• Journal of applied mathematics & informatics
• /
• v.27 no.1_2
• /
• pp.441-452
• /
• 2009

In this paper, a fifth order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been using for delay. Similar boundary value problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, first we use Taylor approximation to tackle terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system and is solved using the boundary conditions. Several numerical examples are solved and compared with exact solution. It is observed that present method approximates the exact solution very well.

• Journal of applied mathematics & informatics
• /
• v.6 no.1
• /
• pp.143-162
• /
• 1999

In this paper we consider the second order generalized difference scheme for the two-point boundary value problem and ob-tain optimal order error estimates in $L^\infty$ and $W^{1,\infty}$ The results in this paper perfect the theory of the second order generalized difference method.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.2
• /
• pp.411-422
• /
• 2010

By using the fixed point index theory, we investigate the existence of at least two positive solutions for p-Laplace equation with sign-changing nonlinear terms $(\varphi_p(u'))'+a(t)f(t,u(t),u'(t))=0$, subject to some boundary conditions. As an application, we also give an example to illustrate our results.

• Bulletin of the Korean Mathematical Society
• /
• v.41 no.3
• /
• pp.483-492
• /
• 2004

Let : [0, 1] $\times$ [0, $\infty$) $\longrightarrow$ [0, $\infty$) be continuous and a ${\in}$ C([0, 1], [0, $\infty$)),and let ${\xi}_{i}$ $\in$ (0, 1) with 0 < {\xi}$_1$ < ${\xi}_2$ < … < ${\xi}_{m-2}$ < 1, $a_{i}$, $b_{i}$ ${\in}$ [0, $\infty$) with 0 < $\Sigma_{i=1}$ /$^{m-2}$ $a_{i}$ < 1 and $\Sigma_{i=1}$$^{m-2}$ < l. This paper is concerned with the following m-point boundary value problem: $\chi$″(t)＋a(t) (t.$\chi$(t))＝0,t ${\in}$(0,1), $\chi$'(0)＝$\Sigma_{i=1}$ $^{m-2}$ /$b_{i}$$\chi$'(${\xi}_{i}$),$\chi$(1)＝$\Sigma_{i=1}$$^{m-2}$$a_{i}$$\chi$(${\xi}_{i}$). The existence, multiplicity and uniqueness of positive solutions of this problem are discussed with the help of two fixed point theorems in cones, respectively.

• Solar Energy
• /
• v.8 no.1
• /
• pp.82-94
• /
• 1988

The hydrodynamic stability equations for natural convection flows adjacent to a vertical isothermal surface in cold or warm water (Boussinesq or non-Boussinesq situation for density relation), constitute a two-point-boundary-value (eigenvalue) problem, which was solved numerically using the simple shooting and the orthogonal collocation method. This is the first instance in which these stability equations have been solved using a computer code COLSYS, that is based on the orthogonal collocation method, designed to solve accurately two-point-boundary-value problem. Use of the orthogonal collocation method significantly reduces the error propagation which occurs in solving the initial value problem and avoids the inaccuracy of superposition of asymptotic solutions using the conventional technique of simple shooting.

• Journal of Ocean Engineering and Technology
• /
• v.37 no.3
• /
• pp.122-128
• /
• 2023

Autonomous berthing is a crucial technology for autonomous ships, requiring optimal trajectory planning to prevent collisions and minimize time and control efforts. This paper presents a two-phase, two-point boundary value problem (TPBVP) strategy for creating an optimal berthing trajectory for a twin-propeller, twin-rudder ship with autonomous berthing capabilities. The process is divided into two phases: the approach and the terminal. Tunnel thruster use is limited during the approach but fully employed during the terminal phase. This strategy permits concurrent optimization of the total trajectory duration, individual phase trajectories, and phase transition time. The efficacy of the proposed method is validated through two simulations. The first explores a scenario with phase transition, and the second generates a trajectory relying solely on the approach phase. The results affirm our algorithm's effectiveness in deciding transition necessity, identifying optimal transition timing, and optimizing the trajectory accordingly. The proposed two-phase TPBVP approach holds significant implications for advancements in autonomous ship navigation, enhancing safety and efficiency in berthing operations.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.1
• /
• pp.71-111
• /
• 2021

We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading. We construct a one-to-one correspondence �� from the set of equivalent well-posed two-point boundary conditions to gl(4, ℂ). Using ��, we derive eigenconditions for the integral operator ��M for each well-posed two-point boundary condition represented by M ∈ gl(4, 8, ℂ). Special features of our eigenconditions include; (1) they isolate the effect of the boundary condition M on Spec ��M, (2) they connect Spec ��M to Spec ����,α,k whose structure has been well understood. Using our eigenconditions, we show that, for each nonzero real λ ∉ Spec ����,α,k, there exists a real well-posed boundary condition M such that λ ∈ Spec ��M. This in particular shows that the integral operators ��M, arising from well-posed boundary conditions, may not be positive nor contractive in general, as opposed to ����,α,k.

• Korean Journal of Mathematics
• /
• v.23 no.4
• /
• pp.665-732
• /
• 2015

Existence results for multiple positive solutions of two classes of boundary value problems for bilateral difference systems are established by using a fixed point theorem under convenient assumptions. It is the purpose of this paper to show that the approach to get positive solutions of boundary value problems of finite difference equations by using multi-fixed-point theorems can be extended to treat the bilateral difference systems with one-dimensional Laplacians. As an application, the sufficient conditions are established for finding multiple positive homoclinic solutions of a bilateral difference system. The methods used in this paper may be useful for numerical simulation. An example is presented to illustrate the main theorems. Further studies are proposed at the end of the paper.

• Journal of applied mathematics & informatics
• /
• v.30 no.3_4
• /
• pp.517-529
• /
• 2012

In this paper, the existence theorem of two positive solutions is established for nonlinear m-point boundary value problem by using an inequality for the following third-order differential equations $$({\phi}(u^{\prime\prime}))^{\prime}+a(t)f(t,u(t))=0,\;t{\in}(0,1)$$, $${\phi}(u^{\prime\prime}(0))=\sum^{m-2}_{i=1}a_i{\phi}(u^{\prime\prime}({\xi}_i)),\;u^{\prime}(1)=0,\;u(0)=\sum^{m-2}_{i=1}b_iu({\xi}_i)$$, where ${\phi}:R{\rightarrow}R$ is an increasing homeomorphism and homomorphism and $\phi(0)=0$. The nonlinear term f may change sign, as an application, an example to demonstrate our results is given.

• 제어로봇시스템학회:학술대회논문집
• /
• 2005.06a
• /
• pp.1228-1233
• /
• 2005

In this paper, we present a new guidance law for a reusable launch vehicle (RLV) that lands vertically after reentry. In our past studies, a guidance law was developed for a vertical/soft landing to a target point. The guidance law, which is analytically obtained, can regenerate a trajectory against disturbances because it is expressed in the form of state feedback. However, the guidance law does not necessarily guarantee a vertical/soft landing when a dynamical system such as an RLV includes a nonlinear phenomenon owing to the atmosphere of the earth. In this study, we introduce a design of the guidance law for a nonlinear system to achieve a vertical/soft landing on the ground using the exact linearization method and solving the two-point boundary-value problem for the derived linear system. Numerical simulation confirmed the validity of the proposed guidance law for an RLV in an atmospheric environment.