• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1285-1303
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• 2018

This note is motivated by gradient estimates of Li-Yau, Hamilton, and Souplet-Zhang for heat equations. In this paper, our aim is to investigate Yamabe equations and a non linear heat equation arising from gradient Ricci soliton. We will apply Bochner technique and maximal principle to derive gradient estimates of the general non-linear heat equation on Riemannian manifolds. As their consequence, we give several applications to study heat equation and Yamabe equation such as Harnack type inequalities, gradient estimates, Liouville type results.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.2
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• pp.149-157
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• 2012

By means of fractional calculus techniques, we find explicit solutions of confluent hypergeometric equations. We use the N-fractional calculus operator $N^{\mu}$ method to derive the solutions of these equations.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.231-243
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• 2007

The existence of solutions of a class of two-point boundary value problems for higher order differential equations is studied. Sufficient conditions for the existence of at least one solution are established. It is of interest that the nonlinearity f in the equation depends on all lower derivatives, and 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 bound of solutions are different enough from known ones. Examples that can not be solved by known results are given to illustrate our theorems.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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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.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1101-1121
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• 2008

One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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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.

• Nonlinear Functional Analysis and Applications
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• v.28 no.1
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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.

• Journal of IKEEE
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• v.23 no.2
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• pp.502-507
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• 2019

We investigated theoretically the quantum optical transition properties of qusi 2-Dinensinal Landau splitting system, in Si. We apply the Quantum Transport theory (QTR) to the system in the confinement of electrons by square well confinement potential. We use the projected Liouville equation method with Equilibrium Average Projection Scheme (EAPS). In order to analyze the quantum transition, we compare the temperature and the magnetic field dependencies of the QTLW and the QTLS on two transition processes, namely, the phonon emission transition process and the phonon absorption transition process. Through the analysis of this work, we found the increasing properties of QTLW and QTLS of Si with the temperature and the magnetic fields. We also found the dominant scattering processes are the phonon emission transition process.

• Journal of IKEEE
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• v.22 no.1
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• pp.61-67
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• 2018

We investigated theoretically the magnetic field dependence of the quantum optical transition of qusi 2-Dimensional Landau splitting system, in CdS and ZnO. In this study, we investigate electron confinement by square well confinement potential in magnetic field system using quantum transport theory(QTR). In this study, theoretical formulas for numerical analysis are derived using Liouville equation method and Equilibrium Average Projection Scheme (EAPS). In this study, the absorption power, P (B), and the Quantum Transition Line Widths (QTLWS) of the magnetic field in CdS and ZnO can be deduced from the numerical analysis of the theoretical equations, and the optical quantum transition line shape (QTLS) is found to increase. We also found that QTLW, ${\gamma}(B)_{total}$ of CdS < ${\gamma}(B)_{total}$ of ZnO in the magnetic field region B<25 Tesla.

• Journal of Korean Society for Quality Management
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• v.20 no.1
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• pp.158-167
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• 1992

This paper represents the new techniques for optimal sampling plans of reliability applying the mathematical complex number(real and imaginary number) in the complex system of reliability. The research formulation represent a mathematical model Which preserves all essential aspects of the main and auxiliary factors of the research objectives. It is important to formule the problem in good agreement with the objective of the research considering the main and auxilary factors which affect the system performance. This model was repeatedly tested to determine the required statistical chatacteristics which in themselves determine the actual and standard distributions. The evaluation programs and techniques are developed for establishing criteria for sampling plans of reliability effectiveness, and the evaluation of system performance was based on the complex stochastic process(derived by the Runge-Kutta method. by kolmogorv's criterion and the transform of a solution to a Sturon-Liouville equation.) The special structure of this mathematical model is exploited to develop the optimal sampling plans of reliability in the complex system.