Generalized Kawahara Equation Research Articles

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

This paper presents a new method to numerically solve the nonlinear time-fractional generalized Kawahara equations (NTFGKE) with uniform initial boundary conditions (IBCs). A class of modified shifted eighth-kind Chebyshev polynomials (MSEKCPs) is introduced to satisfy the given IBCs. The proposed method is based on using the operational matrices (OMs) for the ordinary derivatives (ODs) and the fractional derivatives (FDs) of MSEKCPs. These OMs are employed together with the spectral collocation method (SCM). Our presented algorithm enables the extraction of efficient and accurate numerical solutions. The convergence of the suggested method and the error analysis have been developed. Three numerical examples are presented to demonstrate the applicability and accuracy of our algorithm. Some comparisons of the presented numerical results with other existing ones are offered to validate the efficiency and superiority of our approach. The presented tables and graphs demonstrate that the proposed approach produces approximate solutions with high accuracy.

Eighth-Kind Chebyshev Polynomials Collocation Algorithm for the Nonlinear Time-Fractional Generalized Kawahara Equation

In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as “Eighth-kind Chebyshev polynomials (CPs)”. These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton’s method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through a series of numerical experiments accompanied by comparative assessments. By undertaking these steps, we seek to communicate our findings comprehensively while ensuring the method’s applicability and precision are demonstrated.

Conservative compact difference scheme based on the scalar auxiliary variable method for the generalized Kawahara equation

In this paper, a conservative compact difference scheme for the generalized Kawahara equation is constructed based on the scalar auxiliary variable (SAV) approach. The discrete conservative laws of mass and Hamiltonian energy and boundedness estimates are studied in detail. The error estimates in discrete ‐norm and ‐norm of the presented scheme are analyzed by using the mathematical induction and discrete energy method. We give an efficient algorithm of the presented scheme, which only needs to solve two decoupled equations.

Inverse Problems for the Generalized Kawahara Equation

Inverse Problems for the Generalized Kawahara Equation

On Initial-Boundary Value Problem on Semiaxis for Generalized Kawahara Equation

In this paper, we consider initial-boundary value problem on a semiaxis for the generalized Kawahara equation with higher-order nonlinearity. We obtain a result on the existence and uniqueness of the global solution. Also, if the equation contains an absorbing term vanishing at infinity, we prove that the solution decays at large time values.

Explicit solutions to the time-fractional generalized dissipative Kawahara equation

In this paper, the generalized dissipative Kawahara equation in the sense of conformable fractional derivative is presented and solved by applying the tanh-coth-expansion and sine-cosine function techniques. The quadratic-case and cubic-case are investigated for the proposed model. Expected solutions are obtained with highlighting to the effect of the presence of the alternative fractional-derivative and the effect of the added dissipation term to the generalized Kawahara equation. Some graphical analysis is presented to support the findings of the paper. Finally, we believe that the obtained results in this work will be important and valuable in nonlinear sciences and ocean engineering.

New Periodic and Localized Traveling Wave Solutions to a Kawahara‐Type Equation: Applications to Plasma Physics

In this study, some new hypotheses and techniques are presented to obtain some new analytical solutions (localized and periodic solutions) to the generalized Kawahara equation (gKE). As a particular case, some traveling wave solutions to both Kawahara equation (KE) and modified Kawahara equation (mKE) are derived in detail. Periodic and soliton solutions to this family are obtained. The periodic solutions are expressed in terms of Weierstrass elliptic functions (WSEFs) and Jacobian elliptic functions (JEFs). For KE, some direct and indirect approaches are carried out to derive the periodic and localized solutions. For mKE, two different hypotheses in the form of WSEFs are used to derive the periodic and localized solutions. Also, the cnoidal wave solutions in the form of JEFs are obtained. As a realistic physical application, the solutions obtained can be dedicated to studying many nonlinear waves that propagate in plasma.

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.

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>

On Initial-Boundary Value Problem on Semiaxis for Generalized Kawahara Equation

In this paper, we consider initial-boundary value problem on semiaxis for generalized Kawahara equation with higher-order nonlinearity. We obtain the result on existence and uniqueness of the global solution. Also, if the equation contains the absorbing term vanishing at infinity, we prove that the solution decays at large time values.

A nonlinear Liouville property for the generalized Kawahara equation

In this paper, by applying the method of Martel and Merle [9] , we prove that if the global solution of the generalized Kawahara equation (gKW) is close to the soliton (constructed by Kabakouala and Molinet [4] ) at initial time in the energy space, moreover if this solution is uniformly localized, then it is identically equal to a soliton all time.

Solutions and conservation laws of a generalized 3D Kawahara equation

We study a nonlinear evolution partial differential equation, namely, the generalized $(3+1)$ -dimensional (3D) Kawahara equation. The Lie symmetry method in conjunction with the Kudryashov method will be employed to derived solutions of the 3D generalized Kawahara equation. Furthermore, the multiplier method will be implemented to construct conservation laws of the generalized 3D Kawahara equation. Thereafter, a brief physical meaning of the derived conserved quantities is discussed.

Well-posedness and unique continuation property for the solutions to the generalized Kawahara equation below the energy space

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Kawahara equation [Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation, Adv Math (China).45(2016),pp.80–88] as follows: with initial data in the Sobolev space Benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488–2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ; Acta Math Sci.37(2017),pp.385C39], first, we show that the local well-posedness is established for the initial data with () and () respectively. Then,using these results and conservation laws, we also prove that the IVP is globally well-posed for the initial data with (). Finally, benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488–2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation,Adv Math (China).45(2016),pp.80–88], i.e. using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property (UCP henceforth) for the IVP.

On the initial-boundary-value problem in a half-strip for a generalized Kawahara equation

The initial-boundary-value problem in a half-strip for a generalized Kawahara equation is considered. For the non-regular initial function, some results on the enhancement of the internal regularity of weak solutions depending on the rate of decrease of the initial function at infinity are obtained. These results are used to prove the long-time decay of solutions.

Fourier Splitting Method for Kawahara Type Equations

In this work, we integrate numerically the Kawahara and generalized Kawahara equation by using an algorithm based on Strang’s splitting method. The linear part is solved using the Fourier transform and the nonlinear part is solved with the aid of the exponential operator method. To assess the accuracy of the solution, we compare known analytical solutions with the numerical solution. Further, we show that as t increases the conserved quantities remain constant.

Application of the (G'/G) -expansion Method to Generalized Kawahara Equation and Generalized KdV Equation

Application of the (G'/G) -expansion Method to Generalized Kawahara Equation and Generalized KdV Equation

Nonlinear self-adjointness of a generalized fifth-order KdV equation

The new concepts of self-adjoint equations formulated by Ibragimov and Gandarias are applied to a class of fifth-order evolution equations. Then, from Ibragimov’s theorem on conservation laws, conservation laws for the generalized Kawahara equation, simplified Kahawara equation and modified simplified Kawahara equation are established.

Approximate homotopy similarity reduction for the generalized Kawahara equation via Lie symmetry method and direct method

This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method. Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders, showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method. The homotopy series solutions to the generalized Kawahara equation are consequently derived.

Application of New Triangular Functions to Nonlinear Partial Differential Equations

The results of some new research on a new class of triangular functions that unite the characteristics of the classical triangular functions are presented. Taking into consideration the great role played by triangular functions in geometry and physics, it is possible to expect that the new theory of the triangular functions will bring new results and interpretations in mathematics, biology, physics and cosmology. New traveling wave solutions of some nonlinear partial differential equations are obtained in a unified way. The main idea of this method is to express the solutions of these equations as a polynomial in the solution of the Riccati equation that satisfy the symmetrical triangular Fibonacci functions. We apply this method to the combined Korteweg-de Vries (KdV) and modified KdV (mKdV) equations, the generalized Kawahara equation, Ito’s 5th-order mKdV equation and Ito’s 7th-order mKdV equation.

Weak solutions of a mixed problem in a half-strip for a generalized Kawahara equation

We consider a mixed problem in a half-strip for a generalized Kawahara equation describing long nonlinear waves in inhomogeneous media with weak dispersion. We study the existence and uniqueness of generalized solutions in Sobolev weight spaces.