Existence and numerical simulation of periodic traveling wave solutions to the Casimir equation for the Ito system

Abbasbandy, Saeid; Gorder, Robert A. Van; Hajiketabi, M.; Mesrizadeh, Mehdi

doi:10.1016/j.cnsns.2015.03.016

Cited by 7 publications

(4 citation statements)

References 24 publications

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“…Recalling that η k > 0, the property of the fixed point is not altered by the map (21). More precisely, the map (21) has the same type of stability that the system (4) has.…”

Section: A Brief Review Of the Gbbmb Equationmentioning

confidence: 98%

“…This means that u k is a fixed point of the discretized mapping (21) if and only if the point u k is an equilibrium (critical, fixed) point of the system (4). We next check the property of the fixed point.…”

Section: A Brief Review Of the Gbbmb Equationmentioning

confidence: 99%

“…From the class of numerical Lie group methods, the so-called group preserving scheme (GPS), introduced by Liu [20], is a numerical approach which utilizes the Cayley transformation and the Padé approximations in the Minkowski space. Avoiding the spurious solutions and ghost fixed points are main benefits of this method [21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains

Hajiketabi

Abbasbandy

Casas

2018

Applied Mathematics and Computation

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The aim of this paper is to introduce a new numerical method for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers (GBBMB) equation. This method is combination of group preserving scheme (GPS) with radial basis functions (RBFs), which takes advantage of two powerful methods, one as geometric numerical integration method and the other meshless method. Thus, we introduce this method as the Lie-group method based on radial basis functions (LG-RBFs). In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. One of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions. To demonstrate this point, we solve nonlinear GBBMB equation on various geometric domains in one, two and three dimension spaces. The results of numerical experiments are compared with analytical solutions and the method presented in [6] to confirm the accuracy and efficiency of the presented method.

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“…Recalling that η k > 0, the property of the fixed point is not altered by the map (21). More precisely, the map (21) has the same type of stability that the system (4) has.…”

Section: A Brief Review Of the Gbbmb Equationmentioning

confidence: 98%

Section: A Brief Review Of the Gbbmb Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains

Hajiketabi

Abbasbandy

Casas

2018

Applied Mathematics and Computation

Self Cite

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show abstract

“…Another area where using numerical integration methods preserving the Lie group structure is of paramount importance is in the context of the augmented dynamical systems technique. This was first proposed by Liu in [6] to integrate numerically any system of k differential equations x = f (t, x), x ∈ R k , and later applied to a variety of settings, ranging from stiff differential equations to boundary value problems and systems with constraints [7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Numerical integrators based on the Magnus expansion for nonlinear dynamical systems

Hajiketabi

Casas

2020

Applied Mathematics and Computation

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Explicit numerical integration algorithms up to order four based on the Magnus expansion for nonlinear non-autonomous ordinary differential equations are presented and tested on problems possessing qualitative (very often, geometric) features that is convenient to preserve under numerical discretization. The range of applications covers augmented dynamical systems, highly oscillatory problems and nonlinear non-autonomous partial differential equations of evolution previously discretized in space.

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The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: Application to the heat equation

Hajiketabi

Abbasbandy

2018

Engineering Analysis with Boundary Elements

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Existence and numerical simulation of periodic traveling wave solutions to the Casimir equation for the Ito system

Cited by 7 publications

References 24 publications

The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains

The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains

Numerical integrators based on the Magnus expansion for nonlinear dynamical systems

The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: Application to the heat equation

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