• Journal of applied mathematics & informatics
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• 제23권1_2호
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• pp.167-182
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• 2007

The existence of solutions of the following multi-point boundary value problem $${x^{(n)}(t)=f(t,\;x(t),\;x'(t),{\cdots}, x^{(n-2)}(t))+r(t),\;0 is studied. Sufficient conditions for the existence of at least one solution of BVP(*) are established. It is of interest that the growth conditions imposed on f are allowed to be super-linear (the degrees of phases variables are allowed to be greater than 1 if it is a polynomial). The results are different from known ones since we don't apply the Green's functions of the corresponding problem and the method to obtain a priori bounds of solutions are different enough from known ones. Examples that can not be solved by known results are given to illustrate our theorems.

• 대한수학회논문집
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• 제33권2호
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• pp.639-650
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• 2018

Finding a solution for the Legendre equation is difficult. Especially if it is as a part of the Laplace equation solving in the electric fields. In this paper, first a problem of the generalized differential transform method (GDTM) is solved by the Sturm-Liouville equation, then the Legendre equation is solved by using it. To continue, the approximate solution is compared with the nth-degree Legendre polynomial for obtaining the inner and outer potential of a sphere. This approximate is more accurate than the previous solutions, and is closer to an ideal potential in the intervals.

• Journal of applied mathematics & informatics
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• 제19권1_2호
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• pp.179-190
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• 2005

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order ${\beta}{\in}$ (0, 2] and the first-order time derivative with Caputo derivative of order ${\beta}{\in}$ (0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.

• Journal of applied mathematics & informatics
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• 제26권1_2호
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• pp.1-14
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• 2008

In this paper, A Riesz fractional diffusion equation with a nonlinear source term (RFDE-NST) is considered. This equation is commonly used to model the growth and spreading of biological species. According to the equivalent of the Riemann-Liouville(R-L) and $Gr\ddot{u}nwald$-Letnikov(G-L) fractional derivative definitions, an implicit difference approximation (IFDA) for the RFDE-NST is derived. We prove the IFDA is unconditionally stable and convergent. In order to evaluate the efficiency of the IFDA, a comparison with a fractional method of lines (FMOL) is used. Finally, two numerical examples are presented to show that the numerical results are in good agreement with our theoretical analysis.

• 대한수학회보
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• 제47권6호
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• pp.1225-1234
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• 2010

Numerical solutions for the fractional differential dispersion equations with nonlinear forcing terms are considered. The backward Euler finite difference scheme is applied in order to obtain numerical solutions for the equation. Existence and stability of the approximate solutions are carried out by using the right shifted Grunwald formula for the fractional derivative term in the spatial direction. Error estimate of order $O({\Delta}x+{\Delta}t)$ is obtained in the discrete $L_2$ norm. The method is applied to a linear fractional dispersion equations in order to see the theoretical order of convergence. Numerical results for a nonlinear problem show that the numerical solution approach the solution of classical diffusion equation as fractional order approaches 2.

• 전자공학회논문지D
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• 제34D권3호
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• pp.93-104
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• 1997

Steady-state I-V characteristics of a resonant tunneling diode (RTD) is calculated by numerical analysis using quantum liouville equation based on wigner function which is derived from density operator. Modifications to the conventional discrete model are made to calculate more accurate quantum correlations. It is pointed out that we must include inelastic processes and the resistivity of the contacting layers to get a much more credible potential which can be theoretically obtained from the simple screening theory. The effects of spatially-varying effective mass is also checked briefly.

• 대한수학회지
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• 제55권6호
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• pp.1459-1468
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• 2018

We show that an umbilic-free minimal surface in ${\mathbb{R}}^3$ belongs to the associate family of the catenoid if and only if the geodesic curvatures of its lines of curvature have a constant ratio. As a corollary, the helicoid is shown to be the unique umbilic-free minimal surface whose lines of curvature have the same geodesic curvature. A similar characterization of the deformation family of minimal surfaces with planar lines of curvature is also given.

• Korean Journal of Mathematics
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• 제27권2호
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• pp.487-503
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• 2019

In this study, the author provided a discussion on one dimensional Laplace and Fourier transforms with their applications. It is shown that the combined use of exponential operators and integral transforms provides a powerful tool to solve space fractional partial differential equation with non - constant coefficients. The object of the present article is to extend the application of the joint Fourier - Laplace transform to derive an analytical solution for a variety of time fractional non - homogeneous KdV. Numerous exercises and examples presented throughout the paper.

• Nonlinear Functional Analysis and Applications
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• 제28권1호
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• pp.237-249
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• 2023

In this paper, by means of the Schauder fixed point theorem and Arzela-Ascoli theorem, the existence and uniqueness of solutions for a class of not instantaneous impulsive problems of nonlinear fractional functional Volterra-Fredholm integro-differential equations are investigated. An example is given to illustrate the main results.

• 대한화학회지
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• 제20권5호
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• pp.340-350
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• 1976

글라스(glass) 전이온도 이하에서 고체 중합체들의 역학적 분광학의 여러 현상들을 이론적으로 기술하는 데 적용된 線形 반응 이론을 제안하였다. 시료에 의한 에너지 분산을 어떤 시간 相關函數들로 나타내었다. Liouville 연산자를 Kirkwood의 확산식 연산자로 대치하면, 여기에서 시간은 결과로부터 다리결합을 한 중합체들의 에너지 상실과 완화에 대한 Kirkwood의 결과가 나옴을 보였다. 완화시간을 계산하는 방법을 보이기 위하여 상관함수들을 계산하는 근사법을 고찰하였다. 여기에서 제안한 이론과 저온에서 매달린 (pendant) 시클로헥실기들을 가진 고체 중합체들의 역학적 에너지 완화 현상을 기술하는 데 사용된 한 모형 이론과의 관련성을 찾아 보기 위해, 이 근사법을 써서 二重우물 퍼텐셜 모형을 고찰하였다.