• Journal of applied mathematics & informatics
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• v.36 no.3_4
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• pp.331-348
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• 2018

In this paper, an initial value method for solving a class of singularly perturbed delay differential equations with layer behavior is proposed. In this approach, first the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay using Taylor series expansion. Then from the modified problem, two explicit Initial Value Problems which are independent of the perturbation parameter, ${\varepsilon}$, are produced: the reduced problem and boundary layer correction problem. Finally, these problems are solved analytically and combined to give an approximate asymptotic solution to the original problem. To demonstrate the efficiency and applicability of the proposed method three linear and one nonlinear test problems are considered. The effect of the delay on the layer behavior of the solution is also examined. It is observed that for very small ${\varepsilon}$ the present method approximates the exact solution very well.

• Journal of the Korean Mathematical Society
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• v.54 no.2
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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.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1081-1096
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• 2011

In this paper, a steady axisymmetric MHD flow of two dimensional incompressible fluids is studied under the influence of a uniform transverse magnetic field. The governing equations are reduced to nonlinear boundary value problem by applying the integribility conditions. Optimal Homotopy Asymptotic Method (OHAM) is applied to obtain solution of reduced fourth order nonlinear boundary value problem. For comparison, the same problem is also solved by Variational Iteration Method (VIM).

• Transactions of the Korean Society of Mechanical Engineers
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• v.7 no.1
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• pp.11-18
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• 1983

For the steady-state solutions of vibratory systems where the dynamics involves nonlinearity and discontinuity, a method of numerical simulation has been normally used. This paper presents a new approach which may overcome some difficulties in the simulation method. This approach is based on the fundamental assumption that the steady-state forced vibration is periodic, so that the problem is formulated as a two-point boundary-value problem and can be solved by Waner's algorithm. This method is demonstrated through the solutions of a linear system, a system with Coulomb friction and an impact pair. It is found that the method gives true solutions well both for linear and nonlinear systems, which convinces us of the usefulness of the method.

• Journal of the Korean Society for Precision Engineering
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• v.12 no.5
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• pp.57-68
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• 1995

A general procedure for the numerical solution of coupled, nonlinear, differential two-point boundary-value problems, solutions of which are crucial to the controller design, has been developed and demonstrated. A fixed-end-points, free-terminal-time, optimal-control problem, which is derived from Pontryagin's Maximum Principle, is solved by an extension of Davidenko's method, a differential form of Newton's method, for algebraic root finding. By a discretization process like finite differences, the differential equations are converted to a nonlinear algebraic system. Davidenko's method reconverts this into a pseudo-time-dependent set of implicitly coupled ODEs suitable for solution by modern, high-performance solvers. Another important advantage of Davidenko's method related to the time-optimal problem is that the terminal time can be computed by treating this unkown as an additional variable and sup- plying the Hamiltonian at the terminal time as an additional equation. Davidenko's method uas used to produce optimal trajectories of a single-degree-of-freedom problem. This numerical method provides switching times for open-loop control, minimized terminal time and optimal input torque sequences. This numerical technique could easily be adapted to the multi-point boundary-value problems.

• Journal of the Korean Mathematical Society
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• v.37 no.5
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• pp.665-685
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• 2000

The aim of this paper is to obtain nonlinear operators in suitable spaces whise fixed point coincide with the solutions of the nonlinear boundary value problems ($\Phi$($\upsilon$'))'=f(t, u, u'), l(u, u') = 0, where l(u, u')=0 denotes the Dirichlet, Neumann or periodic boundary conditions on [0, T], $\Phi$: N N is a suitable monotone monotone homemorphism and f:[0, T] N N is a Caratheodory function. The special case where $\Phi$(u) is the vector p-Laplacian $\mid$u$\mid$p-2u with p>1, is considered, and the applications deal with asymptotically positive homeogeneous nonlinearities and the Dirichlet problem for generalized Lienard systems.

• East Asian mathematical journal
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• v.22 no.2
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• pp.171-188
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• 2006

We consider the problem of an optimal control of the wave equation with a localized nonlinear dissipation. An optimal control is used to bring the state solutions close to a desired profile under a quadratic cost of control. We establish the existence of solutions of the underlying initial boundary value problem and of an optimal control that minimizes the cost functional. We derive an optimality system by formally differentiating the cost functional with respect to the control and evaluating the result at an optimal control.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1211-1220
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• 2009

In this paper, we consider the multipoint boundary value problem for the one-dimensional p-Laplacian $({\phi}_p(u'))'$(t)+q(t)f(t,u(t),u'(t))=0, t $\in$ (0, 1), subject to the boundary conditions: $u(0)=\sum\limits_{i=1}^{n-2}{\alpha}_iu({\xi}_i),\;u(1)=\sum\limits_{i=1}^{n-2}{\beta}_iu({\xi}_i)$ where $\phi_p$(s) = $|s|^{n-2}s$, p > 1, $\xi_i$ $\in$ (0, 1) with 0 < $\xi_1$ < $\xi_2$ < $\cdots$ < $\xi{n-2}$ < 1 and ${\alpha}_i,\beta_i{\in}[0,1)$, 0< $\sum{\array}{{n=2}\\{i=1}}{\alpha}_i,\sum{\array}{{n=2}\\{i=1}}{\beta}_i$<1. Using a fixed point theorem due to Bai and Ge, we study the existence of at least three positive solutions to the above boundary value problem. The important point is that the nonlinear term f explicitly involves a first-order derivative.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.65-71
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• 2019

This paper is concerned with the existence of positive solutions of the nonlinear Neumann boundary value problems $$\{u^{{\prime}{\prime}}+a(t)u={\lambda}b(t)f(u),\;t{\in}(0,1),\\u^{\prime}(0)=u^{\prime}(1)=0$$, where $a,b{\in}C[0,1]$ with $a(t)>0,\;b(t){\geq}0$ and the Green's function of the linear problem $$\{u^{{\prime}{\prime}}+a(t)u=0,\;t{\in}(0,1),\\u^{\prime}(0)=u^{\prime}(1)=0$$ may change its sign on $[0,1]{\times}[0,1]$. Our analysis relies on the Leray-Schauder fixed point theorem.

• East Asian mathematical journal
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• v.19 no.2
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• pp.189-205
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• 2003

In this paper we prove the existence of solution and uniform decay rates of the energy to viscoelastic problems with nonlinear boundary damping term. To obtain the existence of solutions, we use Faedo-Galerkin's approximation, and also to show the uniform stabilization we use the perturbed energy method.