• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제17권4호
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• pp.295-306
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• 2013

In this paper, we consider the adaptive multigrid method for solving the Black-Scholes equation to improve the efficiency of the option pricing. Adaptive meshing is generally regarded as an indispensable tool because of reduction of the computational costs. The Black-Scholes equation is discretized using a Crank-Nicolson scheme on block-structured adaptively refined rectangular meshes. And the resulting discrete equations are solved by a fast solver such as a multigrid method. Numerical simulations are performed to confirm the efficiency of the adaptive multigrid technique. In particular, through the comparison of computational results on adaptively refined mesh and uniform mesh, we show that adaptively refined mesh solver is superior to a standard method.

• 대한수학회지
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• 제61권4호
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• pp.621-657
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• 2024

In this study, the numerical solution of a space-fractional Burgers' equation with initial and boundary conditions is considered. This equation is the simplest nonlinear model for diffusive waves in fluid dynamics. It occurs in a variety of physical phenomena, including viscous sound waves, waves in fluid-filled viscous elastic pipes, magneto-hydrodynamic waves in a medium with finite electrical conductivity, and one-dimensional turbulence. The proposed QBS/CNG technique consists of the Galerkin method with a function basis of quadratic B-splines for the spatial discretization of the space-fractional Burgers' equation. This is then followed by the Crank-Nicolson approach for time-stepping. A linearized scheme is fully constructed to reduce computational costs. Stability analysis, error estimates, and convergence rates are studied. Finally, some test problems are used to confirm the theoretical results and the proposed method's effectiveness, with the results displayed in tables, 2D, and 3D graphs.

• Journal of Astronomy and Space Sciences
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• 제3권1호
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• pp.29-40
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• 1986

적층재료로 된 로켓 노즐벽에서의 열전도 방정식을 Crank Nicolson의 방법을 이용하여 수치해석 하였으며 정해진 제한조건에 대하여 최적화 방법에 의하여 각층의 재료와 두께를 선택하였다. 로켓 노즐의 운전조건에 대하여 여러가지 입력한 재료중 각층의 최적의 두께 및 재료는 표 3과 같다.

• 한국지반공학회:학술대회논문집
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• 한국지반공학회 2000년도 봄 학술발표회 논문집
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• pp.631-638
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• 2000

Pioneering work by Terzaghi imparted scientific and mathematical bases to many aspects of this subject and many people use this theory to measure the consolidation settlement until now. In this paper, Finite Difference Methods for consolidation are considered. First, it is shown the stability criterion of Explicit scheme and the Crank-Nicolson scheme, although unconditionally stable in the mathematical sense, produces physically unrealistic solutions when the time step is large. it is also shown that The Fully Implicit scheme shows more satisfactory behavior, but is less accurate for small time steps. and then we need to decide what scheme is more proper to consolidation. The purpose of this paper is to suggest the pertinent scheme to consolidation.

• 물과 미래
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• 제21권3호
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• pp.299-307
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• 1988

돌제 주위의장기간 해빈변형을 예측하기 위하여 수위의 변동, 파의굴절과 회절을 고려하여 해석하는 수치적 모델이다. 이러한 문제에 대한 수치해석은 연안표사랑에 대한 경계조건을 가지는 방정식을 고려함으로써 해석된다. 해석방법으로는 음해법 중의 하나인 Crank-Nicolson Scheme을 사용하였다. 이 모델을 현지에 적용시킨 결과, 굴절과 회절계수의 근사해로 인하여 돌제 내부 영역에서는 실측치와 예측치가 차이가 있으나, 돌제 외부 영역에서의 경향은 잘 일치하고 있음을 알 수 있다.

• Journal of applied mathematics & informatics
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• 제8권3호
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• pp.773-793
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• 2001

The explicit expressions for the 2n+1 primitive idempotents in R/sub pⁿ/ = F[x]/< x/sup pⁿ/ -1>, where F is the field of prime power order q and the multiplicative order of q modulo pⁿ is ø(pⁿ)/2(n≥1 and p is an odd prime), are obtained. An algorithm for computing the generating polynomials of the minimal QR cyclic codes of length pⁿ, generated by these primitive idempotents, is given and hence some bounds on the minimum distance of some QR codes of prime length over GF(q)(q=2, 3, ...) are obtained.

• Journal of the Korean Wood Science and Technology
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• 제24권2호
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• pp.36-41
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• 1996

목재(木材)의 건조과정(乾燥過程) 중에 발생하는 목재 내부의 수분경사(水分傾斜)를 예측하기 위해 비정상상태(非定常狀態)의 확산(擴散)모델을 지배방정식(支配方程式)으로 적용하였으며, 목재 표면에서의 증발저항(蒸發抵抗)과 내부의 대칭적 수분분포를 경계조건(境界條件)으로 채택하였다. 주어진 경계조건에서의 지배방정식에 대한 일반해(一般解)가 무한수열 형태로 이루어지기 때문에, 유한차분법(有限差分法)을 이용하여 수치해석(數値解析)하였으며, 유한차분법(有限差分法) 중 오차범위(誤差範圍)가 안정한 상태인 Crank-Nicolson Scheme 알고리즘을 채택하였다.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제17권3호
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• pp.197-207
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• 2013

The Cahn-Hilliard equation was proposed as a phenomenological model for describing the process of phase separation of a binary alloy. The equation has been applied to many physical applications such as amorphological instability caused by elastic non-equilibrium, image inpainting, two- and three-phase fluid flow, phase separation, flow visualization and the formation of the quantum dots. To solve the Cahn-Hillard equation, many numerical methods have been proposed such as the explicit Euler's, the implicit Euler's, the Crank-Nicolson, the semi-implicit Euler's, the linearly stabilized splitting and the non-linearly stabilized splitting schemes. In this paper, we investigate each scheme in finite-difference schemes by comparing their performances, especially stability and efficiency. Except the explicit Euler's method, we use the fast solver which is called a multigrid method. Our numerical investigation shows that the linearly stabilized stabilized splitting scheme is not unconditionally gradient stable in time unlike the known result. And the Crank-Nicolson scheme is accurate but unstable in time, whereas the non-linearly stabilized splitting scheme has advantage over other schemes on the time step restriction.

• 한국지하수토양환경학회지:지하수토양환경
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• 제7권4호
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• pp.10-16
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• 2002

본 연구의 목적은 누수대수층으로 분리된 다층-대수층에 대한 지하수 해석이다. Crank-Nicolson방법에 의한 유한차분법을 적용하여 1차원이며 정상상태인 2중 대수층 구조에 대해 해석해와 비교하였다. 수치해와 해석해는 거의 일치하였으므로 수치해를 2차원의 확장된 다층-대수층 구조에 적용하였다. 이는 한 개 또는 여러 개의 대수층에서 양수하는 경우에 각 대수층에서의 수두값을 계산할 수 있게 하였다. 본 연구는 지하수의 효율적인 운영에 도움이 될 것이다.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1477-1487
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• 2011

We propose a numerical method for solving Allen-Cahn equation, in both one-dimensional and two-dimensional cases. The new scheme that is explicit, stable, and easy to compute is obtained and the proposed method provides a straightforward and effective way for nonlinear evolution equations.