• East Asian mathematical journal
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• v.34 no.5
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• pp.703-713
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• 2018

Let $N{\geq}3$, $2^*=2N/(N-2)$ and $p{\in}(2,2^*)$. Our purpose in this paper is to consider multiple existence of solutions of problem $$-{\Delta}u-{\frac{\mu}{{\mid}x{\mid}^2}}+{\alpha}u={\mid}u{\mid}^{p-2}u+{\lambda}f\;u{\in}H^1({\mathbb{R}}^n)$$, where a, ${\lambda}$ > 0, ${\mu}{\in}(0,(N-2)^2/4)f{\in}H^{-1}({\mathbb{R}}^N)$, $f{\geq}0$ and $f{\neq}0$.

• Journal of Advanced Marine Engineering and Technology
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• v.15 no.4
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• pp.46-53
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• 1991

In the present study, Navier-Stokes equation is numerically solved by use of a Finite analytic method to obtain the 2-dimensional flow field in the square cavity. The basic idea of F.A.M. is the incorporation of local analytic solutions in the numerical solution of linear or non-linear partial differential equations. In the F.A.M., the total problem is subdivided into a number of all elements. The local analytic solution is obtained for the small element in which the governing equation, if non-linear, to be linearized. The local analytic solutions are then expressed in algebraic form and are overlapped to cover the entire region of the problem. The assembly of these local analytic solutions, which still preserve the overall nonlinearity of the governing equations, results in a system of linear algebraic equations. The system of algebraic equations is then solved to provide the numerical solutions of the total problem. The computed flow field shows the same characteristics to physical concept of flow phenomena.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.411-421
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• 2016

In this paper, we investigate the problem of transcendental entire functions that share two values with one of their derivative. Let f be a transcendental entire function, n and k be two positive integers. If $f^n-Q_1$ and $(f^n)^{(k)}-Q_2$ share 0 CM, and $n{\geq}k+1$, then $(f^n)^{(k)}{\equiv}{\frac{Q_2}{Q_1}}f^n$. Furthermore, if $Q_1=Q_2$, then $f=ce^{\frac{\lambda}{n}z}$, where $Q_1$, $Q_2$ are polynomials with $Q_1Q_2{\not\equiv}0$, and c, ${\lambda}$ are non-zero constants such that ${\lambda}^k=1$. This result shows that the Conjecture given by W. $L{\ddot{u}}$, Q. Li and C. Yang [On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281-1289.] is true. Also we exhibit some examples to show that the conditions of our result are the best possible.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.447-456
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• 2015

In this paper, we investigate the differential-difference equation $(f(z+c)-f(z))^2+P(z)^2(f^{(k)}(z))^2=Q(z)$, where P(z), Q(z) are nonzero polynomials. In addition, we also investigate Fermat type q-difference differential equations $f(qz)^2+(f^{(k)}(z))^2=1$ and $(f(qz)-f(z))^2+(f^{(k)}(z))^2=1$. If the above equations admit a transcendental entire solution of finite order, then we can obtain the precise expression of the solution.

• Kyungpook Mathematical Journal
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• v.64 no.1
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• pp.185-196
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• 2024

In this article, we investigate the growth of meromorphic solutions of $${\alpha}(z)(\frac{{\Delta}_c{\eta}}{{\eta}})^2\,+\,(b_2(z){\eta}^2(z)\;+\;b_1(z){\eta}(z)\;+\;b_0(z))\frac{{\Delta}_c{\eta}}{{\eta}} \atop =d_4(z){\eta}^4(z)\;+\;d_3(z){\eta}^3(z)\;+\;d_2(z){\eta}^2(z)\;+\;d_1(z){\eta}(z)\;+\;d_0(z),$$ where a(z), b_i(z) for i = 0, 1, 2 and d_j (z) for j = 0, ..., 4 are given functions, △_cη = η(z + c) - η(z) with c ∈ ℂ\{0}. In particular, when the a(z), the b_i(z) and the d_j(z) are polynomials, and d₄(z) ≡ 0, we shall show that if η(z) is a transcendental entire solution of finite order, and either deg a(z) ≠ deg d₀(z) + 1, or, deg a(z) = deg d₀(z) + 1 and ρ(η) ≠ ½, then ρ(η) ≥ 1.

• Journal of applied mathematics & informatics
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• v.32 no.5_6
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• pp.721-736
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• 2014

We consider the equation ${\Delta}_mu=p(x)u^{\alpha}+q(x)u^{\beta}$ on $R^N(N{\geq}2)$, where p, q are nonnegative continuous functions and 0 < ${\alpha}{\leq}{\beta}$. Under several hypotheses on p(x) and q(x), we obtain existence and nonexistence of blow-up solutions both for the superlinear and sublinear cases. Existence and nonexistence of entire bounded solutions are established as well.

• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1095-1113
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• 2020

In this paper, we investigate the transcendental meromorphic solutions for the nonlinear differential equations $f^nf^{(k)}+Q_{d_*}(z,f)=R(z)e^{{\alpha}(z)}$ and fⁿf^(k) + Q_d(z, f) = p₁(z)e^α₁(z) + p₂(z)e^α₂(z), where $Q_{d_*}(z,f)$ and Q_d(z, f) are differential polynomials in f with small functions as coefficients, of degree d_* (≤ n - 1) and d (≤ n - 2) respectively, R, p₁, p₂ are non-vanishing small functions of f, and α, α₁, α₂ are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of these kinds of meromorphic solutions and their possible forms of the above equations.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2006.11a
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• pp.573-576
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• 2006

This paper deals with multi-objective optimization problem of finding a set of well-distributed solutions close to the true Pareto optimal solutions. In this paper, we present a two-leveled symbiotic evolutionary algorithm to efficiently solve the problem. Most of the existing multi-objective evolutionary algorithms (MOEAs) operate one population that consists of individuals representing the complete solution to the problem. The proposed algorithm maintains several populations, each of which represents a partial solution to the entire problem, and has a structure with two levels. The parallel search and the structure are intended to improve the capability of searching diverse and good solutions. The performance of the proposed algorithm is compared with those of the existing algorithms in terms of convergence and diversity. The experimental results confirm the effectiveness of the proposed algorithm.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.795-814
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• 2021

In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: fⁿ + P(f) = R(z)e^α(z) and fⁿ + P_*(f) = p₁(z)e^α₁(z) + p₂(z)e^α₂(z) in the complex plane, where P(f) and P_*(f) are differential polynomials in f of degree n - 1 with coefficients being small functions and rational functions respectively, R is a non-vanishing small function of f, α is a nonconstant entire function, p₁, p₂ are non-vanishing rational functions, and α₁, α₂ are nonconstant polynomials. Particularly, we consider the solutions of the second equation when p₁, p₂ are nonzero constants, and deg α₁ = deg α₂ = 1. Our results are improvements and complements of Liao ([9]), and Rong-Xu ([11]), etc., which partially answer a question proposed by Li ([7]).

• Journal of the Korean Operations Research and Management Science Society
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• v.35 no.3
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• pp.25-43
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• 2010

We consider a multi-objective balancing and sequencing problem in mixed model assembly lines, which is important for an efficient use of the assembly lines. In this paper, we present a neighborhood symbiotic evolutionary algorithm to simultaneously solve the two problems of balancing and model sequencing under multiple objectives. We aim to find a set of well-distributed solutions close to the true Pareto optimal solutions for decision makers. The proposed algorithm has a two-leveled structure. At Level 1, two populations are operated : One consists of individuals each of which represents a partial solution to the balancing problem and the other consists of individuals for the sequencing problem. Level 2, which is an upper level, works one population whose individuals represent the combined entire solutions to the two problems. The process of Level 1 imitates a neighborhood symbiotic evolution and that of Level 2 simulates an endosymbiotic evolution together with an elitist strategy to promote the capability of solution search. The performance of the proposed algorithm is compared with those of the existing algorithms in convergence, diversity and computation time of nondominated solutions. The experimental results show that the proposed algorithm is superior to the compared algorithms in all the three performance measures.