Superconvergent Deferred Correction Methods For First Order Systems of Nonlinear Two-Point Boundary Value Problems

The extrapolation of Numerov's scheme for nonlinear two-point boundary value problems

Solution of nonlinear boundary value problems — I method of third order convergence for solution of nonlinear two-point boundary value problems single second order equation

A collocation method for solving a class of singular nonlinear two-point boundary value problems

A flexible rod is rotated from one end. The equilibrium equation is a fourth order nonlinear two‐point boundary value problem which depends on two parameters λ and α representing the importance of centrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, respectively. Previous studies on the existence and uniqueness of solution of the equilibrium equation assumed α = 0. Among the findings of these studies is the existence of a critical value λc beyond which the uniqueness of the trivial solution is lost. The computations of λc required the solution of a nonlinear bifurcation problem. On the other hand, this work is concerned with the existence and uniqueness of solution of the equilibrium equation when α ≠ 0 and in particular in the computations of a critical value λc such that the equilibrium equation has a unique solution for each α ≠ 0 provided λ < λc. For small α ≠ 0 this requires the solution of a nonlinear perturbed bifurcation problem.

An optimal trajectory design of a nonlinear satellite injection problem for transfer to a final target orbit by minimizing the time was investigated. Indeed, this design was an exact solution to the nonlinear two-point boundary value problem which determined optimal control history as well as optimal state trajectories in the open-loop form. Furthermore, the obtained optimal guidance strategy was exerted in the closed-loop form against the environment disturbances using neighboring optimal control method in the exact solution. Neighboring Optimal Control (NOC) law could produce time-variant feedback gains minimizing the performance measure to second order for perturbations from a nominal optimal path. Generally, this law was a function of perturbations appeared in the states and constraints and could be computed utilizing the backward sweep method. The simulation results indicated that the presented methodology was successfully utilized in the real world applications with good robustness to each noise or disturbance produced in each state variable.

Stability and Consistency Analysis of Difference Methods for Singular Perturbation Problems

Methods are developed to determine the refracted propagation paths of high-frequency RF signals in the ionosphere and to determine the sensitivities of these paths to changes of the input model parameters. These techniques are being developed to support the assimilation of data from mono-static and multi-static ionosondes with the goal of improving parameterized estimates of ionosphere electron density profiles. An additional application area is that of navigation using signals from a ground-based network of high-frequency beacons. A nonlinear two-point boundary value problem solver is developed using the shooting method with Newton updates. Robust convergence is achieved by seeding the algorithm with carefully designed first guesses of the wave vector’s initial aiming angles and terminal group delay. Partial derivative sensitivities of the ray-path solution are calculated using the adjoint of the two-point boundary value problem. This approach speeds the calculations when the partial derivatives of many ionosphere model parameters need to be computed. The new algorithm has been applied successfully to determine the paths of O-mode and X-mode radio waves between known transmitter and receiver locations and to spitze reflection points. Spitze singularities pose no difficulties for the new algorithm because it uses a Hamiltonian raypath formulation that remains non-singular at a spitze.

The optimal output tracking control problem for nonlinear large-scale dynamic systems is considered. A series-based approximate approach for designing the optimal controller is proposed with respect to quadratic performance indexes. By using the approach, the high order, coupling, nonlinear two-point boundary value problem is transformed into a sequence of linear decoupling two-point boundary value problems. By intercepting the frontal finite terms of the solution series, we obtained a suboptimal tracking control law. A simulation example illustrates the efficiency of the optimal output tracking control law designed.

We study a priori estimates and solvability of a nonlinear two-point boundary value problem for systems of second-order ordinary differential equations with leading positively homogeneous nonlinearity of order > 1 vanishing on a single surface. Assuming that an a priori estimate holds, we prove the invariance of the solvability of the problem under a continuous change of the leading nonlinear homogeneous terms and under arbitrary perturbations that do not affect the behavior of the leading nonlinear homogeneous terms at infinity.

For certain nonlinear two-point boundary value problems of the fourth order an estimation theory is developed which yields simultaneous estimates of the solution and its second derivative. Methods for computing numerical error bounds for approximate solutions are described and tested. The theory provides also uniqueness and existence statements. The results can be applied to many problems for which a corresponding theory on two-sided bounds is not suitable.

We continue to study the problem of inventory control, with simultaneous pricing optimization in continuous time. In our previous paper[8], we considered the case without set up cost, and established the optimality of the base stock-list price (BSLP) policy. In this paper we consider the situation of fixed price. We prove that the discrete time optimal strategy (see[11]), i.e., the (s,S,p) policy can be extended to the continuous time case using the framework of quasi-variational inequalities (QVIs) involving the value function. In the process we show that an associated second order, nonlinear two-point boundary value problem for the value function has a unique solution yielding the triplet (s,S,p). For application purposes the explicit knowledge of this solution is needed to specify the optimal inventory and pricing strategy. Selecting a particular demand function we are able to formulate and implement a numerical algorithm to obtain good approximations for the optimal strategy.

Using a new fixed point theorem of generalized concave operators, we present in this paper criteria which guarantee the existence and uniqueness of positive solutions to nonlinear two-point boundary value problems for second-order impulsive differential equations with concave or convex nonlinearities.

The boundary value problems of ordinary differential equations play an important role in many fields. Here, we implement the He-Laplace method for the solution of linear and nonlinear two-point boundary value problems. The aim of this paper is to compare the performance of the He-Laplace method with shooting method. As a result, for the same number of terms, our method provides relatively more accurate results with rapid convergence than other methods.

A two-point boundary value problem (TP-BVP) occurs during the process of solving a single differential equation or a set of differential equations whose solution has to satisfy both the given initial and final boundary conditions. The author shows that zeroing the discrepancy function is the crucial step in solving nonlinear TP-BVPs and uses M.P. Kennedy and L.O. Chua's (1988) neural network model to solve this problem. The advantages of this approach include its suitability for VLSI implementation. >

This paper presents the approximate solution of the non-linear two-point boundary value problem for maximum a posteriori estimation. Filtering, fixed point smoothing, fixed interval smoothing, and fixed lag smoothing algorithms are obtained by the discrete invariant embedding procedure. These are tabulated as well as the associated approximate error variances.