• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1057-1069
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• 2008

We consider singularly perturbed Boundary Value Problems (BVPs) for third and fourth order Ordinary Differential Equations(ODEs) of convection-diffusion type with discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions(BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equation does not have the small parameter but the second contains it. In this paper a computational method named as 'An asymptotic finite element method' for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a fitted mesh Finite Element Method (FEM). Numerical experiments support our theoritical results.

• Journal of applied mathematics & informatics
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• 제17권1_2_3호
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• pp.229-244
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• 2005

In this paper, we study the third order ordinary differential equation : $$x'(t)=f(t,x(t),x'(t),x'(t)),t{\in}(0,1)$$ subject to the boundary value conditions: $$x'(0)=x'(\xi),x'(1)=^{m-3}{\Sigma}_{i=1}{{\beta}x'({\eta}i),x'(1)=0}$$. Here ${\beta}_{i}{\in}R,\;^{m-3}{\Sigma}_{i=1}\;{\beta}_{i}\;=\;1,\;0<{\eta}_1<{\eta}_2<{\cdots}<{\eta}_{m-3}<1,\;0<\xi<1$. This is the case dimKer L = 2. When the ${\beta}_i$ have different signs, we prove some existence results for the m-point boundary value problem at resonance by use of the coincidence degree theory of Mawhin [12, 13]. Since all the existence results obtained in previous papers are for the case dimKerL = 1, our work is new.

• Journal of applied mathematics & informatics
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• 제36권1_2호
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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.

• 한국기계가공학회지
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• 제10권4호
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• pp.8-15
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• 2011

In this paper, the influence of a crack on the natural frequency of cracked cantilever L-shaped beam with coupled bending and torsional vibrations by analytically and experimentally is analyzed. The L-shaped beam with a crack is modeled by Hamilton's principle with consideration of bending and torsional energy. The two coupled governing differential equations are reduced to one sixth-order ordinary differential equation in terms of the flexural displacement. The crack is assumed to be in the first, second and third mode of fracture and to be always opened during the vibrations. The theoretical results are validated by a comparison with experimental measurements. The maximal difference between the theoretical results and experimental measurements of the natural frequency is less than 7.5% in the second vibration mode.

• 대한기계학회논문집A
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• 제21권2호
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• pp.337-345
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• 1997

An analysis is presented for the response of a beam constrained by a nonlinear spring to a harmonic excitation. The system is governed by a linear partial differential equation with a nonlinear boundary condition. The method of multiple scales is used to reduce the nonlinear boundary value problem to a system of autonomous ordinary differential equations of the amplitudes and phases. The case of the third-order subharmonic resonance is considered in this study. The autonomous system is used to determine the steady-state responses and their stability.

• 한국전산유체공학회지
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• 제12권4호
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• pp.20-27
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• 2007

Two-dimensional stagnation flow toward a plane wall coated with magnetic fluid of uniform thickness is investigated. The flow field is represented as a similarity solution of the Navier-Stokes equation for this incompressible laminar flow. The resulting third order ordinary differential equation is solved numerically by using the shooting method and by determining two shooting parameters so as to satisfy the boundary and interface conditions. Features of the flow including streamline patterns are investigated for the varying values of density ratio, viscosity ratio, and Reynolds number. An adverse flow with double eddy pair in magnetic fluid region is found to emerge as the Reynolds number becomes higher than a threshold value. The results for the interface velocity, interface and wall shear stress, and boundary layer and displacement thickness are also presented.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제19권3호
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• pp.327-364
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• 2015

In this paper, a generalized guidance law with an arbitrary pair of guidance coefficients for impact angle control is proposed. Under the assumptions of a stationary target and a lag-free missile with constant speed, necessary conditions for the guidance coefficients to satisfy the required terminal constraints are obtained by deriving an explicit closed-form solution. Moreover, optimality of the generalized impact-angle control guidance law is discussed. By solving an inverse optimal control problem for the guidance law, it is found that the generalized guidance law can minimize a certain quadratic performance index. Finally, analytic solutions of the generalized guidance law for a first-order lag system are investigated. By solving a third-order linear time-varying ordinary differential equation, the blowing-up phenomenon of the guidance loop as the missile approaches the target is mathematically proved. Moreover, it is found that terminal misses due to the system lag are expressed in terms of the guidance coefficients, homing geometry, and the ratio of time-to-go to system time constant.

• International Journal of Automotive Technology
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• 제7권5호
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• pp.535-545
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• 2006

A nonlinear analysis of torsional self-excited vibration in the driveline system for wheeled towing tractors was presented, with a 2-DOF mathematical model. The vibration system was described as a second-order ordinary differential equation. An analytical approach was proposed to the solution of the second-order ODE. The mathematical neighborhood concept was used to construct the interior boundary and the exterior boundary. The ODE was proved to have a limit cycle by using $Poincar\'{e}-Bendixson$ Annulus Theorem when two inequalities were satisfied. Because the two inequalities are easily satisfied, the self-excited vibration is inevitable and even the initial slip rate is little. However, the amplitude will be almost zero when the third inequality is satisfied. Only in a few working modes of the towing tractor the third inequality is not satisfied. It is shown by experiments that the torsional self-excited vibration in the driveline of the vehicle is obvious.

• Ocean Systems Engineering
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• 제2권3호
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• pp.229-238
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• 2012

We discuss the bi - stability that is possibly exhibited by a liquid free surface in a parametrically - driven two-dimensional (2D) rectangular tank with finite liquid depth. Following the method of adaptive mode ordering, assuming two dominant modes and retaining polynomial nonlinearities up to third-order, a nonlinear finite-dimensional nonlinear modal system approximation is obtained. A "continuation method" of nonlinear dynamics is then used in order to elicit efficiently the instability boundary in parameters' space and to predict how steady surface elevation changes as the frequency and/or the amplitude of excitation are varied. Results are compared against those of the linear version of the system (that is a Mathieu-type model) and furthermore, against an intermediate model also derived with formal mode ordering, that is based on a second - order ordinary differential equation having nonlinearities due to products of elevation with elevation velocity or acceleration. The investigation verifies that, in parameters space, there must be a region, inside the quiescent region, where liquid surface instability is exhibited. There, behaviour depends on initial conditions and a wave form would be realised only if the free surface was substantially disturbed initially.