Numerical solution of one-dimensional Sine–Gordon equation using high accuracy multiquadric quasi-interpolation

Jiang, Ziwu; Wang, Ren-Hong

doi:10.1016/j.amc.2011.12.095

Cited by 27 publications

(22 citation statements)

References 15 publications

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“…Dehghan and Shokri [13] solved the equation using collocation points and approximate the solution using radial basis functions; Dehghan and Mirzaei [14] used a boundary integral equation method; Rashidinia and Mohammadi [15] developed two implicit finite difference schemes, by using spline function approximations. Li-Min and Zong-Min [16] presented a meshless scheme by using a multiquadric quasiinterpolation without solving a large-scale linear system of equations, but a polynomial was needed to improve the accuracy of the scheme, while Jiang and Wang [17] proposed meshless approach by directly using high accuracy MQ quasiinterpolation without using any polynomial. A modified decomposition method for explicit and numerical solutions of the sine-Gordon equation in the form of convergent power series has been proposed by Kaya [18].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

Mittal

Bhatia

2014

International Journal of Partial Differential Equations

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Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.

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Section: Introductionmentioning

confidence: 99%

Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

Mittal

Bhatia

2014

International Journal of Partial Differential Equations

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“…In 2011, Wu presented an new approach to construct the so‐called shape preserving interpolation curves based on MQ quasi‐interpolation

{scriptL}_{scriptD}

. Hon and Wu , Wu , Chen and Wu , Jiang and Wang , and other researches provided some successful examples using MQ quasi‐interpolation operators to solve different types of partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

A new multiquadric quasi‐interpolation operator with interpolation property

2013

Math Methods in App Sciences

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In this article, we discuss a class of multiquadric quasi‐interpolation operator that is primarily on the basis of Wu–Schaback's quasi‐interpolation operator scriptLscriptD and radial basis function interpolation. The proposed operator possesses the advantages of linear polynomial reproducing property, interpolation property, and high accuracy. It can be applied to construct flexible function approximation and scattered data fitting from numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.

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“…In [8], the authors constructed two high accuracy multiquadric quasi-interpolation operators named L W and L W 2 by using multi-level method via inverse multiquadric radial basis function (IMQ-RBF) interpolation, and Wu and Schaback's MQ quasi-interpolation operator L D . Moreover, in [9], L D was utilized to solve a one-dimensional Sine-Gordon equation with good numerical accuracy. Since the quasi-interpolation operator L D only have an O(h 2 ) error if at least c 2 | log c| = O(h 2 ), in this paper, B-spine quasi-interpolation with high order accuracy is considered.…”

Section: Introductionmentioning

confidence: 99%

A New Approximation Method with High Order Accuracy

Jiang

2017

MCA

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Abstract:In this paper, we propose a new multilevel univariate approximation method with high order accuracy using radial basis function interpolation and cubic B-spline quasi-interpolation. The proposed approach includes two schemes, which are based on radial basis function interpolation with less center points, and cubic B-spline quasi-interpolation operator. Error analysis shows that our method produces higher accuracy compared with other approaches. Numerical examples demonstrate that the proposed scheme is effective.

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Numerical solution of one-dimensional Sine–Gordon equation using high accuracy multiquadric quasi-interpolation

Cited by 27 publications

References 15 publications

Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

A new multiquadric quasi‐interpolation operator with interpolation property

A New Approximation Method with High Order Accuracy

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