• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1737-1751
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• 2013

An efficient and stable point collocation scheme based on a meshfree method is studied for the stationary incompressible Navier-Stokes equations. We describe the diffuse derivatives associated with the moving least square method. Using these diffuse derivatives, we propose a point collocation method to fit in solving the Navier-Stokes equations which improves the stability of the direct point collocation scheme. The convergence of the numerical solution is investigated from numerical examples. The driven cavity ow and the backward facing step ow are implemented for the reliability of the scheme. Also, the viscous ow on complicated geometry is successfully calculated such as the ow past a circular cylinder in duct.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.36 no.2
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• pp.363-370
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• 1999

A new and simple estimate using a convex function for convergence factors of Schwarz algorithms orthogonal spline collocation method is presented. The estimated convergence factors in this context depend only on a way to split a given domain.

• Kyungpook Mathematical Journal
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• v.57 no.2
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• pp.287-299
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• 2017

In this paper, we develop an enhanced Chebyshev collocation method based on an integration scheme of the generalized Chebyshev interpolations for solving stiff initial value problems. Unlike the former error embedded Chebyshev collocation method (CCM), the enhanced scheme calculates the solution and its truncation error based on the interpolation of the derivative of the true solution and its integration. In terms of concrete convergence and stability analysis, the constructed algorithm turns out to have the $7^{th}$ convergence order and the A-stability without any loss of advantages for CCM. Throughout a numerical result, we assess the proposed method is numerically more efficient compared to existing methods.

• Interaction and multiscale mechanics
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• v.2 no.3
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• pp.295-319
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• 2009

In this work, we discuss a reproducing kernel collocation method (RKCM) for solving $2^{nd}$ order PDE based on strong formulation, where the reproducing kernel shape functions with compact support are used as approximation functions. The method based on strong form collocation avoids the domain integration, and leads to well-conditioned discrete system of equations. We investigate the convergence and the computational complexity for this proposed method. An important result obtained from the analysis is that the degree of basis in the reproducing kernel approximation has to be greater than one for the method to converge. Some numerical experiments are provided to validate the error analysis. The complexity of RKCM is also analyzed, and the complexity comparison with the weak formulation using reproducing kernel approximation is presented.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2002.10a
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• pp.599-603
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• 2002

We propose a new meshfree method to be called the fast moving least square reproducing kernel collocation method(FCM). This methodology is composed of the fast moving least square reproducing kernel(FMLSRK) approximation and the point collocation scheme. Using point collocation makes the meshfree method really come true. In this paper, FCM Is shown to be a good method at least to calculate the numerical solutions governed by second order elliptic partial differential equations with geometric singularity or geometric multi-scales. To treat such problems, we use the concept of variable dilation parameter.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.20 no.3
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• pp.261-275
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• 2016

In this paper, an integro-differential equation which arises in oscillating magnetic fields is studied. The generalized fractional order Chebyshev orthogonal functions (GFCF) collocation method used for solving this integral equation. The GFCF collocation method can be used in applied physics, applied mathematics, and engineering applications. The results of applying this procedure to the integro-differential equation with time-periodic coefficients show the high accuracy, simplicity, and efficiency of this method. The present method is converging and the error decreases with increasing collocation points.

• Proceedings of the Korean Society For Composite Materials Conference
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• 2001.10a
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• pp.13-16
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• 2001

This paper presents the results of an elastic buckling analysis of isosceles trapezoidal orthotropic plate. In this study, all edges of plate are assumed to be simply supported and the difference of the applied loads are assumed to be taken out by shear of constant intensity along the sloping sides. For the buckling analysis, collocation method is employed. Finite element analysis is also conducted and the results are compared with theoretical ones.

• Journal of applied mathematics & informatics
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• v.8 no.2
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• pp.371-386
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• 2001

In this work we investigate the finite element preconditioning method for the $C^1$-cubic spline collocation discretizations for an elliptic operator A defined by $Au := -{\Delta}u + a_1u_x+a_2u_y+a_0u$ in the unit square with some boundary conditions. We compute the condition number and the numerical solution of the preconditioning system for the several example problems. Finally, we compare the this preconditioning system with the another preconditioning system.

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.17-27
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• 2001

We discuss a finite difference preconditioner for the$C^1$ Lagrance quadratic spline collocation method for a uniformly elliptic operator with homogeneous Dirichlet boundary conditions. Using the generalized field of values argument, we analyzed eigenvalues of the matrix preconditioned by the matrix corresponding to a finite difference operator with zero boundary condition.