Regularized Long Wave Equation Research Articles

Interpolating element-free Galerkin method for the regularized long wave equation and its error analysis

Combining the simplified interpolating moving least-squares (IMLS) method with the Galerkin weak form, an interpolating element-free Galerkin (IEFG) method is presented for the numerical study of the regularized long wave (RLW) equation. The trial function in the IMLS method has one less unknown coefficient than in the traditional MLS approximation, and the shape functions satisfy the interpolating property at nodes. Then the IEFG method can apply the essential boundary conditions naturally and directly, and has high computational accuracy. And then this paper studies the error estimates of the IEFG method for RLW equation. The error estimate of the semi-discrete IEFG method for the RLW equation on the influence domain radius is first studied, and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied. The theoretical results show that the IEFG method for RLW equation has good convergence rate. For the purpose of demonstration, three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper.

Optical solitons of some fractional differential equations in nonlinear optics

This paper deals with two different forms of the conformable fractional Benjamin–Bona–Mahony (BBM) equations by an analytical method. These physical models have important applications for describing the propagation of optical pulses in non-linear media. The conformable fractional symmetric BBM equation and the conformable time fractional Equal-width (EW) equation are considered. The extended Jacobi’s elliptic function expansion scheme are used to extract explicit solitons.

Conservation laws, soliton solutions for modified Camassa–Holm equation and (2 + 1)-dimensional ZK–BBM equation

By using the multiplier approach, we construct the conservation laws and the corresponding conserved quantities for the modified Camassa–Holm equation and (2 + 1)-dimensional Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation for each multipliers. We also deduce the soliton solutions for the equations using semi-inverse variational principle. We give the discussion of the properties of the soliton waves obtained numerically via some figures and the physical interpretation to complete these studies. Finally, we compare the solutions obtained with other solutions obtained in previous papers to prove that the results in this paper cannot appear anywhere.

Soliton-type solutions for two models in mathematical physics

In this paper, the generalised Klein-Gordon and Kadomtsov–Petviashvili Benjamin–Bona–Mahony equations with power law nonlinearity are investigated. Our study is based on reducing the form of both equations to a first-order ordinary differential equation having the travelling wave solutions. Subsequently, soliton-type solutions such as compacton and solitary pattern solutions are obtained analytically. Additionally, the peaked soliton has been derived where it exists under a specific restrictions. In addition to the soliton solutions, the mathematical method which is exploited in this work also creates a few amount of travelling wave solutions.

Linearized difference schemes for a BBM equation with a fractional nonlocal viscous term

This paper is concerned with the efficient finite difference schemes for a Benjamin–Bona–Mahony equation with a fractional nonlocal viscous term. By using the weighted-shift Grünwald–Letnikov and the fractional centered difference formulae to approximate the nonlocal fractional operators, we design a class of linearized finite difference schemes for the presented nonlocal model. The existence, stability and convergence of the proposed numerical schemes are rigorously derived with the help of functional analysis. Theoretical analysis shows that the proposed numerical schemes are stable with second order accuracy. Numerical examples are presented to verify our theoretical analysis and to demonstrate the efficiency of the proposed numerical schemes.

A note on the Gaussons of some new logarithmic evolution equations

In this paper, we study the Gaussian solitary waves for some nonlinear evolution equations with logarithmic nonlinearities. These studied logarithmic evolution equations are the generalized logarithmic BBM equations, the logarithmic (2+1)-dimensional KP-like equations, the logarithmic (3+1)-dimensional KP-like equations, the generalized logarithmic (2+1)-dimensional Klein-Gordon equations and the generalized logarithmic (3+1)-dimensional Klein-Gordon equations. We not only prove that they possess Gaussons: solitary wave solutions of Gaussian shape but also derive the relationships among the parameters.

A new approach for solving highly nonlinear partial differential equations by successive differentiation method

In this work successive differentiation method is applied to solve highly nonlinear partial differential equations (PDEs) such as Benjamin–Bona–Mahony equation, Burger's equation, Fornberg–Whitham equation, and Gardner equation. To show the efficacy of this new technique, figures have been incorporated to compare exact solution and results of this method. Wave variable is used to convert the highly nonlinear PDE into ordinary differential equation with order reduction. Then successive differentiation method is utilized to obtain the numerical solution of considered PDEs in this paper. Copyright © 2017 John Wiley & Sons, Ltd.

An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a $C^0$ finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.

Elliptic solutions and solitary waves of a higher order KdV–BBM long wave equation

We provide conditions for the existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass ℘ functions of both third and fifth-order KdV–BBM (Korteweg-de Vries–Benjamin-Bona-Mahony) regularized long wave equation. An analysis for the initial value problem is developed together with a local and global well-posedness theory for the third-order KdV–BBM equation. Traveling wave reduction is used together with zero boundary conditions to yield solitons and periodic unbounded solutions, while for nonzero boundary conditions we find solutions in terms of Weierstrass elliptic ℘ functions. For the fifth-order KdV–BBM equation we show that the parameter γ=112, which leads to a Hamiltonian, represents a restriction in where there are constraint curves that never intersect a region of unbounded solitary waves. This in turn shows that only dark or bright solitons and no unbounded solutions exist. Motivated by the lack of a Hamiltonian structure for γ≠112 we develop Hk bounds, and we show for the non-Hamiltonian system that dark and bright solitons coexist together with unbounded periodic solutions. For nonzero boundary conditions, due to the complexity of the nonlinear algebraic system of coefficients of the elliptic equation we construct Weierstrass solutions for a particular set of parameters only.

Unified Riccati equation expansion method and its application to two new classes of Benjamin–Bona–Mahony equations

The Backlund transformations and the superposition formulas of the Riccati equation with constant coefficients are constructed. Two fractional type solutions of the Riccati equation are obtained from its Backlund transformations. The equivalence relations between fractional solutions and previous known solutions are proved. A so-called unified Riccati equation expansion method for generating infinite number of exact traveling wave solutions for nonlinear evolution equations is then developed on the basis of the fractional solutions. With the method, infinitely many exact traveling wave solutions of two new classes of Benjamin–Bona–Mahony equations are presented.

Soliton solutions of (3 + 1)-dimensional Korteweg-de Vries Benjamin–Bona–Mahony, Kadomtsev–Petviashvili Benjamin–Bona–Mahony and modified Korteweg de Vries–Zakharov–Kuznetsov equations and their applications in water waves

In this article, the analytical solution of (3+1)-dimensional Korteweg-de Vries Benjamin–Bona–Mahony equation, Kadomtsev–Petviashvili Benjamin–Bona–Mahony equation and modified Korteweg-de Vries–Zakharov–Kuznetsov equation have been extracted. These results hold numerous traveling wave solutions that are of key importance in elucidating some physical circumstance. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research.

Generalized regularized long wave equation with white noise dispersion

In this article, we address the generalized BBM equation with white noise dispersion which reads $$\begin{aligned} du-du_{xx}+u_x \circ dW+ u^pu_xdt=0, \end{aligned}$$ in the Stratonovich formulation, where W(t) is a standard real valued Brownian motion. We first investigate the well-posedness of the initial value problem for this equation. We then prove theoretically and numerically that for a deterministic initial data, the expectation of the \(L^\infty _x\) norm of the solutions decays to zero at \(O(t^{-\frac{1}{6}})\) as t approaches to \(+\infty \), by assuming that \(p>8\) and that the initial data is small in \(L^1_x\cap H^4_x\). This decay rate matches the one for solutions of the linear equation with white noise dispersion.

Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels

Backward compact dynamics is deduced for a non-autonomous Benjamin-Bona-Mahony (BBM) equation on an unbounded 3D-channel. A backward compact attractor is defined by a time-dependent family of backward compact, invariant and pullback attracting sets. The theoretical existence result for such an attractor is derived from the backward flattening property, and this property is proved to be equivalent to the backward asymptotic compactness in a uniformly convex Banach space. Finally, it is shown that the BBM equation has a backward compact attractor in a Sobolev space under some suitable assumptions, such as, backward translation boundedness and backward small-tail. Both spectrum decomposition and cut-off technique are used to give all required backward uniform estimates.

Complete characterization of algebraic traveling wave solutions for the Boussinesq, Klein–Gordon and Benjamin–Bona–Mahony equations

In this paper, using a new method provided in [4] we characterize all algebraic traveling wave solutions of the fourth order Boussinesq equation, the nonlinear Klein–Gordon equation and a generalized Benjamin–Bona–Mahony equation.

A Meshless Method Using Radial Basis Functions for the Numerical Solution of Two-Dimensional ZK–BBM Equation

The main idea of this article is obtain the numerical solution of the two-dimensional ZK–BBM equation. We construct the solution by using different approach, that is based on using collocation points. The solution is based on using the thin plate splines radial basis function, which builds an approximated solution with discretizing the time and the space to small steps. we use a predictor–corrector scheme to avoid solving the nonlinear system. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method

In this paper, combining with a new generalized ansätz and the fractional Jacobi elliptic equation, an improved fractional Jacobi elliptic equation method is proposed for seeking exact solutions of space‐time fractional partial differential equations. The fractional derivative used here is the modified Riemann‐Liouville derivative. For illustrating the validity of this method, we apply it to solve the space‐time fractional Fokas equation and the the space‐time fractional BBM equation. As a result, some new general exact solutions expressed in various forms including the solitary wave solutions, the periodic wave solutions, and Jacobi elliptic functions solutions for the two equations are found with the aid of mathematical software Maple. Copyright © 2016 John Wiley & Sons, Ltd.

Periodic traveling wave solution for non-homogeneous BBM and KdVB equations

In this paper the Green’s function method and results about fixed point are used to get existence results on periodic traveling wave solution for non-homogeneous problems of generalized versions of the BBM and KdVB equations. It is shown through the constructions of explicit Green’s functions that the periodic boundary value problems for the traveling wave solutions of the BBM and KdVB equations are equivalent to integral equations which generate compact operators in the space of periodic functions. These integral representations allowed us to prove that if the speed of the wave propagation is suitably chosen, then the BBM and KdVB equations will admit periodic traveling wave solution.

On the Solutions of the Space and Time Fractional Benjamin–Bona–Mahony Equation

In this paper, the extended trial equation method is used for constructing exact solutions of fractional partial differential equations (FPDEs) in the sense of Jumarie’s modified Riemann–Liouville derivative. To convert a fractional partial differential equation into its ordinary differential equation of integer order is used the fractional complex transformation. With the aid of symbolic computation, we choose the space and time fractional Benjamin–Bona–Mahony (BBM) equation to illustrate the validity and advantages of this method. As a result, some new exact solutions including elliptic integral function solutions and soliton solutions to this problem are successfully established. The proposed approach can also be applied to other fractional partial differential equations (FPDEs).

HAAR WAVELETS BASED TIME DISCRETIZATION TECHNIQUE FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

Abstract: A new numerical technique is developed to find the solutions of general nonlinear partial differential equations. The technique is based on the time discretization of Haar wavelet series approximations with quasilinearization process. In order to test the efficiency of the proposed technique, it is applied on well known nonlinear partial differential equations such as the generalized regularized long wave equation, the Benjamin Bona-Mahony equation and the Fitzhugh-Nagumo equation. Numerical results are obtained by preparing MATLAB codes of proposed techniques. The beautiful concentration profiles of u and v are shown by figures at different time level and error norms L2 and L∞ are calculated.

Norm-inflation results for the BBM equation

Considered here is the periodic initial-value problem for the regularized long-wave (BBM) equationut+ux+uux−uxxt=0. Adding to previous work in the literature, it is shown here that for any s<0, there is smooth initial data that is small in the L2-based Sobolev spaces Hs, but the solution emanating from it becomes arbitrarily large in arbitrarily small time. This so called norm inflation result has as a consequence the previously determined conclusion that this problem is ill-posed in these negative-norm spaces.