• Title, Summary, Keyword: weak solution

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A WEAK SOLUTION OF A NONLINEAR BEAM EQUATION

  • Choi, Q.H.;Choi, K.P.;Jung, T.;Han, C.H.
    • Korean Journal of Mathematics
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    • v.4 no.1
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    • pp.51-64
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    • 1996
  • In this paper we investigate the existence of weak solutions of a nonlinear beam equation under Dirichlet boundary condition on the interval $-\frac{\pi}{2}<x<\frac{\pi}{2}$ and periodic condition on the variable $t$, $u_{tt}+u_{xxxx}=p(x,t,u)$. We show that if $p$ satisfies condition $(p_1)-(p_3)$, then the nonlinear beam equation possesses at least one weak solution.

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GLOBAL ATTRACTOR FOR A SEMILINEAR PSEUDOPARABOLIC EQUATION WITH INFINITE DELAY

  • Thanh, Dang Thi Phuong
    • Communications of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.579-600
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    • 2017
  • In this paper we consider a semilinear pseudoparabolic equation with polynomial nonlinearity and infinite delay. We first prove the existence and uniqueness of weak solutions by using the Galerkin method. Then, we prove the existence of a compact global attractor for the continuous semigroup associated to the equation. The existence and exponential stability of weak stationary solutions are also investigated.

BOUNDED WEAK SOLUTION FOR THE HAMILTONIAN SYSTEM

  • Choi, Q-Heung;Jung, Tacksun
    • Korean Journal of Mathematics
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    • v.21 no.1
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    • pp.81-90
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    • 2013
  • We investigate the bounded weak solutions for the Hamiltonian system with bounded nonlinearity decaying at the origin and periodic condition. We get a theorem which shows the existence of the bounded weak periodic solution for this system. We obtain this result by using variational method, critical point theory for indefinite functional.

A Study on Performance Characteristics of A Single Effect LiBr/Water Refrigeration Cycle (단효용 LiBr/물 흡수식 냉동사이클의 성능특성에 관한 연구)

  • 연제문;임삼택;오주원;이경우
    • Proceedings of the Korean Society of Marine Engineers Conference
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    • pp.95-99
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    • 2001
  • As a way to use energy effectively, the present study is aimed at investigating the performance characteristics of a Single Effect LiBr/Water Absorption Refrigerator using a low temperature driving heat-source. It was carried out by changing the driving heat-source temperature, the cold water outlet temperature(the refrigeration load), the cooling water inlet temperature, and the weak solution flow rate and this study compares the performance characteristics of refrigerator against the existence and non-existence of the Recirculation of the Weak solution which is used as a method to improve the performance of refrigerator. In case of Recirculation of the weak solution, more improved the Refrigeration Capacity and COP was obtained, and these effects became more larger in the high temperature of driving heat-source and large quantity of solution.

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Flapwise and non-local bending vibration of the rotating beams

  • Mohammadnejad, Mehrdad;Saffari, Hamed
    • Structural Engineering and Mechanics
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    • v.72 no.2
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    • pp.229-244
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    • 2019
  • Weak form integral equations are developed to investigate the flapwise bending vibration of the rotating beams. Rayleigh and Eringen nonlocal elasticity theories are used to investigate the rotatory inertia and Size-dependency effects on the flapwise bending vibration of the rotating cantilever beams, respectively. Through repetitive integrations, the governing partial differential equations are converted into weak form integral equations. The novelty of the presented approach is the approximation of the mode shape function by a power series which converts the equations into solvable one. Substitution of the power series into weak form integral equations results in a system of linear algebraic equations. The natural frequencies are determined by calculation of the non-trivial solution for resulting system of equations. Accuracy of the proposed method is verified through several numerical examples, in which the influence of the geometry properties, rotatory inertia, rotational speed, taper ratio and size-dependency are investigated on the natural frequencies of the rotating beam. Application of the weak form integral equations has made the solution simpler and shorter in the mathematical process. Presented relations can be used to obtain a close-form solution for quick calculation of the first five natural frequencies of the beams with flapwise vibration and non-local effects. The analysis results are compared with those obtained from other available published references.

A STRONG SOLUTION FOR THE WEAK TYPE II GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS

  • Kim, Won-Kyu;Kum, Sang-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.3
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    • pp.599-610
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    • 2006
  • The aim of this paper is to give an existence theorem for a strong solution of generalized vector quasi-equilibrium problems of the weak type II due to Hou et al. using the equilibrium existence theorem for 1-person game, and as an application, we shall give a generalized quasivariational inequality.

NOTES ON NEW SINGULAR FUNCTION METHOD FOR DOMAIN SINGULARITIES

  • Kim, Seok-Chan;Pyo, Jae-Hong;Xie, Shu-Sen;Yi, Su-Cheol
    • Honam Mathematical Journal
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    • v.29 no.4
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    • pp.701-721
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    • 2007
  • Recently, a new singular function(NSF) method was posed to get accurate numerical solution on quasi-uniform grids for two-dimensional Poisson and interface problems with domain singularities by the first author and his coworkers. Using the singular function representation of the solution, dual singular functions, and an extraction formula for stress intensity factors, the method poses a weak problem whose solution is in $H^2({\Omega})$ or $H^2({\Omega}_i)$. In this paper, we show that the singular functions, which are not in $H^2({\Omega})$, also satisfy the integration by parts and note that this fact suggests the possibility of different choice of the weak formulations. We show that the original choice of weak formulation of NSF method is critical.

UNIQUENESS OF POSITIVE STEADY STATES FOR WEAK COMPETITION MODELS WITH SELF-CROSS DIFFUSIONS

  • Ko, Won-Lyul;Ahn, In-Kyung
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.2
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    • pp.371-385
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    • 2004
  • In this paper, we investigate the uniqueness of positive solutions to weak competition models with self-cross diffusion rates under homogeneous Dirichlet boundary conditions. The methods employed are upper-lower solution technique and the variational characterization of eigenvalues.

WEAK SOLUTIONS OF THE EQUATION OF MOTION OF MEMBRANE WITH STRONG VISCOSITY

  • Hwang, Jin-Soo;Nakagiri, Shin-Ichi
    • Journal of the Korean Mathematical Society
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    • v.44 no.2
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    • pp.443-453
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    • 2007
  • We study the equation of a membrane with strong viscosity. Based on the variational formulation corresponding to the suitable function space setting, we have proved the fundamental results on existence, uniqueness and continuous dependence on data of weak solutions.