Order KdV Equation Research Articles

General boundary value problems of a class of fifth order KdV equations on a bounded interval

General boundary value problems of a class of fifth order KdV equations on a bounded interval

Data-driven solutions and parameter estimations of a family of higher-order KdV equations based on physics informed neural networks

Physics informed neural network (PINN) demonstrates powerful capabilities in solving forward and inverse problems of nonlinear partial differential equations (NLPDEs) through combining data-driven and physical constraints. In this paper, two PINN methods that adopt tanh and sine as activation functions, respectively, are used to study data-driven solutions and parameter estimations of a family of high order KdV equations. Compared to the standard PINN with the tanh activation function, the PINN framework using the sine activation function can effectively learn the single soliton solution, double soliton solution, periodic traveling wave solution, and kink solution of the proposed equations with higher precision. The PINN framework using the sine activation function shows better performance in parameter estimation. In addition, the experiments show that the complexity of the equation influences the accuracy and efficiency of the PINN method. The outcomes of this study are poised to enhance the application of deep learning techniques in solving solutions and modeling of higher-order NLPDEs.

An efficient computational technique for higher order KdV equation arising in shallow water waves

This paper focuses on the computational study of seventh-order KdV equations via the quintic B-spline (QBS) finite element method (FEM). The KdV equations contain higher-order derivative terms. We transform the higher order into lower order and obtain a coupled system to ease the solution procedure. Forward difference combined with the theta-weighted scheme is used to discretize the temporal part. In addition, the quintic B-spline approximates the functions and their derivatives. The Von Neumann method is used to check stability of the scheme. The performance of the proposed method is evaluated using four test problems. The numerical results are presented in the tables to evaluate the accuracy and proficiency of the scheme at various collocation points. Furthermore, the approximate and exact solutions are compared graphically which match each other, showing validation of the proposed method.

Computational and numerical wave solutions of the Caudrey–Dodd–Gibbon equation

The Caudrey–Dodd–Gibbon (CDG) model, a variation of the fifth–order KdV equation (fKdV) with significant practical consequences, is solved in this study using a precise and numerical technique. This model shows how gravity-capillary waves, shallow–water waves driven by surface tension, and magneto-acoustic waves move through a plasma medium. With a focus on accuracy, new computational and approximation methods have been made possible by recent improvements in analytical and numerical methods. Numeric information is represented visually in the tables. All simulation results are shown in two and three dimensions to show both the numerical and fundamental behavior of the single soliton. Recent research shows that this method is the best way to solve nonlinear equations that are common in mathematical physics.

Wave solutions to an integrable negative order KdV equation

In this paper, we study the traveling wave solutions to the second representative equation of the negative-order Korteweg–de Vries hierarchy. The corresponding traveling wave system is a singular planar dynamical system with a singular straight line. By using the method of dynamical systems, explicit solutions are given. But this integrable negative order Korteweg–de Vries flow has no traveling solitary wave solution.

Non-homogeneous boundary value problems of the Kawahara equation posed on a finite interval

Considered in this paper is the initial boundary value problem (IBVP) of the Kawahara equation, a class of the fifth order KdV equation, posed on a finite interval, (0.1)ut+ux+βuxxx+uxxxxx+uux=0,0<x<L,t>0,u(x,0)=ϕ(x), subject to the non-homogeneous boundary conditions (0.2)Bju=hj(t),j=1,2,3,4,5t>0where Bju=∑k=04(ajk∂xku(0,t)+bjk∂xku(L,t)),j=1,2,3,4,5and ajk,bjk(k=0,1,2,3,4andj=1,2,3,4,5) are real constants. Under some general assumptions imposed on the coefficients ajk,bjk, the IBVP (0.1)–(0.2) is shown to be locally well posed in the space Hs(0,L) for any s≥0 with naturally compatible ϕ∈Hs(0,L) and boundary values hj,j=1,2,3,4,5 belonging to some appropriate spaces with optimal regularity. The sharp Kato smoothing properties (due to Kenig, Ponce and Vega (Kenig et al., 1991; Kenig et al., 1993)) of the pure initial value problem (IVP) of the linear inhomogeneous fifth order KdV equation posed on the whole line R, vt+βvxxx+vxxxxx=g(x,t),v(x,0)=ψ(x),x,t∈R,have played an important role in establishing the well-posedness of the IBVP (0.1)–(0.2).

Lie Symmetry Analysis of Seventh Order Caudrey-Dodd- Gibbon Equation

In the present paper, seventh order Caudrey-Dodd-Gibbon (CDG) equation is solved by Lie symmetry analysis. All the geometry vector fields of seventh order KdV equations are presented. Using Lie transformation seventh order CDG equation is reduced into ordinary differential equations. These ODEs are solved by power series method to obtain exact solution. The convergence of the power series is also discussed.

Closed form solutions for the generalized0fifth-order KDV equation by using the modified exp-function method

• The new approach of modified exp-function method has been effectively executed. • Find exact soliton solutions to the non-linear partial differential equation. • We have applied successfully modified exponential function method to the generalized fifth- order KDV equation. We investigate solitary wave solutions of the generalized fifth-order Korteweg–de Vries equation, which includes additional terms such as cubic nonlinearity and fifth order linear dispersion, as well as two nonlinear dispersive terms, in addition to the traditional quadratic nonlinearity and third-order dispersion. The projected modified exponential function approach is used to find exact solitary wave solutions to this equation, and the dependence of their amplitude, width, and speed on the parameters of the governing equation is investigated. Depending on the equation parameters, the resultant solution can represent either an embedded or regular soliton. Hyperbolic, trigonometric, and rational function solutions are obtained as a result. The symbolic computation software Maple verifies the accuracy of all obtained solutions. Finally, 3D, 2D, and contour plots of some of the derived solutions are shown to demonstrate the model's physical look. Many scientific and oceanographic applications involving ocean gravity waves and other associated phenomena require our acquired conclusions in this work concerning our examined equation.

On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform

In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite difference (RBF-FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain. Laplace transform is utilized for obtaining the equivalent time-independent equation in Laplace space and also valuable to handle time-fractional derivatives in the Caputo sense. Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non-linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and resultsRBFs Methods

Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV Equations

In this study, a new modification of the newly developed semi-analytical method, optimal auxiliary function method (OAFM) is used for fractional-order KdVs equations. This method is called the fractional optimal auxiliary function method (FOAFM). The time fractional derivatives are treated with Caputo sense. A rapidly convergent series solution is obtained from the FOAFM and is validated by comparing with other results. The analysis proves that our method is simplified and applicable, contains less computational work, and has fast convergence. The beauty of this method is that there is no need to assume a small parameter such as in the perturbation method. The effectiveness and accuracy of the method is proven by numerical and graphical results.

On the Solution of Fractional Order KdV Equation and Its Periodicity on Bounded Domain Using Radial Basis Functions

The Coimbra concept of fractional order derivative is used to build a numerical approach using radial functions in this paper. The Coimbra derivative is capable of modelling a dynamic system with varying fractional order behaviour over time. The proposed scheme’s stability and convergence are investigated. In one and two space dimensions, the developed approach is validated for the given model. By applying a periodic boundary condition on a bounded domain, the model’s periodicity is shown statistically. The acquired findings demonstrate the new numerical scheme’s potency and, as a result, its high order accuracy.

The Analysis of Fractional‐Order Nonlinear Systems of Third Order KdV and Burgers Equations via a Novel Transform

In this article, we solve nonlinear systems of third order KdV Equations and the systems of coupled Burgers equations in one and two dimensions with the help of two different methods. The suggested techniques in addition with Laplace transform and Atangana–Baleanu fractional derivative operator are implemented to solve four systems. The obtained results by implementing the proposed methods are compared with exact solution. The convergence of the method is successfully presented and mathematically proved. The results we get are compared with exact solution through graphs and tables which confirms the effectiveness of the suggested techniques. In addition, the results obtained by employing the proposed approaches at different fractional orders are compared, confirming that as the value goes from fractional order to integer order, the result gets closer to the exact solution. Moreover, suggested techniques are interesting, easy, and highly accurate which confirm that these methods are suitable methods for solving any partial differential equations or systems of partial differential equations as well.

Soliton Solutions of Kaup-Kupershimdt Equation Using HPTM

In this paper, we employ the idea of homotopy perturbation transform method(HPTM), which is a combination of Laplace transformation and homotopy perturbation method(HPM) for solving nonlinear ordinary and partial differential equations. The equations are Laplace transformed and nonlinear terms are represented by He’s polynomials. The solutions are obtained in the series form which converges at fast rate with easily computable terms. Comparison of standard perturbation method shows that HPTM is appropriate even for system without any small/large parameters and therefore it can be applied more extensively than traditional perturbation techniques. HPTM is used to obtain soliton solution of Kaup-kupershimdt(KK) equation, which is modified form of fifth order KdV equation. Numerical results are presented in tabular form to show the accuracy, efficiency and reliability of this method.

Negative Order KdV Equation with No Solitary Traveling Waves

We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces different evolution equations. We apply the traveling wave setting to study one of these equations. Assuming a particular type of solution leads us to solve a cubic equation. New solutions are found, but none of these are classical solitary traveling wave solutions.

A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel

In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.

A fifth-order variable coefficient KdV equation arising from a fluid system

Starting from the model describing nonlinear phenomenon in atmosphere and ocean, a fifth order KdV equation with some arbitrary parameters is derived by using a long-wave approximation. Different to the usual treatment, the variable coefficient fifth KdV equation is obtained by the sum of the lower order term and the higher order term in the derivation procedure. Applying a novel travelling wave method to the derived model, it is found to possess many different and abundant travelling wave structures, such as the general solitary waves, the plateau soliton, the double-peak soliton, periodic waves, some type of soliton molecules (SMs) and two types of few-cycle-pulse (FCP) solitons.

Analysing the nature of plasma and blood flow using exact roving wave solutions of fifth order rove-time equation

In this paper, the nonlinear higher order differential scrupulous equations share in countless spectacles in Physics such as Fluid dynamics, Hydro dynamics, Optics, Plasma physics and so on. The exact solutions of fifth order Kaup-Kupershmit equation describe the excitations and dynamics of ionised particles such as blood and plasmas which is an essential bodily function. Here the generalised Kudrayashov method is entertained for finding the exact solutions of fifth order KdV equation and for discovering precise roving wave solutions mutable to the solitary wave solutions for Kaup-Kupershmit equation. Among the voluminous methods, generalised Kudrayashov method is very imperative and produces the knowledge of Plasma dynamics.

Novel travelling wave structures: few-cycle-pulse solitons and soliton molecules

We discuss a fifth order KdV (FOKdV) equation via a novel travelling wave method by introducing a background term. Results show that the background term plays an essential role in finding new abundant travelling wave structures, such as the soliton induced by negative background, the periodic travelling wave excited by the positive background, the few-cycle-pulse (FCP) solitons with and without background, the soliton molecules excited by the background. The FCP solitons are first obtained for the FOKdV equation.

Invariant measure of stochastic higher order KdV equation driven by Poisson processes

The current paper is devoted to stochastic damped higher order KdV equation driven by Poisson process. We establish the well-posedness of stochastic damped higher-order KdV equation, and prove that there exists an unique invariant measure for non-random initial conditions. Some discussion on the general pure jump noise case are also provided. Some numerical simulations of the invariant measure are provided to support the theoretical results.

Ergodicity of stochastic damped higher-order KdV equation driven by white noise

The current paper is devoted to the stochastic damped higher-order KdV equation driven by white noise. We prove the existence of invariant measure for stochastic damped (2n+1)-th order KdV equation with random initial value, and the ergodicity with deterministic initial data.