Time-Fractional KdV Equation: Formulation and Solution using Variational Methods

In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then, the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He's variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure

A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: generalized Korteweg-de Vries equation

This work deals with the constitute of numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with Petrov-Galerkin finite element approach utilising a cubic B-spline function as the trial function and a quadratic function as the test function. Accurateness and effectiveness of the submitted methods are shown by employing propagation of single solitary wave. The L2, L∞error norms and I1, I2and I3invariants are used to validate the applicability and durability of our numerical algorithm. Implementing the Von-Neumann theory, it is manifested that the suggested method is marginally stable. Furthermore, supernonlinear traveling wave solution of the GKdV equation is presented using phase plots. It is seen that the GKdV equation supports superperiodic traveling wave solution only and it is significantly affected by velocity and nonlinear parameters. Also, considering a superficial periodic forcing multistability of traveling waves of perturbed GKdV equation is presented. It is found that the perturbed GKdV equation supports coexisting chaotic and various quasiperiodic features with same parametric values at different initial condition

New exact solutionsand numerical approximations of the generalized kdv equation

This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element method based on cubic B-splines over finite elements. Propagation of single solitary wave is investigated to show the efficiency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L∞ error norms. Also, three invariants I1, I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann’s theory is surveyed and indicated that our method is unconditionally stable

Analytical solutions of some special nonlinear partial differential equations using Elzaki-Adomian decomposition method

We apply the Elzaki-Adomian Decomposition Method (EADM) in this study to solve nonlinear Benjamin-Bona-Mahony (BBM) and Fisher's partial differential equations (PDE). This method, being an integral transform, is a hybrid of two well-known and efficient methods: the Elzaki transform and the Adomian decomposition method. The method is demonstrated by solving two special cases of the BBM Equation and one special case of Fisher's partial differential equation. Because of its high convergence rate in approximating exact solutions, this approach is very dependable. The method can also produce numerical solutions without the usage of restrictive assumptions or the discretization typical of numerical methods; making it free of round-off errors. The Elzaki-Adomian Decomposition method employs a straightforward computation that leads to effectiveness. The efficiency of EADM is demonstrated in the significant reduction of number of numerical computations. The effectiveness and efficiency of EADM account for its broad application, particularly for higher order PDEs

Numerical simulation of the nonlinear coupled jaulent-miodek equation by elzaki transform-adomian polynomial method

The Elzaki transform which is an integral transform used to obtain solutions of linear differential equations is coupled with Adomian polynomial to solve nonlinear coupled Jaulent-Miodek (JM) equation. The Adomian polynomial is used to linearise the nonlinear functions in the partial differential equation before the scheme of the Elzaki transform was used to iteratively generate each term of the series solution. The solutions obtained were compared with the exact solutions and were found to give a very small error, the graphical representation of the solutions which give the shape of the solitons also agree with that of the Adomian decomposition method when a comparison is made. The method is powerful and effective as it does not involve large computer memory and does not involve discretizing the independent variables to achieve the required solution.publishedVersio

Algorithm for solving a generalized Hirota-Satsuma Coupled KdV equation using homotopy perturbation transform method

In this paper, we merge homotopy perturbation method with He’s polynomials and Laplace transformation method to produce a highly effective algorithm for finding approximate solutions for generalized Hirota-Satsuma Coupled KdV equations. This technique is called the Homotopy Perturbation Transform Method (HPTM). With this technique, the solutions are obtained without any discretization or restrictive assumptions, and devoid of roundoff errors. This technique solved a generalized Hirota-Satsuma Coupled KdV equation without using Adomian’s polynomials which can be considered as a clear advantage over the decomposition method. MAPLE software was used to calculate the series generated from the algorithm. The results reveal that the homotopy perturbation transform method (HPTM) is very efficient, simple and can be applied to other nonlinear problems.Keywords: Coupled KdV equations, Homotopy perturbation transform method, Laplace transform method, Maple software, He’s polynomia

Approximate Method for Studying the Waves Propagating along the Interface between Air-Water

This paper is devoted to consider the approximate solutions of the nonlinear water wave problem for a fluid layer of finite depth in the presence of gravity. The method of multiple-scale expansion is employed to obtain the Korteweg-de Vries (KdV) equations for solitons, which describes the behavior of the system for free surface between air and water in a nonlinear approach. The solutions of the water wave problem split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as the solutions of KdV equations. The solution of KdV equations is obtained analytically by using a reliable modification of Laplace decomposition method (LDM), namely, the modified Laplace decomposition method (MLDM) is presented. This procedure is a powerful tool for solving large amount of nonlinear problems. The proposed method provides the solution as a series which may converge to the exact solution of the problem. Also, the convergence analysis of the proposed method is given. Finally, we observe that the elevation of the water waves is in form of traveling solitary waves. The horizontal and vertical of the velocity components have nonlinear characters