On the numerical solution of Korteweg–de Vries equation by the iterative splitting method

Gücüyenen, Nurcan; Tanoğlu, Gamze

doi:10.1016/j.amc.2011.03.084

Cited by 5 publications

(5 citation statements)

References 12 publications

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“…The same parameters ϵ = 6 and μ = 1 are used. The exact solution [21] is given as follows

U false(x, t false) = 0.5 \sec h^{2} false(0.5 false(x - t false) false),

and the initial condition is taken from the exact solution by using t = 0 as

U false(x, 0 false) = 0.5 \sec h^{2} false(0.5 x false) .

…”

Section: Test Problems and Discussionmentioning

confidence: 99%

“…As a result of increasing amplitude value and change of the initial position, the velocity is increasing and solution domain should be changed to [−15, 15]. First, small‐time solutions are obtained and then the time interval is extended to the [0, 5] to be able to compare with earlier methods iterative splitting method and Lie‐Trotter splitting method [21] for long‐time solution with the same parameters Δ t = 0.0005, N = 101. Since, small‐time solutions of this problem do not existing in the literature, numerical results for small times t = 0.005 and t = 0.01 are given with exact values in Table 3.…”

Section: Test Problems and Discussionmentioning

confidence: 99%

“…Since the numerical results are too close to exact solutions the graphs are indistinguishable. For the long‐time solution, the simulation is run up to time t = 5 and error values L 2 and L ∞ are compared with earlier methods such as iterative splitting method and Lie‐Trotter splitting method [21] in Table 4. The present error values are the least of the given ones.…”

Section: Test Problems and Discussionmentioning

confidence: 99%

“…Besides the exact solutions, different base functions are applied for many methods. Among others, collocation method [15][16][17][18][19][20], iterative splitting method [21], Galerkin method [22][23][24], heat balance integral method [25], analytical-numerical method [26], spline approximation [27], the method of lines [28], exponential finite difference method [29], Haar wavelet method [30], differential quadrature method [31,32], Taylor-Galerkin method [33] and modified Petrov-Galerkin method [34].…”

mentioning

confidence: 99%

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Single soliton and double soliton solutions of the quadratic‐nonlinear Korteweg‐de Vries equation for small and long‐times

Başhan

Esen

2020

Numerical Methods Partial

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In this article, numerical solutions of the seven different forms of the single soliton and double soliton solutions of the Korteweg‐de Vries equation are investigated. Since numerical solution of the six test problems for small‐times do not exist in the literature, the present numerical results firstly are reported with exact solutions. Besides small‐time solutions, long‐time solutions of all test problems are obtained and compared with some of the earlier works. Present algorithm which is based on combination of finite difference method and differential quadrature method have obtained superior results than those in the given literature. Numerical and exact solutions for small‐time of all test problems are plotted together for all test problems. Since the numerical results are too close to exact solutions the graphs are indistinguishable. Numerical simulations for long‐time solutions are plotted and the error graphs are plotted for the end of the simulations of all test problems.

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“…The same parameters ϵ = 6 and μ = 1 are used. The exact solution [21] is given as follows

U false(x, t false) = 0.5 \sec h^{2} false(0.5 false(x - t false) false),

and the initial condition is taken from the exact solution by using t = 0 as

U false(x, 0 false) = 0.5 \sec h^{2} false(0.5 x false) .

…”

Section: Test Problems and Discussionmentioning

confidence: 99%

Section: Test Problems and Discussionmentioning

confidence: 99%

Section: Test Problems and Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Single soliton and double soliton solutions of the quadratic‐nonlinear Korteweg‐de Vries equation for small and long‐times

Başhan

Esen

2020

Numerical Methods Partial

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show abstract

“…Idrees et al [27] use the optimal homotopic asymptotic technique to solve this equation. To solve the KdV equation numerically, Gücüyenen and Tanoğlu [28] used the iterative splitting approach. Sarma [29] provided a solitary wave solution for this equation.…”

Section: Introductionmentioning

confidence: 99%

Differential Quadrature Method to Examine the Dynamical Behavior of Soliton Solutions to the Korteweg-de Vries Equation

Mishra

Arora

Emadifar

et al. 2022

Advances in Mathematical Physics

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Nonlinear evolution equations are crucial for understanding the phenomena in science and technology. One such equation with periodic solutions that has applications in various fields of physics is the Korteweg-de Vries (KdV) equation. In the present work, we are concerned with the implementation of a newly defined quintic B-spline basis function in the differential quadrature method for solving the Korteweg-de Vries (KdV) equation. The results are presented using four experiments involving a single soliton and the interaction of solitons. The accuracy and efficiency of the method are presented by computing the L 2 and L ∞ norms along with the conservational quantities in the forms of tables. The results show that the proposed scheme not only gives acceptable results but also consumes less time, as shown by the CPU for the elapsed time in two examples. The graphical representations of the obtained numerical solutions are compared with the exact solution to discuss the nature of solitons and their interactions for more than one soliton.

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Modification of quintic B-spline differential quadrature method to nonlinear Korteweg-de Vries equation and numerical experiments

Başhan

2021

Applied Numerical Mathematics

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On the numerical solution of Korteweg–de Vries equation by the iterative splitting method

Cited by 5 publications

References 12 publications

Single soliton and double soliton solutions of the quadratic‐nonlinear Korteweg‐de Vries equation for small and long‐times

Single soliton and double soliton solutions of the quadratic‐nonlinear Korteweg‐de Vries equation for small and long‐times

Differential Quadrature Method to Examine the Dynamical Behavior of Soliton Solutions to the Korteweg-de Vries Equation

Modification of quintic B-spline differential quadrature method to nonlinear Korteweg-de Vries equation and numerical experiments

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