An Unconventional Finite Difference Scheme for Modified Korteweg-de Vries Equation

Köroğlu, Canan; Aydın, Ayhan

doi:10.1155/2017/4796070

Cited by 5 publications

(3 citation statements)

References 11 publications

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“…In recent years, NSFD models have been proven to be one of the efficient numerical algorithms for large step sizes. They are not exact models, but they do not possess numerical instabilities when they are compared with standard finite difference models (see [16] and the references therein). In the construction of an NSFD model, it is assumed that the orders of the derivatives must be exactly equal to the orders of the corresponding derivatives of the differential equations [38].…”

Section: Nsfd Scheme For the Kdvb(2 1 2) Equationmentioning

confidence: 99%

Exact and nonstandard finite difference schemes for the generalized KdV–Burgers equation

Köroğlu

2020

Adv Differ Equ

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We consider the generalized KdV-Burgers KdVB(p, m, q) equation. We have designed exact and consistent nonstandard finite difference schemes (NSFD) for the numerical solution of the KdVB(2, 1, 2) equation. In particular, we have proposed three explicit and three fully implicit exact finite difference schemes. The proposed NSFD scheme is linearly implicit. The chosen numerical experiment consists of tanh function. The NSFD scheme is compared with a standard finite difference(SFD) scheme. Numerical results show that the NSFD scheme is accurate and efficient in the numerical simulation of the kink-wave solution of the KdVB(2, 1, 2) equation. We see that while the SFD scheme yields numerical instability for large step sizes, the NSFD scheme provides reliable results for long time integration. Local truncation error reveals that the NSFD scheme is consistent with the KdVB(2, 1, 2) equation.

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Section: Nsfd Scheme For the Kdvb(2 1 2) Equationmentioning

confidence: 99%

Exact and nonstandard finite difference schemes for the generalized KdV–Burgers equation

Köroğlu

2020

Adv Differ Equ

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“…can be calculated by compatible condition φ n,t = φ t,n , which is subject to equation ( 1). If we assume that q n = εqðnεÞ in equation (1) and to rescale time t ↦ t/2ε 2 , then, to send ε ⟶ 0, thus, equation ( 1) can be reduced to the following continuous mKdV equation [26,31,32]:…”

Section: Introductionmentioning

confidence: 99%

Various Soliton Solutions and Asymptotic State Analysis for the Discrete Modified Korteweg-de Vries Equation

Lin

Wen

Qin

2021

Advances in Mathematical Physics

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Under investigation is the discrete modified Korteweg-de Vries (mKdV) equation, which is an integrable discretization of the continuous mKdV equation that can describe some physical phenomena such as dynamics of anharmonic lattices, solitary waves in dusty plasmas, and fluctuations in nonlinear optics. Through constructing the discrete generalized m , N − m -fold Darboux transformation for this discrete system, the various discrete soliton solutions such as the usual soliton, rational soliton, and their mixed soliton solutions are derived. The elastic interaction phenomena and physical characteristics are discussed and illustrated graphically. The limit states of diverse soliton solutions are analyzed via the asymptotic analysis technique. Numerical simulations are used to display the dynamical behaviors of some soliton solutions. The results given in this paper might be helpful for better understanding the physical phenomena in plasma and nonlinear optics.

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“…Aydin et al [2] have proposed and studied a linearly implicit NSFD method for the numerical solution of modified KdV equation. Koroglu et.al [9] have presented a NSFD scheme with theta method which includes the implicit Euler and a Crank-Nicolson type discretization for the numerical solution of the modified Korteweg-de Vries (MKdV) equation. In [10], the author have designed exact and consistent nonstandard finite difference schemes for the numerical solution of the KdVB( 2, 1, 2 ) equation.…”

Section: Introductionmentioning

confidence: 99%

Exact and nonstandard finite difference schemes for the Burgers equationB(2,2)

Köroğlu¹,

Aydın²

2021

Turk J Math

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In this paper, we consider the Burgers equation B(2, 2). Exact and nonstandard finite difference schemes (NSFD) for the Burgers equation B(2, 2) are designed. First, two exact finite difference schemes for the Burgers equation B(2, 2) are proposed using traveling wave solution. Then, two NSFD schemes are represented for this equation. These two NSFD schemes are compared with a standard finite difference(SFD) scheme. Numerical results show that the NSFD schemes are accurate and efficient in the numerical simulation of the kink-wave solution of the B(2, 2) equation. We see that although the SFD scheme yields numerical instability for large step sizes, NSFD schemes provide reliable results for long time integration. Local truncation errors show that the NSFD schemes are consistent with the B(2, 2) equation.

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An Unconventional Finite Difference Scheme for Modified Korteweg-de Vries Equation

Cited by 5 publications

References 11 publications

Exact and nonstandard finite difference schemes for the generalized KdV–Burgers equation

Exact and nonstandard finite difference schemes for the generalized KdV–Burgers equation

Various Soliton Solutions and Asymptotic State Analysis for the Discrete Modified Korteweg-de Vries Equation

Exact and nonstandard finite difference schemes for the Burgers equationB(2,2)

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