• Journal of the Korean Mathematical Society
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• v.45 no.3
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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.

• Structural Engineering and Mechanics
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• v.15 no.5
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• pp.535-550
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• 2003

In this paper, a boundary-type meshfree method, the boundary radial point interpolation method (BRPIM), is presented for solving boundary value problems of two-dimensional solid mechanics. In the BRPIM, the boundary of a problem domain is represented by a set of properly scattered nodes. A technique is proposed to construct shape functions using radial functions as basis functions. The shape functions so formulated are proven to possess both delta function property and partitions of unity property. Boundary conditions can be easily implemented as in the conventional Boundary Element Method (BEM). The Boundary Integral Equation (BIE) for 2-D elastostatics is discretized using the radial basis point interpolation. Some important parameters on the performance of the BRPIM are investigated thoroughly. Validity and efficiency of the present BRPIM are demonstrated through a number of numerical examples.

• 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 the Korean Mathematical Society
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• v.50 no.4
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• pp.715-725
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• 2013

In this paper, a finite difference scheme for a two-point boundary value problem of 2nth-order ordinary differential equations is presented. The convergence and uniqueness of the solution for the scheme are proved by means of theories on matrix eigenvalues and norm. Numerical examples show that our method is very simple and effective, and that this method can be used effectively for other types of boundary value problems.

• Nonlinear Functional Analysis and Applications
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• v.28 no.2
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• pp.537-555
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• 2023

In this paper, we consider two types of fractional boundary value problems, one of them is an implicit type and the other will be an integro-differential type with nonlocal integral multi-point boundary conditions in the frame of generalized Hilfer fractional derivatives. The existence and uniqueness results are acquired by applying Krasnoselskii's and Banach's fixed point theorems. Some various numerical examples are provided to illustrate and validate our results. Moreover, we get some results in the literature as a special case of our current results.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1585-1596
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• 2016

In this paper we investigate the existence of nontrivial solutions for the following fractional boundary value problem (FBVP) $$\{_tD_T^{\alpha}(_0D_t^{\alpha}u(t))={\nabla}W(t,u(t)),\;t{\in}[0,T],\\u(0)=u(T)=0,$$ where ${\alpha}{\in}(1/2,1)$, $u{\in}{\mathbb{R}}^n$, $W{\in}C^1([0,T]{\times}{\mathbb{R}}^n,{\mathbb{R}})$ and ${\nabla}W(t,u)$ is the gradient of W(t, u) at u. The novelty of this paper is that, when the nonlinearity W(t, u) involves a combination of superquadratic and subquadratic terms, under some suitable assumptions we show that (FBVP) possesses at least two nontrivial solutions. Recent results in the literature are generalized and significantly improved.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.335-350
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• 2016

Based on the notion of $q_k$-derivative introduced by the authors in [17], we prove in this paper existence and uniqueness results for nonlinear second-order impulsive $q_k$-difference equations with anti-periodic boundary conditions. Two results are obtained by applying Banach's contraction mapping principle and Krasnoselskii's fixed point theorem. Some examples are presented to illustrate the results.

• Nuclear Engineering and Technology
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• v.4 no.1
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• pp.11-22
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• 1972

Determination of a two-point boundary value problem is the key of finding the control function u(t) with the application of the fundamental idea of Minimum principle. The late development shows the discovery of the initial costate vector for the solution of a two-point value problem. As a new technique of determining the optimal control function, Newton's Sequential method is examined about a number of engineering problems and found available.

• Transactions of the Korean Society of Mechanical Engineers
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• v.15 no.2
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• pp.630-643
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• 1991

The hydrodynamic stability equations are formulated for buoyancy-induced flows adjacent to a vertical, planar, isothermal surface in cold pure water. The resulting stability equations, when reduced to ordinary differential equation by a similarity transformation, constitute a two-point boundary-value(eigenvalue) problem, which was numerically solved for various values of the density extremum parameter R=( $T_{m}$ - $T_.inf./) / ( $T_{o}$ - $T_.inf./). These stability equations have been solved using a computer code designed to accurately solve two-point boundary-value problems. The present numerical study includes neutral stability results for the region of the flows corresponding to 0.0.leq. R. leq.0.15, where the outside buoyancy force reversals arise. The results show that a small amount of outside buoyancy force reversal causes the critical Grashof number $G^*/ to increase significantly. A further increase of the outside buoyancy force reversal causes the critical Grashof number to decrease. But the dimensionless frequency parameter $B^*/ at $G^*/ is systematically decreased. When the stability results of the present work are compared to the experimental data, the numerical results agree in a qualitative way with the experimental data.erimental data.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.457-469
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• 2008

In this paper, we study the existence of positive solutions for the following nonlinear m-point boundary value problem with p-Laplacian: $\{{{{(\phi_p(u'))'\;+\;f(t,u(t))=0, \;0<t<1,} \atop u'(0)={\sum}{^{m-2}_{i=1}}\;a_iu'(\xi_i),} \atop u(1)={\sum}{^k_{i=1}}\;b_iu(\xi_i)\;-\;{\sum}{^s_{i=k+1}}\;b_iu(\xi_i)\;-\;{\sum}{^{m-2}_{i=s+1}}\;b_iu'(xi_i),}$ where ${\phi}_p(s)$ is p-Laplacian operator, i.e., ${\phi}_p(s)=\mid s\mid^{p-2}s$, p>1, ${\phi}_q\;=\;({\phi}_p)^{-1}$, $\frac{1}{p}+\frac{1}{q}=1$, $1\;{\leq}\;k\;{\leq}\;s\;{\leq}m\;-\;2$, $b_i\;{\in}\;(0,+{\infty})$ with $0\;<\;{\sum}{^k_{k=1}}\;b_i\;-\;{\sum}{^s_{i=k+1}}\;b_i\;<\;1$, $0\;<\;{\sum}{^{m-2}_{i=1}}\la_i\;<\;1$, $0\;<\;{\xi}_1\;<\;{\xi}_2\;<\;{\cdots}\;<\;{\xi}_{m-2}\;<\;1$, $f\;{\in}\;C([0,\;1]\;{\times}\;[0,\;+{\infty}),\;[0,\;+{\infty}))$. We show that there exists one or two positive solutions by using fixed-point theorem for operator on a cone. The conclusions in this paper essentially extend and improve the known results.