Obtaining soliton solutions of equations combined with the Burgers and equal width wave equations using a novel method

In this paper, more recent method, called the Exp-function, is used to find generalized solutions of the important soliton equations which is called equal width wave (EW) and modified equal width wave (MEW) equations. It is shown that the Exp-function method with the help of symbolic computation provides a straight forward and powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics. Additionally, this paper studies the EW/MEW equation by the aid of He’s semi-inverse variational principle. AMS codes: 78A60, 37K10, 35Q51, 35Q55. Key words: Exp- function method; semi-inverse variational principle; equal width wave (EW), modified equal width wave (MEW) equations.

Exact solutions and invariants of motion for general types of regularized long wave equations

Analyzing modified equal width (MEW) wave equation using the improved element-free Galerkin method

The equal-width wave (EW) equation is a model partial differential equa- tion for the simulation of one-dimensional wave propagation in media with nonlinear wave steepening and dispersion processes. The background of the EW equation is reviewed and this equation is solved by using an advanced numerical method of lines with an adaptive grid whose node movement is based on an equidistribution princi- ple. The solution procedure is described and the performance of the solution method is assessed by means of computed solutions and error measures. Many numerical solutions are presented to illustrate important features of the propagation of a solitary wave, the inelastic interaction between two solitary waves, the breakup of a Gaussian pulse into solitary waves, and the development of an undular bore.

In this paper, we analyse the equal width (EW) wave equation by using the mesh-free reproducing kernel particle Ritz (kp-Ritz) method. The mesh-free kernel particle estimate is employed to approximate the displacement field. A system of discrete equations is obtained through the application of the Ritz minimization procedure to the energy expressions. The effectiveness of the kp-Ritz method for the EW wave equation is investigated by numerical examples in this paper.

Although the homotopy analysis method (HAM) is, by now, a well-known analytic method for handling functional equations, there is no general proof of its applicability to all kinds of equations. In this paper, by using this method to solve equal-width wave (EW) and modified equal-width wave (MEW) equations, we have made a new contribution to this field of research. Our goal is to emphasize on two points: one is the efficiency of HAM in handling these important family of equations and its superiority over other analytic methods like homotopy perturbation method (HPM), variational iteration method (VIM), and Adomian decomposition method (ADM). The other point is that although the considered two equations have different nonlinear terms, we have used the same auxiliary elements to solve them.

 Abstract—In this paper, the Reduced Differential Transform method (RDTM) is used to find the numerical solution of the equal width wave (EW) equation and the exact analytical solution of the inviscid Burgers' equation with initial conditions. The method has been used successfully to investigate the motion of a single solitary wave that is governed by the EW equation. The solution is obtained in the form of convergent power series. The results obtained show that the error norms to the exact solutions are reasonably small and that the present method is easier and powerful than some other known techniques. It is commonly known that the equations of gas dynamics are the mathematical expressions of conservation laws which exist in engineering physics such as conservation of mass, conservation of momentum, conservation of energy etc. The inviscid equation of gas dynamics can be written in the conservation form which is a nonlinear partial differential equation. Partial differential equations have numerous applications in various fields such as fluid mechanics, physics, thermodynamics etc. Most of these equations are nonlinear partial differential equations. A broad class of analytical solution methods and numerical solution methods were applied to solve these partial differential equations by Morrison et al. (1), Gardner and Gardner (2), Smith (3), Courant and Friedrichs (4), Evans and Bulut (5), Wazwaz ((6) and (7)), He ((8) and (9)), Keskin (10), Peregrine (11), Bluman and Kumei (12), Whitham (13), Hunter and Keller (14), He and Moodie (15), Sharma and Radha (16), Sharma and Srinivasan (17) and Arora and Sharma (18) etc.wave equation. The equal width wave equation, introduced by Morrison et al. (1), is of great importance since it is used as a model partial differential equation for the simulation of one dimensional wave propagation in nonlinear media with dispersion process. In this Paper, we shall consider the Inviscid Burgers' equation

The equal width wave (EW) equation is a model partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. The EW-Burgers equation models the propagation of nonlinear and dispersive waves with certain dissipative effects. In this work, we derive exact solitary wave solutions for the general form of the EW equation and the generalized EW-Burgers equation with nonlinear terms of any order. We also derive analytical expressions of three invariants of motion for solitary wave solutions of the generalized EW equation.

A numerical solution of the equal width wave equation by a lumped Galerkin method

In this paper, we consider conformable equal width wave (EW) equation in order to construct its exact solutions. This equation plays an important role in physics and gives an interesting model to define change waves with weak nonlinearity. The aim of this paper is to present new exact solutions to conformable EW equation. For this purpose, we use an effective method called Improved Bernoulli Sub-Equation Function Method (IBSEFM). Based on the values of the solutions, the 2D and 3D graphs and contour surfaces are plotted with the aid of mathematics software. The obtained results confirm that IBSEFM is a powerful mathematical tool to solve nonlinear conformable partial equations arising in mathematical physics.

Three different methods for numerical solution of the EW equation

This article applies the sextic B-spline collocation scheme to obtain the approximate solution of the generalized equal width (GEW) wave equation. The accuracy of the proposed technique is discussed over three test applications including the single soliton wave, interaction of soliton waves and Maxwellian initial problem while we are getting the three invariant A 1, A 2, A 3 and two error norms referred as to L 2 and L ∞ . Applying the Von Neumann algorithm, the linearized technique is unconditionally stable. Our computational data show the superiority of results over those existing results in the literature review.

In this paper, the Exp-function method is used to find an exact solution of the equal-width wave (EW) equation. The method is straightforward and concise, and its applications are promising. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving the EW equation.

The equal width (EW) equation governs nonlinear wave phenomena like waves in shallow water. Numerical solution of the (EW) equation is obtained by using the method of lines (MOL) based on Runge-Kutta integration. Using von Neumann stability analysis, the scheme is found to be unconditionally stable. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Accuracy of the proposed method is discussed by computing theL2andL∞error norms. The results are found in good agreement with exact solution.

Objectives: In this paper, a very light and straightforward algorithm is proposed for customs fraud detection. Methods/Analysis: in order to fraud detection we have proposed our algorithm based on unsupervised methods. Our approach is a combination of data clustering methods, Mahalanobis distance classifier, K Nearest Neighbor (KNN) method, and density-based methods. Findings: The results showed that the proposed method was able to accurately identify frauds, as more than 73 percent of high-risk goods that the proposed method is detected, has been violated. It is faster and more rapid than the other methods. The method requires less processing than other methods, and more than 30 percent CPU usage has been improved. The approach is independent of distribution and scattering of data samples. It also has the ability to work with samples by different clusters, densities, and no limitation on dimension of data. Novelty of the Study: For the first time, an unsupervised method is used for finding the frauds in customs. Application/Improvements: One of the most important applications of the results of this study is the Customs Risk Management System. Also, the proposed approach will enhance the ability of fraud detection in trade.