of test problems.

Low regularity for the Kawahara equation on the half‐line

We study the low‐regularity properties of the Kawahara equation on the half line. We obtain the local existence, uniqueness, and continuity of the solution. Moreover, we obtain that the nonlinear terms of the solution are smoother than the initial data.

THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES

The Cauchy problem for the Kawahara equation with data in analytic Gevrey spaces on the circle is considered and its local well-posedness in these spaces is proved. Using Bourgain-Gevrey type analytic spaces and appropriate bilinear estimates, it is shown that local in time wellposedness holds when the initial data belong to an analytic Gevrey spaces of order σ. Moreover, the solution is not necessarily Gσ in time. However, it belongs to G5σ near zero for every x on the circle. We study a Cauchy problem for Kawahara equation . With data in analytic Gevrey spaces on the circal, we prove that the problem is well defined. We also treat the regularity in time which belongs to $G^{5\sigma}$ .

Traveling Wave Solutions using Tanh Method for solving Kawahara and Modified Kawahara Equations

This paper implemented very reliable technique which is called the Tanh method for solving evolution equations. The Tanh method has been successfully tested on two important Kawahara and modified Kawahara equations. The calculations demonstrate the effectiveness and convenience of Tanh method for solving nonlinearPDEs.

Solitary wave solution and a linear mass-conservative difference scheme for the generalized Korteweg–de Vries–Kawahara equation

Solitary wave solution and a linear mass-conservative difference scheme for the generalized Korteweg–de Vries–Kawahara equation

Novel anlytical solution to the damped Kawahara equation and its application for modeling the dissipative nonlinear structures in a fluid medium

A novel approximate analytical solution to the linear damped Kawahara equation using a suitable hypothesis is reported for the first time. Based on the exact solutions (such as solitary waves, cnoidal waves, etc.) of the undamped Kawahara equation, the dissipative nonlinear structures like dissipative solitons and cnoidal waves are investigated. The obtained solution is considered a general solution, i.e., it can be applied for studying the properties of all dissipative traveling waves described by the linear damped Kawahara equation. Our technique is not limited to solve the linear damped Kawahara equation only, but it can be used for solving a large number of non-integrable evolution equations related to the realistic natural phenomena. Moreover, the maximum global residual error in the whole space-time domain is estimated for checking the accuracy of the obtained solutions. The obtained solutions can help many researchers in explaining the ambiguities about the mechanisms of propagation of nonlinear waves in complex systems such as seas, oceans, plasma physics, and much more.

ANALYSIS OF TIME-FRACTIONAL KAWAHARA EQUATION UNDER MITTAG-LEFFLER POWER LAW

In this paper, we study a newly updated nonlinear fractional Kawahara equation (KE) using Atangana–Baleanu fractional operator in the sense of Caputo (ABC). To find the approximate solution, one of the famous techniques of the Laplace Adomian decomposition method (LADM) is used along with a time-fractional derivative. For evaluation, the required quantity is decomposing into small particles along with the application of Adomian polynomial to the nonlinear term. By the addition of the first few evaluating terms, the required convergent quantity is obtained. To explain the authenticity and the manageability of the procedure, few examples are present at different fractional orders both in three and two dimensions. Further, to compare the obtained results between fractional derivative and integer derivative, some graphical presentations are given. So, the newly updated version of the KE equation is analyzed in fraction operator providing the whole density of the total dynamics at any fractional value between two different integers.

Approximation and Eventual periodicity of Generalized Kawahara equation using RBF-FD method

In engineering and mathematical physics nonlinear evolutionary equations play an important role. Kawahara equation is one of the famous nonlinear evolution equation appeared in the theories of shallow water waves possessing surface tension, capillary-gravity waves and also magneto-acoustic waves in a plasma. Another specific subjective parts of arrangements for some of evolution equations demonstrated by investigations, which connect alongwith their large-time behavior named as eventual time periodicity uncovered across solutions to IBVPs (initialboundary-value problems). In this study eventual periodicity of solutions for the generalized fifth order Kawahara equation (IBVP) on bounded domain coupled with periodic boundary condition will explored numerically utilizing meshless technique called as Radial basis function generated finite difference (RBF-FD) method.

Analysis of fractional Kawahara and modified Kawahara equations based on Caputo-Fabrizio derivative operator

In this paper, nonlinear time fractional Kawahara and modified Kawahara equations based on Caputo-Fabrizio derivative operator is analysed using iterative Laplace transform method to obtain approximate solutions. The substantive features of the manuscript is to offer the stability conditions of solution for proposed technique. The acquired approximate solutions are in comparison with the precise solutions to confirm the applicability, performance and accuracy of the method. Moreover, the 3D plots of obtained numerical solution of the concerned equations for various specific cases are presented.

Solitary and Rogue Wave Solutions to the Conformable Time Fractional Modified Kawahara Equation in Mathematical Physics

Utilizing of illustrative scheming programming, the study inspects the careful voyaging wave engagements from the nonlinear time fractional modified Kawahara equation (mKE) by employing the advanced exp − φ ξ -expansion policy in terms of trigonometric, hyperbolic, and rational function through some treasured fractional order derivative and free parameters. The undercurrents of nonlinear wave answer are scrutinized and confirmed by MATLAB in 3D and 2D plots, and density plot by specific values of the convoluted parameters is designed. Our preferred advanced exp − φ ξ -expansion technique which is parallel to ( G ′ / G ) expansion technique is trustworthy dealing for searching significant nonlinear waves that progress a modification of dynamic depictions that ascend in mathematical physics and engineering grounds.

Numerical solution by quintic B-spline collocation finite element method of generalized Rosenau–Kawahara equation

Numerical solution by quintic B-spline collocation finite element method of generalized Rosenau–Kawahara equation

Unconditional well-posedness for the Kawahara equation

This article is concerned with the unconditional well-posedness for the Kawahara equation on the real line and shows that this holds true for initial data in L2(R). This is achieved by applying an infinite iteration scheme of normal form reductions.

Novel analytical cnoidal and solitary wave solutions of the Extended Kawahara equation

In this work, some novel analytic traveling wave solutions including the cnoidal and solitary wave solutions of the planar Extended Kawahara equation are deduced. Four different analytical methods (the Jacobian elliptic function, Weierrtrass elliptic function, the traditional tanh method and the sech-square) are devoted for solving this equation. By means of the Jacobian elliptic function ansatz, the cnoidal and soliary wave solutions are obtained. Also, new cnoidal wave solutions are derived via a new hypothesis in the form of the Weierrtrass elliptic function. Moreover, the standard tanh method is utilized to get a new set of solitary wave solutions for the evolution equation. Over and above, the hyperbolic ansatz method (a new ansatz in the form of squre-sech) is employed to get a new set of solitary wave solutions for the evolution equation. Furthermore, all obtained solutions of the planar Extended Kawahara equation cover the traveling wave solutions of the planar modified Kawahara equation. These solutions maybe useful to many researchers interested in studying the propagation of nonlinear waves in nonlinear dispersion mediums like plasma physics, optical fibers, fluid mechanics, and many different branches of science.

Conservation laws and series solutions of variable coefficient time fractional Kawahara equation

This paper investigates explicit power series solutions and conservation laws of variable coefficient time fractional Kawahara equation with Riemann–Liouville derivative which describs oscillatory solitary wave propagation in dispersive media. The similarity reductions and explicit solutions are derived by using Lie symmetry method. Further, using Ibragimov's theorem conservation laws for this equation are discussed.

A novel approach for fractional Kawahara and modified Kawahara equations using Atangana-Baleanu derivative operator

In this manuscript, iterative laplace transform method is applied to obtain approximate solutions of the nonlinear time fractional Kawahara and modified Kawahara equations based on Atangana-Baleanu derivative operator. The noticeable features of the manuscript is to providing the existence and uniqueness conditions of solution for proposed technique and the graphical presentations of numerical solution of the concerned equations for various specific cases. The obtained approximate solutions are compared with the exact solutions to verify the applicability, efficiency and accuracy of the method.

The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation

<p style='text-indent:20px;'>In this paper, we are concerned with the existence of solitary waves for a generalized Kawahara equation, which is a model equation describing solitary-wave propagation in media. We obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the generalized Kawahara equation without delay and perturbation by employing the phase space analysis. Furthermore the existence of solitary wave solutions for the equation with two types of special delay convolution kernels is proved by combining the geometric singular perturbation theory, invariant manifold theory and Fredholm orthogonality. We also discuss the asymptotic behaviors of traveling wave solutions by means of the asymptotic theory. Finally, some examples are given to illustrate our results.</p>

Advance in compact structure‐preserving manner to the Rosenau–Kawahara model of shallow‐water wave

In this paper, we pursue further analysis of the performance of a compact structure‐preserving finite difference scheme. The convergence, stability, and accuracy of the approximate solution with respect to grid refinement are discussed. The compact difference approach to precisely preserving invariants on any time‐space regions gives a three‐level linear‐implicit scheme with the spatial accuracy, found to be fourth order on a uniform grid. The method is verified by comparison with a solution of the Rosenau–Kawahara equation just obtained with second‐order finite difference schemes recently. Also, the efficiency of the present algorithm is confirmed by simulations of the problem at a long time. Details of CPU time are examined in order to assess the usefulness of the compact scheme for determining an approximate solution.

Well-posedness and controllability of Kawahara equation in weighted Sobolev spaces

We consider the Kawahara equation, a fifth order Korteweg–de Vries type equation, posed on a bounded interval. The first result of the article is related to the well-posedness in weighted Sobolev spaces, which one was shown using a general version of the Lax–Milgram Theorem. With respect to the control problems, we will prove two results. First, if the control region is a neighborhood of the right endpoint, an exact controllability result in weighted Sobolev spaces is established. Lastly, we show that the Kawahara equation is controllable by regions on L2 Sobolev space, the so-called regional controllability, that is, the state function is exact controlled on the left part of the complement of the control region and null controlled on the right part of the complement of the control region.

Lower bounds on the radius of spatial analyticity for the Kawahara equation

In this paper we obtain lower bounds on the radius of spatial analyticity of solutions to the Kawahara equation \(u_t + uu_x + \alpha u_{xxx} + \beta u_{xxxxx} = 0\), \(\beta \ne 0\), given initial data which is analytic with a fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time t can decay no faster than 1/|t| as \(|t|\rightarrow \infty \).

Analysis of fractional Kawahara and modified Kawahara equations based on Caputo-Fabrizio derivative operator

Analysis of fractional Kawahara and modified Kawahara equations based on Caputo-Fabrizio derivative operator