• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.983-993
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• 2022

The main purpose of this paper is to solve the second kind Volterra integral equations by Clenshaw-Curtis spectral collocation method. First of all, we can transform the integral interval from [-1, x] to [-1, 1] through a simple linear transformation, and discretize the integral term in the equation by Clenshaw-Curtis quadrature formula to obtain the collocation equations. Then we provide a rigorous error analysis for the proposed method. At last, several numerical example are used to verify the results of theoretical analysis.

• 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.

• Journal of Aerospace System Engineering
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• v.14 no.3
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• pp.17-23
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• 2020

In this paper, the efficient periodic unsteady flow analysis is developed by using a Chebyshev collocation operator applied to the time differential term of the governing equations. The partial implicit time integration method was also applied in the governing equation for a fluid, which means flux terms were implicitly processed for a time integration and the time derivative terms were applied explicitly in the form of the source term by applying the Chebyshev collocation operator. To verify this method, we applied the 1D unsteady Burgers equation and the 2D oscillating airfoil. The results were compared with the existing unsteady flow frequency analysis technique, the Harmonic Balance Method, and the experimental data. The Chebyshev collocation operator can manage time derivatives for periodic and non-periodic problems, so it can be applied to non-periodic problems later.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.595-612
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• 2014

The spectral collocation method for a second order elliptic boundary value problem on a domain ${\Omega}$ with curved boundaries is studied using the Gordon and Hall transformation which enables us to have a transformed elliptic problem and a square domain S = [0, h] ${\times}$ [0, h], h > 0. The preconditioned system of the spectral collocation approximation based on Legendre-Gauss-Lobatto points by the matrix based on piecewise bilinear finite element discretizations is shown to have the high order accuracy of convergence and the efficiency of the finite element preconditioner.

• Interaction and multiscale mechanics
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• v.2 no.4
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• pp.333-352
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• 2009

We introduce a radial basis collocation method to solve axially moving beam problems which involve $2^{nd}$ order differentiation in time and $4^{th}$ order differentiation in space. The discrete equation is constructed based on the strong form of the governing equation. The employment of multiquadrics radial basis function allows approximation of higher order derivatives in the strong form. Unlike the other approximation functions used in the meshfree methods, such as the moving least-squares approximation, $4^{th}$ order derivative of multiquadrics radial basis function is straightforward. We also show that the standard weighted boundary collocation approach for imposition of boundary conditions in static problems yields significant errors in the transient problems. This inaccuracy in dynamic problems can be corrected by a statically condensed semi-discrete equation resulting from an exact imposition of boundary conditions. The effectiveness of this approach is examined in the numerical examples.

• Structural Engineering and Mechanics
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• v.76 no.4
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• pp.435-449
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• 2020

This work presents the formulation of the isogeometric collocation method to solve the strong form equation of a unified and integrated approach of Reissner Mindlin plate theory (UI-RM). In this plate theory model, the total displacement is expressed in terms of bending and shear displacements. Rotations, curvatures, and shear strains are represented as the first, the second, and the third derivatives of the bending displacement, respectively. The proposed formulation is free from shear locking in the Kirchhoff limit and is equally applicable to thin and thick plates. The displacement field is approximated using the B-splines functions, and the strong form equation of the fourth-order is solved using the collocation approach. The convergence properties and accuracy are demonstrated with square plate problems of thin and thick plates with different boundary conditions. Two approaches are used for convergence tests, e.g., increasing the polynomial degree (NELT = 1×1 with p = 4, 5, 6, 7) and increasing the number of element (NELT = 1×1, 2×2, 3×3, 4×4 with p = 4) with the number of control variable (NCV) is used as a comparable equivalent variable. Compared with DKMQ element of a 64×64 mesh as the reference for all L/h, the problem analysis with isogeometric collocation on UI-RM plate theory exhibits satisfying results.

• Structural Engineering and Mechanics
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• v.22 no.1
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• pp.1-16
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• 2006

This paper presents a virtual boundary element-equivalent collocation method (VBEM) for the plane magnetoelectroelastic solids, which is based on the fundamental solutions of the plane magnetoelectroelastic solids and the basic idea of the virtual boundary element method for elasticity. Besides all the advantages of the conventional boundary element method (BEM) over domain discretization methods, this method avoids the computation of singular integral on the boundary by introducing the virtual boundary. In the end, several numerical examples are performed to demonstrate the performance of this method, and the results show that they agree well with the exact solutions. So the method is one of the efficient numerical methods used to analyze megnatoelectroelastic solids.

• Structural Engineering and Mechanics
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• v.17 no.5
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• pp.671-690
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• 2004

A meshless method with novel variation of point collocation by finite mixture approximation is developed in this paper, termed the meshless finite mixture (MFM) method. It is based on the finite mixture theorem and consists of two or more existing meshless techniques for exploitation of their respective merits for the numerical solution of partial differential boundary value (PDBV) problems. In this representation, the classical reproducing kernel particle and differential quadrature techniques are mixed in a point collocation framework. The least-square method is used to optimize the value of the weight coefficient to construct the final finite mixture approximation with higher accuracy and numerical stability. In order to validate the developed MFM method, several one- and two-dimensional PDBV problems are studied with different mixed boundary conditions. From the numerical results, it is observed that the optimized MFM weight coefficient can improve significantly the numerical stability and accuracy of the newly developed MFM method for the various PDBV problems.

• Journal of the Korean Society for Aeronautical & Space Sciences
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• v.34 no.5
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• pp.56-64
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• 2006

Station collocation of closely placed multiple GEO spacecraft is required to avoid the problem of collision risk, attitude sensor interference and/or occultation. This paper presents the method of obtaining the orbit correction scheme for collocating two GEO spacecraft within a small station-keeping box. The relative motion of each spacecraft with respect to the virtual geostationary satellite is precisely expressed in terms of power and trigonometry functions. This closed-form orbit propagator is used to define the constraint conditions which meet the requirements for the station collocation. Finally, the technique of constrained optimization is used to find the orbit maneuver sequence. Nonlinear simulations are performed and their results are compared with those of the classical method.

• Journal of applied mathematics & informatics
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• v.7 no.2
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• pp.439-451
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• 2000

To compute derivatives using matrix vector multiplication method, new algorithms were introduced in [1.2]n By these algorithms, we reduced roundoff error in computing derivative using Chebyshev collocation methods (CCM). In this paper, some applications of these algorithms ar presented.