A spline collocation approach for the numerical solution of a generalized nonlinear Klein–Gordon equation

Khuri, Sami; Sayfy, A.

doi:10.1016/j.amc.2010.01.122

Cited by 33 publications

(32 citation statements)

References 10 publications

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“…The accuracy as can be seen from Table 4 is rather good in the order of O(10 −2 )−O(10 −5 ). We compare with the numerical results in [23] with that from the finite element collocation approach with the cubic B-splines (see Table 10 of the above cited paper), our numerical results are better than the above-mentioned studies.…”

Section: Examplesupporting

confidence: 55%

“…Bülbül and Sezer [41] have calculated this experiment by the Taylor matrix method presented that this approach does not give good results with terminal times T = 1, 2, 3, 4 and 5 as displayed in their Table 1. Upon comparing with the numerical results in [23] with that from the finite element collocation approach with the cubic B-splines (see Table 8 of the above cited paper), our numerical results are better than the above-mentioned references.…”

Section: Examplesupporting

confidence: 51%

“…Dehghan and Shokri [38] have calculated this example by the thin plate splines radial basis function and shown that this method does not give satisfactory results with terminal times T = 1, 2, 3, 4 and 5 as shown in their Table 2. Upon comparing with the numerical results in [23] with that from the finite element collocation approach with the cubic B-splines (see Table 5 of the above cited paper), and in [17] with that from the mixed finite difference and collocation methods (see Table 3 of the above cited paper), our numerical results are better than the above-mentioned literature. Table 1 Numerical errors in the solution of Example 1.…”

Section: Examplesupporting

confidence: 45%

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An implicit Lie-group iterative scheme for solving the nonlinear Klein–Gordon and sine-Gordon equations

Chang

Liu

2016

Applied Mathematical Modelling

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a b s t r a c tIn this article, the nonlinear Klein-Gordon and sine-Gordon equations are solved by pondering the semi-discretization numerical schemes and then, the resulting ordinary differential equations at the discretized spaces are numerically integrated toward the time direction by using the implicit Lie-group iterative method to find the unknown physical quantity. When six numerical experiments are examined, we reveal that the present implicit Lie-group iterative scheme is applicable to the nonlinear Klein-Gordon and sine-Gordon equations and convergent very fast at each time marching step, and the accuracy is raised several orders, of which the numerical results are rather accurate, effective and stable.

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Section: Examplesupporting

confidence: 55%

Section: Examplesupporting

confidence: 51%

Section: Examplesupporting

confidence: 45%

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An implicit Lie-group iterative scheme for solving the nonlinear Klein–Gordon and sine-Gordon equations

Chang

Liu

2016

Applied Mathematical Modelling

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“…Over the last four decades, spectral methods [1][2][3][4][5][6][7] have been developed rapidly. They have a good reputation compared with others numerical tools due to their wide applications in many fields.…”

Section: Introductionmentioning

confidence: 99%

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

2014

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Abstract:In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach. PACS (2008): 35K10, 65M70, 33C05Keywords: non-local boundary conditions • integral conservation condition • collocation method • shifted Jacobi-GaussLobatto quadrature • system of differential equations © Versita sp. z o.o.

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“…Some compact finite difference approaches for the solution of KG problems are given in [22 -24]. A spline collocation approach for the solution of the KG equation is presented in [25]. The method of lines approach is used in [5] to transform the sine-Gordon equation into a first-order nonlinear initial-value problem and then replacing the matrix exponential term in a recurrence relation by rational approximation which leads to the second-order methods in both space and time variables.…”

Section: Introductionmentioning

confidence: 99%

Fast and High Accuracy Numerical Methods for the Solution of Nonlinear Klein–Gordon Equations

Mohebbi

Asgari

Shahrezaee

2011

Zeitschrift Für Naturforschung A

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and a mohebbi@kashanu.ac.ir Z. Naturforsch. 66a, 735 -744 (2011) / DOI: 10.5560/ZNA.2011-0038 Received April 13, 2011 / revised July 6, 2011 In this work we propose fast and high accuracy numerical methods for the solution of the onedimensional nonlinear Klein-Gordon (KG) equations. These methods are based on applying fourthorder time-stepping schemes in combination with discrete Fourier transform to numerically solve the KG equations. After transforming each equation to a system of ordinary differential equations, the linear operator is not diagonal, but we can implement the methods such as for the diagonal case which reduces the time in the central processing unit (CPU). In addition, the conservation of energy in KG equations is investigated. Numerical results obtained from solving several problems possessing periodic, single, and breather-soliton waves show the high efficiency and accuracy of the mentioned methods.

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A spline collocation approach for the numerical solution of a generalized nonlinear Klein–Gordon equation

Cited by 33 publications

References 10 publications

An implicit Lie-group iterative scheme for solving the nonlinear Klein–Gordon and sine-Gordon equations

An implicit Lie-group iterative scheme for solving the nonlinear Klein–Gordon and sine-Gordon equations

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

Fast and High Accuracy Numerical Methods for the Solution of Nonlinear Klein–Gordon Equations

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