• 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.

• 충청수학회지
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• 제27권3호
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• pp.455-467
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• 2014

In this paper, we consider the additive set-valued functional equation $nf(\sum_{i=1}^{n}x_i)=\sum_{i=1}^{n}f(x_i){\oplus}\sum_{1{\leq}i<j{\leq}n}f(x_i+x_j)$ where $n{\geq}2$ is an integer, and prove the Hyers-Ulam stability of the functional equation.

• 한국수학교육학회지시리즈A:수학교육
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• 제10권2호
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• pp.4-8
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• 1972

We consider the class (equation omitted) of all k-degenerate graphs, for k a non-negative integer. The class (equation omitted) and (equation omitted) are exactly the classes of totally disconnected graphs and of forests, respectively; the classes (equation omitted) and (equation omitted) properly contain all outerplanar and planar graphs respectively. The advantage of this view point is that many of the known results for chromatic number and point arboricity have natural extensions, for all larger values of k. The purpose of this note is to show that a graph G is (P$^3$)-realizable if G is planar and 3-degenerate.

• 대한수학회보
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• 제33권4호
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• pp.513-519
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• 1996

The stability problem of functional equations has been originally raised by S. M. Ulam. In 1940, he posed the following problem: Give conditions in order for a linear mapping near an approximately additive mapping to exist (see [9]).

• 대한수학회논문집
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• 제31권1호
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• pp.101-113
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• 2016

In this article, we considered the stability of the following (${\alpha}$, ${\beta}$, ${\gamma}$)-derivation $${\alpha}D[x,y]={\beta}[D(x),y]+{\gamma}[x,D(y)]$$ and homomorphisms associated to the quadratic type functional equation $$f(kx+y)+f(kx+{\sigma}(y))=2kg(x)+2g(y),\;x,y{\in}A$$, where ${\sigma}$ is an involution of the Lie $C^*$-algebra A and k is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

• Journal of applied mathematics & informatics
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• 제18권1_2호
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• pp.339-350
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• 2005

A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order $\alpha{\in}(0,1]$, and the second-order space derivative is replaced with a Riesz-Feller derivative of order $\beta{\in}0,2]$. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.

• 한국수학교육학회지시리즈D:수학교육연구
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• 제14권1호
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• pp.33-50
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• 2010

The mathematical thinking which transforms important mathematical content and developed into mathematical structure is a vital process in building up mathematical ability as mathematical knowledge based on structure. Such process based on students' recognition of mathematical concept. Developing mathematical thinking into mathematical structure happens when different cognitive units are connected and compressed to form schema of solution, which could happen through some guided problems. The effort of arithmetic approach in problem solving did not necessarily provide students the structure schema of solution. The using of equation to solve the problem is based on the schema of building equation, and is not necessary recognizing the structure of the solution, as the recognition of structure may be lost in the process of simplification of algebraic expressions, leaving only the final numeric answer of the problem.

• 대한수학회논문집
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• 제19권3호
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• pp.495-506
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• 2004

Some oscillation criteria are given for second order nonlinear differential equations by means of integral averaging technique.

• 대한수학회논문집
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• 제14권2호
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• pp.353-364
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• 1999

We found the characterizations for convolution operators in the Beurling`s generalized distribution space to be hyperbolic.