• Journal of applied mathematics & informatics
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• 제35권3_4호
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• pp.277-302
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• 2017

In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of reaction-diffusion type of third order Ordinary Differential Equations (ODEs). The SPBVP is reduced into a weakly coupled system of one first order and one second order ODEs, one without the parameter and the other with the parameter ${\varepsilon}$ multiplying the highest derivative subject to suitable initial and boundary conditions, respectively. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. The weakly coupled system is decoupled by replacing one of the unknowns by its zero-order asymptotic expansion. Finally the present numerical method is applied to the decoupled system. In order to get a numerical solution for the derivative of the solution, the domain is divided into three regions namely two inner regions and one outer region. The Shooting method is applied to two inner regions whereas for the outer region, standard finite difference (FD) scheme is applied. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing. The main advantage of this method is that due to decoupling the system, the computation time is very much reduced.

• Korean Journal of Mathematics
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• 제29권2호
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• pp.227-237
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• 2021

Opial inequality and its consequences are useful in establishing existence and uniqueness of solutions of initial and boundary value problems for differential and difference equations. In this paper we analyze Opial-type inequalities for convex functions. We have studied different versions of these inequalities for a generalized integral operator. Further difference of Opial-type inequalities are utilized to obtain generalized mean value theorems, which further produce various interesting derivations for fractional and conformable integral operators.

• Environmental Engineering Research
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• 제11권3호
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• pp.164-171
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• 2006

Numerical modeling of the flow velocity fields for the near corona wire electrohydrodynamic (EHD) flow was conducted. The steady, two-dimensional momentum equations have been computed for a wire-plate type electrostatic precipitator (ESP). The equations were solved in the conservative finite-difference form on a fine uniform rectilinear grid of sufficient resolution to accurately capture the momentum boundary layers. The numerical procedure for the differential equations was used by SIMPLEST algorithm. The Phoenics (Version 3.5.1) CFD code, coupled with Poisson's electric field, ion transport equations and the momentum equation with electric body force were used for the numerical simulation and the Chen-Kim ${\kappa}-{\varepsilon}$ turbulent model numerical results that an EHD secondary flow was clearly visible in the downstream regions of the corona wire despite the low Reynolds number for the electrode ($Re_{cw}=12.4$). Secondary flow vortices caused by the EHD increases with increasing discharge current or EHD number, hence pressure drop of ESP increases.

• Structural Engineering and Mechanics
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• 제17권1호
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• pp.1-14
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• 2004

Numerical solution to linear bending analysis of circular plates is obtained by the method of harmonic differential quadrature (HDQ). In the method of differential quadrature (DQ), partial space derivatives of a function appearing in a differential equation are approximated by means of a polynomial expressed as the weighted linear sum of the function values at a preselected grid of discrete points. The method of HDQ that was used in the paper proposes a very simple algebraic formula to determine the weighting coefficients required by differential quadrature approximation without restricting the choice of mesh grids. Applying this concept to the governing differential equation of circular plate gives a set of linear simultaneous equations. Bending moments, stresses values in radial and tangential directions and vertical deflections are found for two different types of load. In the present study, the axisymmetric bending behavior is considered. Both the clamped and the simply supported edges are considered as boundary conditions. The obtained results are compared with existing solutions available from analytical and other numerical results such as finite elements and finite differences methods. A comparison between the HDQ results and the finite difference solutions for one example plate problem is also made. The method presented gives accurate results and is computationally efficient.

• 대한수학회지
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• 제50권4호
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• pp.715-725
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• 2013

In this paper, a finite difference scheme for a two-point boundary value problem of 2nth-order ordinary differential equations is presented. The convergence and uniqueness of the solution for the scheme are proved by means of theories on matrix eigenvalues and norm. Numerical examples show that our method is very simple and effective, and that this method can be used effectively for other types of boundary value problems.

• 대한수학회논문집
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• 제37권3호
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• pp.939-955
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• 2022

In this paper, we study a first-order non-linear singularly perturbed Volterra integro-differential equation (SPVIDE). We discretize the problem by a uniform difference scheme on a Bakhvalov-Shishkin mesh. The scheme is constructed by the method of integral identities with exponential basis functions and integral terms are handled with interpolating quadrature rules with remainder terms. An effective quasi-linearization technique is employed for the algorithm. We establish the error estimates and demonstrate that the scheme on Bakhvalov-Shishkin mesh is O(N^-1) uniformly convergent, where N is the mesh parameter. The numerical results on a couple of examples are also provided to confirm the theoretical analysis.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 2008년도 정기 학술대회
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• pp.2-7
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• 2008

A highly efficient moving least squares finite difference method (MLS FDM) for heat transfer analysis of composite material with interface. In the MLS FDM, governing differential equations are directly discretized at each node. No grid structure is required in the solution procedure. The discretization of governing equations are done by Taylor expansion based on moving least squares method. A wedge function is designed for the modeling of the derivative jump across the interface. Numerical examples showed that the numerical scheme shows very good computational efficiency together with high aocuracy so that the scheme for heat transfer problem with different heat conductivities was successfully verified.

• 한국정보과학회논문지:시스템및이론
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• 제32권9호
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• pp.445-456
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• 2005

본 논문은 물리적인 힘을 기반으로 유체의 흐름을 실시간으로 시뮬레이션하기 위하여 유체 의 흐름을 지배하는 Wavier-Stokes 방정식에 대한 빠르고 정확한 풀이 기법을 제안한다 본 논문에서는 Navier-Stokes 방정식에 있는 비선형 항의 속도에 대한 초기값을 Stokes 방정식의 해로써 추정한다. 주어진 비선형 미분방정식의 해에 근사하게 초기값을 추정함으로써 정확하고 안정적인 풀이 기법을 만들 수 있었다. 또한 유한차분법(finite difference method)의 암시적(implicit) 방법 중에서 방대한 계산량을 피할 수 있는 ADI(Alternating Direction Implicit) 방법을 사용함으로써 큰 시간 간격(time-step)에 대해서 시스템이 안정적이며 계산속도 또한 빠르다. 실험 결과들은 특히 연기, 구름과 같이 큰 레이놀드 수(Reynolds number)를 가지는 유체에 대해서 탁월한 성능을 보여주었다.

• Nuclear Engineering and Technology
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• 제16권2호
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• pp.47-57
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• 1984

2상유동을 해석하기 위한 3차원 코드인 THERMIT-6S의 미분 방정식을 세우기 위해, 수학적으로 정확하게 유도된 시간과 공간에 대해 평균한 보존 방정식을 단순화했다. 미분 방정식을 불연속화(discretization)하여 THERMIT-6S의 차분방정식을 얻는다. First-order spatial scheme, donor cell method, 그리고, staggered mesh layout을 써서 공간에 대한 불연속화를 한다. 그리고 시간에 대한 불연속화는 first-order semi-implicit scheme으로써, sonic terms와 국부적인 전달 현상에 관계되는 항들은 implicit하게 그리고 대류 전달 항들은 explicit하게 취급한다. 이렇게 얻어진 방정식들은Newton-Raphson 방법으로 선형화된다. 축소된 압력 방정식을 만들기 위해 모든 변수들이 mesh cells사이에서 단지 압력 변수를 통해서만 결부되도록, 선형화된 방정식들을 처리한다. OPERA-15 실험을 수치해석적으로 모의실험하여 본 결과, THERMIT-6S가 flow coastdown, 역류, 유체진동(flow oscillation) 등을 포함하고, sodium boiling 후의 원자로내의 변화를 예측하는데 매우 유효하다는 것이 밝혀졌다.

• 한국대기환경학회지
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• 제7권3호
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• pp.189-196
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• 1991

In recent years spectral methods have been found to be a powerful tool for the numerical solution of hynamic differential equations. The main attraction of spectral method is accuracy even though it is generally difficult to implement and solve the complex problems using spectral method. We introduced diffusion equations describing the state of air pollution and solved by pseutospectral method in dimensionless form. The results were compared with both those of other numerical methods and analytical solutions. Comparing with finite difference method and finite element method, spectral method shows the highest accuracy for one dimension problem in this study. Also, the results of two dimensional diffusion problems show good agreement with analytical solutions.