Numerical KDV Equation by the Adomian Decomposition Method

Akdi, Mahmoud; Sedra, M. B.

doi:10.11648/j.ajmp.20130203.13

Cited by 8 publications

(7 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When α = 1, this problem has a soliton solution in the form of u (x, t) = 0.5 sech 2 (0.5 (x − t)) in case ξ = 6 and δ = 1 (see [5]). …”

Section: Numerical Results and Analysismentioning

confidence: 99%

“…The KdV equation was firstly derived as an evolution equation that governs small-amplitude, long-surface gravity waves propagating in a shallow channel of water. However, the KdV equation can only be solved analytically in selected initial and boundary conditions due to its nonlinearity [2][3][4][5]. With the development of computational techniques, the KdV equation gained much attention of mathematicians and physicists in the past few years.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation

Cao

Zheng

2018

Applied Sciences

View full text Add to dashboard Cite

Abstract:In this paper, we studied the numerical solution of a time-fractional Korteweg-de Vries (KdV) equation with new generalized fractional derivative proposed recently. The fractional derivative employed in this paper was defined in Caputo sense and contained a scale function and a weight function. A finite difference/collocation scheme based on Jacobi-Gauss-Lobatto (JGL) nodes was applied to solve this equation and the corresponding stability was analyzed theoretically, while the convergence was verified numerically. Furthermore, we investigated the behavior of solution of the generalized KdV equation depending on its parameter δ, scale function z(t) in fractional derivative. We found that the full discrete scheme was effective to obtain a numerical solution of the new KdV equation with different conditions. The wave number δ in front of the third order space derivative term played a significant role in splitting a soliton wave into multiple small pieces.

show abstract

“…When α = 1, this problem has a soliton solution in the form of u (x, t) = 0.5 sech 2 (0.5 (x − t)) in case ξ = 6 and δ = 1 (see [5]). …”

Section: Numerical Results and Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation

Cao

Zheng

2018

Applied Sciences

View full text Add to dashboard Cite

show abstract

“…Once created, the researchers can predict the evolution of the solitary waves with respect to the spatial and time variables. The following figure gives an idea about the numerical calculation obtained [3] and [4]. There exist other numerical methods that can also be used depending on their calculation effectiveness [5] and [6].…”

Section: The Adomian Decomposition Methods Applied To the Kd-v Equationmentioning

confidence: 99%

Solitons and the new renewable energy approach

Akdi

Sedra²

2015

Ren. Ener. & Sust. Dev.

View full text Add to dashboard Cite

This article aims at deepening research on the new renewable energy production from the dams reservoir. Coupled to the solitary waves, the wind generated waves present an attractive natural phenomenon that deserves to be emphasized. The Adomain numerical method is used to model the practical observation regarding the soliton propagation. The actual cases contained in this document provide an illustration of the given principals.

show abstract

“…A Legendre-Petrov-Galerkin method was designed to solve the KdV equation by Ma and Sun [16]. Akdi and Sedra [1] gave the Adomian decomposition method for the numerical solution of the KdV equation. Yan and Shu presented a local discontinuous Galerkin method for solving KdV type equations [38].…”

Section: Introductionmentioning

confidence: 99%

Wavelet-based Numerical Approaches for Solving the Korteweg-de Vries (KdV) Equation

Özdemir

Seçer

2022

Turkish Journal of Mathematics and Computer Science

View full text Add to dashboard Cite

In this research work, we examine the Korteweg-de Vries equation (KdV), which is utilized to formulate the propagation of water waves and occurs in different fields such as hydrodynamics waves in cold plasma acoustic waves in harmonic crystals. This research work presents two efficient computational methods based on Legendre wavelets to solve the Korteweg-de Vries. The three-step Taylor method is first applied to the Korteweg-de Vries equation for time discretization. Then, the Galerkin method and the collocation method are used for spatial discretization. With these approaches, bringing the approximate solutions of the Korteweg-de Vries equation turns into getting the solution of the algebraic equation system. The solution of this system gives the Legendre wavelet coefficients. Substituting the obtained coefficients into the Legendre wavelet series expansion, the approximate solution can be obtained. The presented wavelet methods are tested by studying different problems at the end of this study.

show abstract

Numerical KDV Equation by the Adomian Decomposition Method

Cited by 8 publications

References 4 publications

Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation

Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation

Solitons and the new renewable energy approach

Wavelet-based Numerical Approaches for Solving the Korteweg-de Vries (KdV) Equation

Contact Info

Product

Resources

About