• Journal of applied mathematics & informatics
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• v.33 no.5_6
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• pp.657-670
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• 2015

In this paper, we provide numerical analysis of so-called Legendre Gauss-Radau and Legendre-Gauss collocation methods for ordinary differential equations. After recasting these collocation methods as Runge-Kutta methods, we prove that the Legendre-Gauss collocation method is equivalent to the well-known Gauss method, while the Legendre-Gauss-Radau collocation method does not belong to the classes of Radau IA or Radau IIA methods in the Runge-Kutta literature. Making use of the well-established theory of Runge-Kutta methods, we study stability and accuracy of the Legendre-Gauss-Radau collocation method. Numerical experiments are conducted to confirm our theoretical results on the accuracy and numerical stability of the Legendre-Gauss-Radau collocation method, and compare Legendre-Gauss collocation method with the Gauss method.

• Journal of applied mathematics & informatics
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• v.32 no.5_6
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• pp.567-584
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• 2014

In this paper, we consider approximation of eigenelements of a two dimensional compact integral operator with a smooth kernel by discrete collocation and iterated discrete collocation methods. By choosing numerical quadrature appropriately, we obtain convergence rates for gap between the spectral subspaces, and also we obtain superconvergence rates for eigenvalues and iterated eigenvectors. We then apply Richardson extrapolation to obtain further improved error bounds for the eigenvalues. Numerical examples are presented to illustrate theoretical estimates.

• Structural Engineering and Mechanics
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• v.45 no.5
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• pp.595-611
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• 2013

This paper aims to compare three collocation point methods associated with the odd order stochastic response surface method (SRSM) in a systematical and quantitative way. The SRSM with the Hermite polynomial chaos is briefly introduced first. Then, three collocation point methods, namely the point method, the root method and the without origin method underlying the odd order SRSMs are highlighted. Three examples are presented to demonstrate the accuracy and efficiency of the three methods. The results indicate that the condition that the Hermite polynomial information matrix evaluated at the collocation points has a full rank should be satisfied to yield reliability results with a sufficient accuracy. The point method and the without origin method are much more efficient than the root method, especially for the reliability problems involving a large number of random variables or requiring complex finite element analysis. The without origin method can also produce sufficiently accurate reliability results in comparison with the point and root methods. Therefore, the origin often used as a collocation point is not absolutely necessary. The odd order SRSMs with the point method and the without origin method are recommended for the reliability analysis due to their computational accuracy and efficiency. The order of SRSM has a significant influence on the results associated with the three collocation point methods. For normal random variables, the SRSM with an order equaling or exceeding the order of a performance function can produce reliability results with a sufficient accuracy. The order of SRSM should significantly exceed the order of the performance function involving strongly non-normal random variables.

• Structural Engineering and Mechanics
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• v.51 no.6
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• pp.893-907
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• 2014

Meshless local collocation method produces much better conditioned matrices than meshless global collocation methods. In this paper, the meshless local collocation method based on thin plate spline radial basis function and first-order shear deformation theory are used to calculate the natural frequencies and mode shapes of laminated composite shells. Through numerical experiments, the accuracy and efficiency of present method are demonstrated.

• Kyungpook Mathematical Journal
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• v.51 no.4
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• pp.435-456
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• 2011

The Chebyshev collocation method in [21] to solve stiff initial-value problems is generalized by using arbitrary degrees of interpolation polynomials and arbitrary collocation points. The convergence of this generalized Chebyshev collocation method is shown to be independent of the chosen collocation points. It is observed how the stability region does depend on collocation points. In particular, A-stability is shown by taking the mid points of nodes as collocation points.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.951-969
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• 2012

In this paper, we propose a class of meshless collocation approaches for the solution of time dependent partial differential algebraic equations (PDAEs) in terms of a radial basis function interpolation numerical scheme. Kansa's method and the Hermite collocation method (HCM) for PDAEs are given. A sensitivity analysis of the solutions from different shape parameter c is obtained by numerical experiments. With use of the random collocation points, we have obtain the more accurate solution by the methods than those by the finite difference method for the PDAEs with index-2, i.e, we avoid the influence from an index jump of PDAEs in some degree. Several numerical experiments show that the methods are efficient.

• English Language & Literature Teaching
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• v.16 no.3
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• pp.141-160
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• 2010

This study aims to analyze the effects of explicit lexical collocation instruction on grammatical collocation knowledge, by focusing on the preposition collocations. The forty participants consisting of twenty men and twenty women are currently freshmen English majors in college. They responded to the questionnaire on their basic lexical learning strategies in pre-post tests, and carried out a lexical collocation test and a preposition collocation test as pre-post tests. They took the open class on the lexical collocations(LC) 5 times, but they did not concretely study any grammatical collocations(GC). Each test is analyzed and compared with pre-post tests by SPSS 12.0. From all the analyses, it was first revealed that most students who didn't know about the concepts, types and uses of collocations became aware of them after the class. The effects of the lexical collocations on the lexical collocation knowledge were quite revealing. Second, the correlations between the lexical collocations and grammatical collocations were not significant, although relations among the lexical collocations were quite significant. Finally, some suggestions for methods of mixed lexical chunks instruction are made for students' productive skills. Also, it is suggested that more profound attention should be paid to giving direct instructions on the basis of the lexico-grammatical continuum between LC and GC.

• Journal of applied mathematics & informatics
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• v.6 no.2
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• pp.513-522
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• 1999

In this paper we present a collocation method based on biquintic splines for a fourth order elliptic problems. To have a better accuracy we formulate the standard collocation method by an appro-priate perturbation on the original differential equations that leads to an optimal approximating scheme. As a result computational results confirm that this method is optimal.

• Kyungpook Mathematical Journal
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• v.50 no.3
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• pp.419-431
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• 2010

A low-order finite element preconditioner is analyzed for a cubic spline collocation method which is used for discretizations of coupled elliptic problems derived from an optimal control problrm subject to an elliptic equation. Some numerical evidences are also provided.

• Honam Mathematical Journal
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• v.43 no.3
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• pp.373-383
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• 2021

In this study, a numerical method deals with the Chebyshev wavelet collocation and Adomian decomposition methods are proposed for solving Korteweg-de Vries equation. Integration of the Chebyshev wavelets operational matrices is derived. This problem is reduced to a system of non-linear algebraic equations by using their operational matrix. Thus, it becomes easier to solve KdV problem. The error estimation for the Chebyshev wavelet collocation method and ADM is investigated. The proposed method's validity and accuracy are demonstrated by numerical results. When the exact and approximate solutions are compared, for non-linear or linear partial differential equations, the Chebyshev wavelet collocation method is shown to be acceptable, efficient and accurate.