• Journal of the Korean Mathematical Society
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• v.50 no.4
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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.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.14 no.3
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• pp.175-187
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• 2010

This paper presents the numerical valuation of the two-asset step-down equitylinked securities (ELS) option by using the operator-splitting method (OSM). The ELS is one of the most popular financial options. The value of ELS option can be modeled by a modified Black-Scholes partial differential equation. However, regardless of whether there is a closedform solution, it is difficult and not efficient to evaluate the solution because such a solution would be represented by multiple integrations. Thus, a fast and accurate numerical algorithm is needed to value the price of the ELS option. This paper uses a finite difference method to discretize the governing equation and applies the OSM to solve the resulting discrete equations. The OSM is very robust and accurate in evaluating finite difference discretizations. We provide a detailed numerical algorithm and computational results showing the performance of the method for two underlying asset option pricing problems such as cash-or-nothing and stepdown ELS. Final option value of two-asset step-down ELS is obtained by a weighted average value using probability which is estimated by performing a MC simulation.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.19 no.1
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• pp.1-21
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• 2015

This paper presents two algorithms based on the Jamshidian equation which is from the Black-Scholes partial differential equation. The first algorithm is for American call options and the second one is for American put options. They compute numerically free boundary and then option price, iteratively, because the free boundary and the option price are coupled implicitly. By the upwind finite-difference scheme, we discretize the Jamshidian equation with respect to asset variable s and set up a linear system whose solution is an approximation to the option value. Using the property that the coefficient matrix of this linear system is an M-matrix, we prove several theorems in order to formulate a bisection method, which generates a sequence of intervals converging to the fixed interval containing the free boundary value with error bound h. These algorithms have the accuracy of O(k + h), where k and h are step sizes of variables t and s, respectively. We prove that they are unconditionally stable. We applied our algorithms for a series of numerical experiments and compared them with other algorithms. Our algorithms are efficient and applicable to options with such constraints as r > d, $r{\leq}d$, long-time or short-time maturity T.

• Journal of the Korean Society for Precision Engineering
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• v.22 no.1
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• pp.89-96
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• 2005

This paper is concerned with a balancing motion formulation and control of the ZMP (Zero Moment Point) for a biped-walking robot that has a prismatic balancing weight or a revolute balancing weight. The dynamic stability equation of a walking robot which have a prismatic balancing weight is conditionally linear but a walking robot's stability equation with a revolute balancing weight is nonlinear. For a stable gait, stabilization equations of a biped-walking robot are modeled as non-homogeneous second order differential equations for each balancing weight type, and a trajectory of balancing weight can be directly calculated with the FDM (Finite Difference Method) solution of the linearized differential equation. In this paper, the 3dimensional graphic simulator is developed to get and calculate the desired ZMP and the actual ZMP. The operating program is developed for a real biped-walking robot IWRⅢ. Walking of 4 steps will be simulated and experimented with a real biped-walking robot. This balancing system will be applied to a biped humanoid robot, which consist legs and upper body, as a future work.

• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.175-190
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• 2007

By employing the Riccati transformation technique some new oscillation criteria for the second-order neutral delay dynamic equation $$(y(t)+r(t)y({\tau}(t)))^{{\Delta}{\Delta}}+p(t)y(\delta(t))=0$$, on a time scale $\mathbb{T}$ are established. Our results as a special case when $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{N}$ improve some well known oscillation criteria for second order neutral delay differential and difference equations, and when $\mathbb{T}=q^{\mathbb{N}}$, i.e., for second-order $q$-neutral difference equations our results are essentially new and can be applied on different types of time scales. Some examples are considered to illustrate the main results.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.6 no.2
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• pp.17-22
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• 1986

The primary objective of this study is to compare the effects that are caused by shrinkage and creep of a concrete bridge deck during its construction. In this study four different bridge structures were compared. Two straight box girders and two curved box girders were compared for stress changes in positive moment region and negative moment region due to the effects of concrete. The effects on displacement behavior by the assumed section length by concrete placement were also studied. The analyses were performed by using Vlasov equation and finite difference numerical method to solve the governing differential equation.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.62 no.1
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• pp.57-62
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• 2013

Because of high torque and easiness of speed control, Direct Current(DC) motors have been used for a long time. But, its applications are limited in circumstance and performance, since they contained brush and commutator. The commutation characteristic gives effect to life and performance of the DC motor. Naturally, the commutation characteristic analysis is strongly required. In this paper, With the result of finite element analysis, The inductance is calculated each rotor position and applied to the voltage equations coupled with commutation equation. Also, contact resistances of brush/commutator assembly are considered using contact area and brush width converted with commutator segments. The time derivative term in the differential equation is solved in time difference method. This algorithm was applied to 2-pole shunt DC motor. We considered commutation characteristic by changing contact resistance between brush and commutator segment.

• The Pure and Applied Mathematics
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• v.17 no.4
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• pp.289-298
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• 2010

In this paper, we propose a second-order prediction/correction (SPC) domain decomposition method for solving one dimensional linear hyperbolic partial differential equation $u_{tt}+a(x,t)u_t+b(x,t)u=c(x,t)u_{xx}+{\int}(x,t)$. The method can be applied to variable coefficients problems and singular problems. Unconditional stability and error analysis of the method have been carried out. Numerical results support stability and efficiency of the method.

• Journal of applied mathematics & informatics
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• v.36 no.1_2
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• pp.135-154
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• 2018

In this paper, an almost second order overlapping Schwarz method for singularly perturbed third order convection-diffusion type problem is constructed. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the combination of classical finite difference scheme and central finite difference scheme on a uniform mesh while on the non-layer region we use the midpoint difference scheme on a uniform mesh. It is shown that the numerical approximations which converge in the maximum norm to the exact solution. We proved that, when appropriate subdomains are used, the method produces convergence of second order. Furthermore, it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantages of this method used with the proposed scheme are it reduce iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.