• 대한수학회보
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• 제35권3호
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• pp.473-484
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• 1998

We consider a finite difference approximation to a singular boundary value problem arising in the study of a nonlinear circular membrane under normal pressure. It is proved that the rate of convergence is $O(h^2)$. To obtain the solution of the finite difference equation, an iterative scheme converging monotonically to the solution of the finite difference equation is introduced. And the numerical experiment of this method is given.

• 대한수학회지
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• 제36권2호
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• pp.299-316
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• 1999

We consider two finite difference approxiamations to a singular boundary value problem arising in the study of a nonlinear circular membrane under normal pressure. It is shown that the rates of convergence are O(h) and O($h^2$), respectively. An iterative scheme is introduced which converges to the solution of the finite difference equations. Finally the numerical experiments are given

• 대한수학회지
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• 제32권4호
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• pp.761-773
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• 1995

The nonlinear boundary value problem $$ y" = f(x, y, y') = -\frac{x}{3}y' - \frac{y^2}{g(x)}, 0 < x < 1, $$ $$ (1.1) y'(0) = 0, and either (H) : y(1) = \lambda > 0 $$ $$ or (S) : y'(1) + (1 - \upsilon)y(1) = 0, 1 - \upsilon > 0, $$ $$g \in C[0, 1], k \leq g(x) \leq K on [0, 1] for some k, K > 0 $$ arises in the nonlinear circular membrane under normal pressure [2, 3]., 3].

• 충청수학회지
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• 제2권1호
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• pp.9-14
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• 1989

Certain type of singular two point boundary value problem is studied. This contains a wider class of differential equations than [5]. An example is provided for comparison with earlier results.

• Journal of applied mathematics & informatics
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• 제28권3_4호
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• pp.763-772
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• 2010

This paper deals with the existence of triple positive solutions for the nonlinear second-order three-point boundary value problem z"(t)+a(t)f(t, z(t), z'(t))=0, t $\in$ (0, 1), $z(0)={\nu}z(1)\;{\geq}\;0$, $z'(\eta)=0$, where 0 < $\nu$ < 1, 0 < $\eta$ < 1 are constants. f : [0, 1] $\times$ [0, $+{\infty}$) $\times$ R $\rightarrow$ [0, $+{\infty}$) and a : (0, 1) $\rightarrow$ [0, $+{\infty}$) are continuous. First, Green's function for the associated linear boundary value problem is constructed, and then, by means of a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of triple positive solutions to the boundary value problem. The interesting point is that the nonlinear term f is involved with the first-order derivative explicitly.

• Journal of applied mathematics & informatics
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• 제15권1_2호
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• pp.237-249
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• 2004

This paper deals with existence of positive solutions of nonlinear elliptic singular boundary value problems in a ball. It is shown that results of Grandall et al. [1] and [2] follow as special cases of our results proved in this article.

• 대한수학회지
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• 제54권2호
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• pp.359-380
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• 2017

This paper is concerned with a kind of boundary value problem for singular second order differential systems with Laplacian operators. Using a multiple fixed point theorem, sufficient conditions to guarantee the existence of at least three positive solutions of this kind of boundary value problem are established. An example is presented to illustrate the main results.

• 대한수학회지
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• 제45권3호
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• pp.631-644
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• 2008

In this paper, we present a new method for solving a nonlinear two-point boundary value problem with finitely many singularities. Its exact solution is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximation $u_n(x)$ to the exact solution u(x) is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.

• Kyungpook Mathematical Journal
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• 제53권2호
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• pp.257-271
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• 2013

In this article, we establish the existence of at least three unbounded positive solutions to a boundary-value problem of the nonlinear singular fractional differential equation. Our analysis relies on the well known fixed point theorems in the cones.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제16권4호
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• pp.327-344
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• 2009

In this paper, we deal with the following system of nonlinear singular boundary value problems(BVPs) on time scale $\mathbb{T}$ $$\{{{{{{x^{\bigtriangleup\bigtriangleup}(t)+f(t,\;y(t))=0,\;t{\in}(a,\;b)_{\mathbb{T}},}\atop{y^{\bigtriangleup\bigtriangleup}(t)+g(t,\;x(t))=0,\;t{\in}(a,\;b)_{\mathbb{T}},}}\atop{\alpha_1x(a)-\beta_1x^{\bigtriangleup}(a)=\gamma_1x(\sigma(b))+\delta_1x^{\bigtriangleup}(\sigma(b))=0,}}\atop{\alpha_2y(a)-\beta_2y^{\bigtriangleup}(a)=\gamma_2y(\sigma(b))+\delta_2y^{\bigtriangleup}(\sigma(b))=0,}}$$ where $\alpha_i$, $\beta_i$, $\gamma_i\;{\geq}\;0$ and $\rho_i=\alpha_i\gamma_i(\sigma(b)-a)+\alpha_i\delta_i+\gamma_i\beta_i$ > 0(i = 1, 2), f(t, y) may be singular at t = a, y = 0, and g(t, x) may be singular at t = a. The arguments are based upon a fixed-point theorem for mappings that are decreasing with respect to a cone. We also obtain the analogous existence results for the related nonlinear systems $x^{\bigtriangledown\bigtriangledown}(t)$ + f(t, y(t)) = 0, $y^{\bigtriangledown\bigtriangledown}(t)$ + g(t, x(t)) = 0, $x^{\bigtriangleup\bigtriangledown}(t)$ + f(t, y(t)) = 0, $y^{\bigtriangleup\bigtriangledown}(t)$ + g(t, x(t)) = 0, and $x^{\bigtriangledown\bigtriangleup}(t)$ + f(t, y(t)) = 0, $y^{\bigtriangledown\bigtriangleup}(t)$ + g(t, x(t)) = 0 satisfying similar boundary conditions.