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We investigate a class of quasi-linear multi-point boundary value problem with one-dimensional p-Laplacian. By applying monotone iterative technique, some sufficient conditions for the existence of twin positive solutions are established, they are the maximal and minimal solutions of the above problem. At the same time, we also establish iterative schemes for approximating this twin solutions, which...
By using shooting method, large deflection bending of the Euler-Bernoulli beam with fixed-free ends subjected to a tip follower force is investigated. Based on geometrically non-linear theory for extensible elastic beams, the governing nonlinear governing equations of elastic beam subjected to a tip-concentrated follower force along the central axis are established. They consist of a boundary value...
Let T be a time scale. In this paper, we study the existence of positive solutions for the following nonlinear second-order m-point boundary value problem with sign changing coefficient on time scales {uΔ∇(t)+f(t,u(t))=0, 0<;t<;T, uΔ(0) = Σi=1m-2 aiuΔ(ξi), u(T)=Σi=1k biu(ξi)- Σi=k+1s biu(ξi) - Σi=s+1m-2 biuΔ(ξi), where 1 ≤ k ≤ s ≤ m - 2, ai, bi ∈ (0, +∞) with 0 <; Σi=1k bi- Σi=k+1s bi<;1,...
In this paper the equation with periodical condition and small parameter is considered. The equation will lose boundary condition and lead to large oscillation in boundary layer. The properties of the equation is studied firstly. The analytical solution is decomposed into the smooth component and the singular component. The asymptotic solution is proposed secondly. The asymptotic solution is used...
The aim of this paper is to show that a special kind of boundary value problem for second-order ordinary differential equations which reduces to the problem of solving a tridiagonal system of linear equations with almost Toeplitz structure can be efficiently solved on modern heterogeneous computer architectures based on CPU and GPU processors using an algorithm based on the divide and conquer method...
Successive approximate design of the optimal tracking controller for linear systems with time-delay is considered. A functional-based transformation is introduced, which transforms the system with control delay into a system without control delay formally. By applying the successive approximation approach of differential equations, the two-point boundary value (TPBV) problem with both time-delay and...
The initial-boundary value problem for a class of inhomogeneous BBM equations with nonlinear term |u|??-2u is studied. For some range of parameter ?? value, The decay estimates of the global solutions for this problem are proved by means of Sobolev inequality.
This paper considers the optimal tracking control (OTC) problem for bilinear systems affected by sinusoidal disturbances, and gives a successive approximation approach to solve analogous problems for general bilinear systems. For a given bilinear OTC problem, a sequence of linear problems is constructed, and their solutions are shown to converge to the desired solution. By iteratively solving the...
A two-sided inverse of the differential operator for the unbounded system of elliptic equations on Lipshitz domains was obtained. It was based on a special convolution of sequences. The Dirichlet and Neumann problems for the unbounded systems were reduced to the systems of Fredholm integral equations of either the first kind or the second kind. All equations in integral systems distinguish only by...
A multi region finite difference method is described and applied to the one dimension, semi linear, singularly perturbed boundary value problem (SPBVP). The process of developing high precision algorithms for this problem is described and it is shown that when the multi region method is combined with the use of these high order algorithms, numerical solutions can achieve accuracies in the range of...
Singularly perturbed ordinary differential equation with two small parameters is considered. Firstly, the solution is decomposed into the smooth component and the singular component. The upper bounds of the smooth component and the singular component are studied. Secondly, the traditional Shishkin's scheme is presented and it is proved to be uniformly convergent with respect to the small parameter...
In this paper, a new difference scheme by using cubic splines for solving a singularly-perturbed two-point boundary-value problem for second-order ordinary differential equations is derived. The proposed scheme is fourth order accurate, which is better than previous published results. Finally, two numerical examples are solved to illustrate the efficiency of our method.
The aim of this paper is to show that a kind of boundary value problem for second-order ordinary differential equations which reduces to the problem of solving tridiagonal systems of linear equations can be efficiently solved on modern multicore computer architectures. A new method for solving such tridiagonal systems of linear equations, based on recently developed algorithms for solving linear recurrence...
This note presents a feedforward and feedback optimal control (FFOC) law for a class of nonlinear time-delay systems affected by external disturbances. A sequence of inhomogeneous linear non-delay state and adjoint vector equations are constructed to approximate to the original two-point boundary value (TPBV) problems from the original optimal control problem. The convergence of the sequences is proved...
In this paper, we consider a kind of boundary value problem of a second order elliptic differential equation of variable coefficient. First, we give the variation inequality which is equal to this boundary value problem, and prove the existence and uniqueness of the solution of the variation inequality of this kind by using Green formula and variation method lemma. Then we can obtain the existence...
An approximate optimal tracking control (OTC) algorithm for sinusoidal disturbances rejections of nonlinear systems is developed in this paper. In the proposed control method, the nonlinear two-point boundary value (TPBV) problem, which is derived from the original OTC problem, is rewritten into a new form through a transformation and is transformed into a sequence of linear TPBV problems by using...
The optimal tracking control problem for nonlinear interconnected large-scale systems with time-delays is considered. By using a successive approximation approach (SAA), a high order, coupled, nonlinear two-point boundary value (TPBV) problem with both time-delay and time-advance terms, which is derived from the necessary condition of the original optimal tracking control problem, is transformed into...
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