• East Asian mathematical journal
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• 제30권1호
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• pp.79-91
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• 2014

We consider the parabolic evolution differential equation such as heat equation and porus-medium equation on a Riemannian manifold M whose Ricci curvature is bounded below by $-(n-1)k^2$ and bounded below by 0 on some amount of M. We derive some bounds of differential quantities for a positive solution and some inequalities which resemble Harnack inequalities.

• 대한수학회보
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• 제52권4호
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• pp.1185-1199
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• 2015

This paper is concerned with the optimal control problem for the viscous weakly dispersive Benjamin-Bona-Mahony (BBM) equation. We prove the existence and uniqueness of weak solution to the equation. The optimal control problem for the viscous weakly dispersive BBM equation is introduced, and then the existence of optimal control to the problem is proved.

• 대한수학회논문집
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• 제13권3호
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• pp.481-487
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• 1998

Let (equation omitted) denote the closure of the set (equation omitted) of dominant operators in the norm topology. We show that the Weyl spectrum of an operator T $\in$ (equation omitted) satisfies the spectral mapping theorem for analytic functions, which is an extension of [5, Theorem 1]. Also we show that an operator approximately equivalent to an operator of class (equation omitted) is of class (equation omitted).

• 대한수학회보
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• 제42권1호
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• pp.57-73
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• 2005

In this paper we solve a generalized quadratic Jensen type functional equation $m^2 f (\frac{x+y+z}{m}) + f(x) + f(y) + f(z) =n^2 [f(\frac{x+y}{n}) +f(\frac{y+z}{n}) +f(\frac{z+x}{n})]$ and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and Gavruta.

• 대한수학회지
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• 제50권1호
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• pp.173-187
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• 2013

This paper studies the properties of solutions of the Riccati equation arising from the quadratic optimal control problem of the general damped second order system. Using the semigroup theory, we establish the weak differential characterization of the Riccati equation for a general class of the second order distributed systems with arbitrary damping terms.

• 대한수학회보
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• 제40권2호
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• pp.253-267
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• 2003

In this paper we deal With the quadratic functional equation (equation omitted) deriving from an inequality of T. Popoviciu for convex functions. We solve this functional equation by proving that its solutions we the polynomials of degree at most two. Likewise, we investigate its stability in the spirit of Hyers, Ulam, and Rassias.

• Journal of applied mathematics & informatics
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• 제13권1_2호
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• pp.233-245
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• 2003

A time fractional advection-dispersion equation is Obtained from the standard advection-dispersion equation by replacing the firstorder derivative in time by a fractional derivative in time of order ${\alpha}$(0 < ${\alpha}$ $\leq$ 1). Using variable transformation, Mellin and Laplace transforms, and properties of H-functions, we derive the complete solution of this time fractional advection-dispersion equation.

• 대한수학회지
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• 제36권5호
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• pp.833-846
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• 1999

In this paper, we will study the relationship between the controlling factor of the solution to a difference equation and the solution of the corresponding differential equation. Many times the controlling factors are the same. But even the controlling factor of the two solutions may be different, we will discover a way to compute, for first order non-linear equations, the controlling factor of the solution to the difference equation using the solution of the differential equation.

• 대한수학회보
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• 제43권4호
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• pp.791-804
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• 2006

This paper is concerned with a second-order iterated differential equation of the form $c_0x'(Z)+c_1x'(z)+c_2x(z)=x(az+bx(z))+h(z)$ with the distinctive feature that the argument of the unknown function depends on the state. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained.

• 충청수학회지
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• 제25권1호
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• pp.1-18
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• 2012

The aim of this paper is to investigate the superstability of the pexider type sine(hyperbolic sine) functional equation $f(\frac{x+y}{2})^{2}-f(\frac{x+{\sigma}y}{2})^{2}={\lambda}g(x)h(y),\;{\lambda}:\;constant$ which is bounded by the unknown functions ${\varphi}(x)$ or ${\varphi}(y)$. As a consequence, we have generalized the stability results for the sine functional equation by P. M. Cholewa, R. Badora, R. Ger, and G. H. Kim.