• Journal of the Computational Structural Engineering Institute of Korea
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• v.20 no.2
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• pp.127-136
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• 2007

In this paper, the equations of the scaled boundary finite element method are derived for non-homogeneous half plane and analyzed numerically In the scaled boundary finite element method, partial differential equations are weaken in the circumferential direction by approximation scheme such as the finite element method, and the radial direction of equations remain in analytical form. The scaled boundary equations of non-homogeneous half plane, its elastic modulus varies as power function, are newly derived by the virtual work theory. It is shown that the governing equation of this problem is the Euler-Cauchy equation, therefore, the logarithm mode used in the half plane problem is not valid in this problem. Two numerical examples are analysed for the verification and the feasibility.

• Journal of applied mathematics & informatics
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• v.34 no.5_6
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• pp.359-367
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• 2016

This paper deals with the Cauchy problem for heat equations with coupling moving reactions of mixed type. After obtaining the infinite Fujita blow-up exponent, we classify optimally the simultaneous and non-simultaneous blow-up for two components of the solutions. Moreover, blow-up rates and set are determined. By using the analogous procedures, one can fill in the gaps for the other two systems, which are studied in the paper 'Australian and New Zealand Industrial and Applied Mathematics Journal' 48(2006)37-56.

• Journal of applied mathematics & informatics
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• v.3 no.2
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• pp.279-290
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• 1996

In the present paper we study the Cauchy problem in a Banach space E for an abstract nonlinear differential equation of form $$\frac{d^2u}{dt^2}=-A{\frac{du}{dt}}+B(t)u+f(t, W)$$ where W=($A_1$(t)u, A_2(t)u)..., A_{\nu}(t)u), A_{i}(t),\;i=1,2,...{\nu}$,(B(t), t{\in}I$=[0, b]) are families of closed operators defined on dense sets in E into E, f is a given abstract nonlinear function on $I{\times}E^{\nu}$ into E and -A is a closed linar operator defined on dense set in e into E which generates a semi-group. Further the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families ($A_{i}$(t), i =1.2...${\nu}$), (B(t), $t{\in}I$) An application and some properties are also given for the theory of partial diferential equations.

• Honam Mathematical Journal
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• v.22 no.1
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• pp.37-46
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• 2000

We prove the stability of a generalized Cauchy functional equation of the form ; $$f(a_1x+a_2y)=b_1f(x)+b_2f(y)+w.$$ That is, we obtain a partial answer for the open problem which was posed by the Th.M Rassias and J. Tabor on the stability for a generalized functional equation.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1291-1302
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• 2011

We establish the local well-posedness for the Cauchy problem of the nonlinear Schr$\ddot{o}$odinger equation with harmonic potential in $H^s(\mathbb{R}^n)$, where $s{\in}\mathbb{R}$, s > 0.

• Journal of the Korean Mathematical Society
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• v.38 no.4
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• pp.843-852
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• 2001

In this lecture I shall discuss some recent progress in the development of methods for attacking the central questions of the formation and structure of singularities and of global regularity for solutions of the Cauchy problem for nonlinear systems of partial differential equations of hyperbolic type. Applications to the Einstein equations of general relativity and to the equations of compressible fluid flow shall be particularly emphasized and detailed.

• Journal of the Korean Statistical Society
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• v.34 no.1
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• pp.35-38
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• 2005

A single, tractable, special case of the problem of continuous mixtures of beta distributions over their parameters is considered. This is the uniform mixture of generalized arc-sine distributions which, curiously, turns out to be linked by transformation to the Cauchy distribution.

• Journal of Soil and Groundwater Environment
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• v.20 no.2
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• pp.10-21
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• 2015

The present study introduces a novel numerical approach for solving dispersion dominated problems with Cauchy boundary condition in an Eulerian-Lagrangian scheme. The study reveals the incapability of traditional Neuman approach to address the dispersion dominated problems with Cauchy boundary condition, even though it can produce reliable solution in the advection dominated regime. Also, the proposed numerical approach is applied to a real field problem of radioactive contaminant migration from radioactive waste repository which is a major current waste management issue. The performance of the proposed numerical approach is evaluated by comparing the results with numerical solutions of traditional FDM (Finite Difference Method), Neuman approach, and the analytical solution. The results show that the proposed numerical approach yields better and reliable solution for dispersion dominated regime, specifically for Peclet Numbers of less than 0.1. The proposed numerical approach is validated by applying to a real field problem of radioactive contaminant migration from radioactive waste repository of varying Peclet Number from 0.003 to 34.5. The numerical results of Neuman approach overestimates the concentration value with an order of 100 than the proposed approach during the assessment of radioactive contaminant transport from nuclear waste repository. The overestimation of concentration value could be due to the assumption that dispersion is negligible. Also our application problem confirms the existence of real field situation with advection dominated condition and dispersion dominated condition simultaneously as well as the significance or advantage of the proposed approach in the real field problem.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.267-296
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• 2018

This paper studies the Cauchy problem and blow-up phenomena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood-Paley decomposition theory, we investigate the local well-posedness of the equation in $B^s_{p,r}$ with s > $max\{{\frac{1}{p}},\;{\frac{1}{2}},\;1-{\frac{1}{p}}\},\;p,\;r{\in}[0,{\infty}]$. Second, we prove that the equation is locally well-posed in $B^s_{2,r}$ with the critical index $s={\frac{1}{2}}$ by virtue of the logarithmic interpolation inequality and the Osgood's Lemma, and it is shown that the data-to-solution mapping is $H{\ddot{o}}lder$ continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of m along the characteristics.

• Journal of the Korean Mathematical Society
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• v.32 no.1
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• pp.41-50
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• 1995