• Journal of the Chungcheong Mathematical Society
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• v.26 no.3
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• pp.501-516
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• 2013

The well-known Vlasov-Poisson equation describes plasma physics as nonlinear first-order partial differential equations. Because of the nonlinear condition from the self consistency of the Vlasov-Poisson equation, many problems occur: the existence, the numerical solution, the convergence of the numerical solution, and so on. To solve the problems, a viscosity term (a second-order partial differential equation) is added. In a viscosity term, the Vlasov-Poisson equation changes into a parabolic equation like the Fokker-Planck equation. Therefore, the Schauder fixed point theorem and the classical results on parabolic equations can be used for analyzing the Vlasov-Poisson equation. The sequence and the convergence results are obtained from linearizing the Vlasove-Poisson equation by using a fixed point theorem and Gronwall's inequality. In numerical experiments, an implicit first-order scheme is used. The numerical results are tested using the changed viscosity terms.

• Bulletin of the Korean Mathematical Society
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• v.44 no.3
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• pp.569-587
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• 2007

In this paper, we establish the generalized Hyers-Ulam-Rassias stability of the Pexider type quadratic equation $f_1(x+y+z)+f_2(x-y)+f_3(x-z)-f_4(x-y-z)-f_5(x+y)-f_6(x+z)=0$ and its general solution.

• Bulletin of the Korean Mathematical Society
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• v.39 no.1
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• pp.81-87
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• 2002

The weighted Hardy＇s inequality is known as (equation omitted) where -$\infty$$\leq$a$\leq$b$\leq$$\infty$ and 1 < p < $\infty$. The purpose of this article is to provide a useful formula to express the weight r(x) in terms of s(x) or vice versa employing a Bernoulli equation having the other weight as coefficients.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.2
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• pp.173-182
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• 2007

In this paper, we prove the Cauchy-Rassias stability of derivations on quasi-Banach algebras associated to the Cauchy functional equation and the Jensen functional equation. We use the Cauchy-Rassias inequality that was first introduced by Th. M. Rassias in the paper "On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300".

• Journal of the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.887-900
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• 2013

In this paper, we investigate the stability for the functional equation f(3x+y)-3f(2x+y)+3f(x+y)-f(y)=0 in the sense of M. S. Moslehian and Th. M. Rassias.

• Journal of the Korean Mathematical Society
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• v.38 no.6
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• pp.1179-1190
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• 2001

Quadratic invariant curve is one of the simplest nonlinear invariant curves and was considered by C. T. Ng and the author in order to study the one-dimensional nonlinear dynamics displayed by a second order delay differential equation with piecewise constant argument. In this paper a functional equation derived from the problem of invariant curves is discussed. Using a different method from what C. T. Ng and the author once used, we define solutions piecewisely and give results in the remaining difficult case left in C. T. Ng and the authors work. A problem of analytic extension given in their work is also answered negatively.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.377-384
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• 2008

In this paper, we prove the stability of the functional equation $$\sum\limits_{i=0}^{3}3Ci(-1)^{3-i}f(ix+y)-3!f(x)=0$$ in the sense of P. $G{\breve{a}}vruta$ on the punctured domain. Also, we investigate the superstability of the functional equation.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.2
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• pp.199-230
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• 2018

In this paper, we introduce the general form of a viginti functional equation. Then, we find the general solution and study the generalized Ulam-Hyers stability of such functional equation in multi-Banach spaces by using fixed point technique. Also, we indicate an example for non-stability case regarding to this new functional equation.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1241-1252
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• 2015

In this paper, we study the dynamical bifurcation of the modified Swift-Hohenberg equation on a periodic interval as the system control parameter crosses through a critical number. This critical number depends on the period. We show that there happens the pitchfork bifurcation under the spatially even periodic condition. We also prove that in the general periodic condition the equation bifurcates to an attractor which is homeomorphic to a circle and consists of steady states solutions.

• Bulletin of the Korean Mathematical Society
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• v.42 no.1
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• pp.75-85
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• 2005

In this paper, the visco-Da-Rios equation; (0.1) ($$\frac{{\partial}{\gamma}}{{\partial}t}=\frac{{\partial}{\gamma}}{{\partial}s}{\bigwedge}\frac{D}{ds}\frac{{\partial}{\gamma}}{{\partial}s}+{\nu}\frac{{\partial}{\gamma}}{{\partial}s}$$) is investigated on 3-dimensional complete orientable Riemannian manifolds. The global existence of solution is discussed by trans-forming (0.1) into a cubic nonlinear Schrodinger equation for complete orient able Riemannian 3-manifolds of constant curvature.