Class Of Two-point Boundary Value Problems Research Articles

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

The exact multiplicity of positive solutions for a class of two-point boundary-value problems with the one-dimensional p-Laplacian

Employing the Kolodner–Coffman method, we show the exact multiplicity of positive solutions for the one-dimensional p-Laplacian that is subject to a Dirichlet boundary condition with a positive convex nonlinearity and an indefinite weight function.

Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach

Using an elementary approach, we prove the existence of three positive and concave solutions of the second-order two-point boundary-value problem $$ \begin{array}{lll}&& x^{\prime\prime}(t)=\alpha(t)f(t,x(t),x^{\prime}(t)),\qquad 0 < t < 1,\\&& \qquad \qquad \qquad x(0)=x(1)=0.\end{array}$$ We rely on the analysis of the corresponding vector field on the phase space, Kneser-type properties of the solution funnel, and the Schauder fixed-point theorem. The obtained results demonstrate the simplicity and efficiency (one could study a problem with more general boundary conditions) of our new approach as compared with the commonly used ones, such as the Leggett–Williams fixed-point theorem and its generalizations.

Solution of Two-Point Boundary-Value Problems Using Lagrange Implicit Function Theorem

TWO-POINT boundary-value problems (TPBVPs) form an important ingredient in the solution of several multiphysics modeling and control analyses, including but not limited to guidance, navigation, and control problems of aerospace engineering. Several techniques to solve TPBVPs have been developed so far and the workhorse at the core of each has been Newton’s method (cf. Bryson and Ho [1]). In this Note, we present an implicit derivative Newton’s shooting method to solve a class of two-point boundary-value problems, most often encountered in the solution of optimal control problems. To motivate the developments of the paper, consider the optimal control problem

Comment on the paper “A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications”

Comment on the paper “A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications” in Applied Mathematics and Computation 174 (2006) 1169–1180. The paper considered a class of two-point boundary-value problems of the form (1) d 4 y d x 4 + f ( x ) y = g ( x ) , a ⩽ x ⩽ b , (2) y ( a ) = A 1 , y ( b ) = A 2 , y ′ ( a ) = B 1 , y ′ ( b ) = B 2 , where f( x) and g( x) are continuous on [ a, b], and A i and B i ( i = 1, 2) are finite real constants. Here we correct some mistake in derivation of non-polynomial spline, boundary formulas, truncation errors, convergence analysis and computational experiments.

One-dimensional models for topographic Rossby waves in elongated basins on thef-plane

Topographic Rossby waves in elongated basins on thef-plane are studied by transforming the linear boundary-value problem for the mass transport stream function into a class of two-point boundary-value problems of which the independent spatial variable is the (curved) basin axis. The procedure for deriving the substitute problems is theMethod of Weighted Residuals. What emerges is a vector differential equation and associated boundary conditions, its dimension indicating the order of the approximate model. It is shown that each substitute problem in the class entails the qualitative features typical of topographic waves, and increasing the order of the model corresponds to higher-order approximations. Equations are explicitly presented for cross-sectional distributions of the lake topography which has a power-law representation and permits the analysis of weak and strong topographies.Straight channels in which the depth profile does not change with position along the axis are studied in detail. The dispersion relation is discussed and dispersion curves are shown for the three lowest-order models. Convergence properties are thereby uncovered and phase speed and group velocity properties are found as they depend on wavenumber and topography. Further, for the lowest two modes, cross-channel stream-function distributions are presented. Apart from further convergence properties these distributions show that forU-shaped channels wave activity is nearer to the shore than forV-shaped channels, important information in the design of mooring systems.An analysis of topographic Rossby wave reflection follows, which emphasizes the importance of the depth profile in the reflecting zone. Based on these results some lake solutions are presented.

A shooting method for solving two-point boundary-value problems arising from non-singular bang-bang optimal control problems

A shooting method for a class of two-point boundary-value problems arising from non-singular bang-bang optimal control problems is developed. The zeros of a switching function are calculated internally, and therefore the introduction of extra unknowns is not required. It is also shown that accurate determination of switching times is essential in order to solve this class of boundary-value problems. The computational algorithm is tested on three example problems.

Obtaining starting values for the shooting method solution of a class of two-point boundary-value problems

A method based on matching a zero of the right-hand side of the differential equations, in a two-point boundary-value problem, to the boundary conditions is suggested. Effectiveness of the procedure is tested on three nonlinear, two-point boundary-value problems.

A numerical method for solving optimal control problems with unspecified terminal time†

This paper presents a numerical technique which automatically computes the unspecified terminal-time of an optimal control problem as a systematic part of the whole procedure for finding the optimal control inputs. In the proposed technique, a given unspecified terminal-time problem is transformed into a fixed terminal-time problem by a time-sealing method. Then, by considering the time-scaling parameter as either an extra control input or as an additional variable, and by applying the necessary conditions of optimality, a fixed terminal-time two-point boundary-value problem is obtained. Three algorithms are proposed to solve such a class of two-point boundary-value problems : two of them are based on the gradient methods, and one on the Newton-Raphson method in function space. Several numerical examples are included. A comparison of the rate of convergence of different algorithms used in the solution of each example problem is given.

Method of solution of a special class of two-point boundary-value problems

A numerical method is presented for the solution of a special class of two-point boundary value problems characterised by state equations of the form x = f(U, t), a nonlinear coupling function and fixed end points.

Monte Carlo path-integral calculations for two-point boundary-value problems

A computational technique based on the method of path integral is studied with a view to finding approximate solutions of a class of two-point boundary-value problems. These solutions are "rough" solutions by Monte Carlo sampling. From the computational point of view, however, once these rough solutions are obtained for any nonlinear cases, they serve as good starting approximations for improving the solutions to higher accuracy. Numerical results of a few examples are also shown.

An application of an analog computer to solve the two-point boundary-value problem for a fourth-order optimal control problem

The application of Pontryagin's maximum principle to trajectory optimization problems results in a two-point boundary-value problem. Computationally, this problem is solved by various digital techniques which are sometimes inconvenient and costly. An analog computer can be used to solve a large class of two-point boundary-value problems if the accuracy is acceptable. In this paper, an analog computer is used in conjunction with a human operator who has a display of the phase planes of the admissible trajectories. The human operator, having a general knowledge of the behavior of the system, adjusts control law parameters until the boundary conditions of the system are satisfied. Apparently this technique has been avoided previously due to the impression that unacceptable errors would be introduced in solving the problem. This technique was applied to a time-optimal rendezvous with bounds on rocket thrust and fuel available and demonstrated that accurate analog computer solutions are possible. Solutions of the rendezvous problem were compared with an exact solution using MIMIC.