• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.53-68
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• 2004

Numerical solutions for the viscous Cahn-Hilliard equation are considered using the Crank-Nicolson type finite difference method which conserves the mass. The corresponding stability and error analysis of the scheme are shown. The decay speeds of the solution in $H^1-norm$ are shown. We also compare the evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation numerically and computationally, which has been given as an open question in Novick-Cohen[13].

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.265-276
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• 2019

For a simple implementation, a linear convex splitting scheme was coupled with the Fourier spectral method for the Cahn-Hilliard equation with a logarithmic free energy. However, an inappropriate value of the splitting parameter of the linear scheme may lead to incorrect morphologies in the phase separation process. In order to overcome this problem, we present a nonlinear convex splitting Fourier spectral scheme for the Cahn-Hilliard equation with a logarithmic free energy, which is an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. Using the nonlinear scheme, we derive a useful formula for the relation between the gradient energy coefficient and the thickness of the interfacial layer. And we present numerical simulations showing the different evolution of the solution using the linear and nonlinear schemes. The numerical results demonstrate that the nonlinear scheme is more accurate than the linear one.

• Journal of the Korean Mathematical Society
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• v.41 no.3
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• pp.435-459
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• 2004

Identification problems for the system governed by abstract nonlinear damped second order evolution equations are studied. Since unknown parameters are included in the diffusion operator, we can not simply identify them by using the usual optimal control theories. In this paper we present how to solve our identification problems via the method of transposition.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.359-370
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• 2011

In this work, we obtain new solitary wave solutions for some nonlinear partial differential equations. The Jacobi elliptic function rational expansion method is used to establish new solitary wave solutions for the combined KdV-mKdV and Klein-Gordon equations. The results reveal that Jacobi elliptic function rational expansion method is very effective and powerful tool for solving nonlinear evolution equations arising in mathematical physics.

• 한국전산유체공학회:학술대회논문집
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• 2008.08a
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• pp.33.1-33.1
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• 2008

• Communications of the Korean Mathematical Society
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• v.11 no.4
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• pp.1003-1013
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• 1996

The existence of a bounded generalized solution on the real line for a nonlinear functional evolution problem of the type $$ (FDE) x'(t) + A(t,x_t)x(t) \ni 0, t \in R $$ in a general Banach spaces is considered. It is shown that (FDE) has a bounded generalized solution on the whole real line with well-known Crandall and Pazy's result and recent results of the functional differential equations involving the operator A(t).

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.501-515
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• 2009

A second order nonlinear ordinary differential equation is obtained by solving the initial-boundary value problem of a class of elas-todynamics equations, which models the radially symmetric motion of a incompressible hyper-elastic solid sphere under a suddenly applied surface tensile load. Some new conclusions are presented. All existence conditions of nonzero solutions of the ordinary differential equation, which describes cavity formation and motion in the interior of the sphere, are presented. It is proved that the differential equation has singular periodic solutions only when the surface tensile load exceeds a critical value, in this case, a cavity would form in the interior of the sphere and the motion of the cavity with time would present a class of singular periodic oscillations, otherwise, the sphere remains a solid one. To better understand the results obtained in this paper, the modified Varga material is considered simultaneously as an example, and numerical simulations are given.

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.547-564
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• 1994

Let X be a real Banach space. We consider the existence of solutions of the abstract nonlinear functional evolution equation : $$ (E) \frac{du(t)}{dt} + A(t)u(t) + F(u)(t) \ni h(t), $$ $$ u(s) = x_o \in D(A(s)), 0 \leq s \leq t \leq T, $$ where u : $[s, T] \to x$ is an unknown function, ${A(t) : 0 \leq t \leq T}$ is a given family of nonlinear (possibly multivalued) operators in X, and $F : C([s, t];X) \to L^{\infty}([s, X];X)$ and $h : [s, T] \to X$ are given functions.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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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.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.265-291
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• 2012

In this paper, we consider the regularity theory and the existence of smooth solution of a degenerate fully nonlinear equation describing the evolution of the rolling stones with nonconvex sides: $\{M(h)=h_t-F(t,z,z^{\alpha}h_{zz})\;in\;\{0<z{\leq}1\}{\times}[0,T] \\ h_t(z,t)=H(h_z(z,t),h)\;{on}\;\{z=0\}$. We establish the Schauder theory for $C^{2,{\alpha}}$-regularity of h.