• 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.

• 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.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.9 no.1
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• pp.153-164
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• 1996

An $O(h^4)$ cubic spline collocation method for biharmonic equation with a special boundary conditions is formulated and a fast direct method is proposed for the linear system arising when the cubic spline collocation method is employed. This method requires $O(N^2\;{\log}\;N)$ arithmatic operations over an $N{\times}N$ uniform partition.

• 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.

• 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.

• The Pure and Applied Mathematics
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• v.13 no.1 s.31
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• pp.19-38
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• 2006

We analyze the spectral collocation approximation for a parabolic partial integrodifferential equations(PIDE) with a weakly singular kernel. The space discretization is based on the spectral collocation method and the time discretization is based on Crank-Nicolson scheme with a graded mesh. We obtain the stability and second order convergence result for fully discrete scheme.

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.867-880
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• 1996

Numerical approximations to the linear elasticity are traditionally based on the finite element method. In this paper we propose a new formulation based on the cubic spline collocation method for linear elastic problem on the unit square. We present several numerical results for the eigenvalues of the matrix represented by cubic collocation method and preconditioner matrix which is preconditioned by FEM and FDM. Finally we present the numerical solution for some example equation.

• Journal of Korea Water Resources Association
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• v.32 no.3
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• pp.237-246
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• 1999

A finite element model is studied to simulate unsteady free surface flow based on dynamic wave equation and collocation method. The collocation method is used in conjunction with Hermite polynomials, and resulting matrix equations are solved by skyline method. The model is verified by applying to hydraulic jump, nonlinear disturbance propagation and dam-break flow in a horizontal frictionless channel. The computed results are compared with those by Bubnov-Galerkin and Petrov-Galerkin methods. It is also applied to the North Han River to simulate the floodwave propagation. The computed results have good agreements with those of DWOPER model in terms of discharge hydrographs. The suggested model has proven to be one of the promising scheme for simulating the gradually and rapidly varied unsteady flow in open channels.

• 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.