Modified Numerov method for solving system of second-order boundary-value problems

We introduce and discuss a new numerical method for solving system of second order boundary value problems, where the solution is required to satisfy some extra continuity conditions on the subintervals in addition to the usual boundary conditions. We show that the present method gives approximations which are better than that produced by other collocation, finite difference and spline methods. Numerical example is presented to illustrate the applicability of the new method.     

A finite element method for solving singular boundary-value problems

It is proved that under certain assumptions on the functions q(t) and f(t), there is one and only one function u₀(t) ∈ at which the functional

A method for solving two-point boundary-value problems of difference equations

The paper proposes an iterative solution method for discrete-time, nonlinear, two-point boundary-value problems (TPBVP) of the form: $$\begin{gathered} x(k) - x(k - 1) = f(k, x(k - 1), p(k)), \hfill \\ p(k) - p(k - 1) = g(k, x(k - 1), p(k)), \hfill \\ \end{gathered} $$ subject to $$h(x(0), p(0)) = 0,e(x(N), p(N)) = 0.$$ It is a counterpart of a method recently proposed by the authors for similar continuous-time TPBVPs with ordinary differential equations. The method, based on invariant imbedding and a generalized Riccati transformation, reduces the TPBVP to a pair of approximate initial-value problems with ordinary difference equations. Numerical tests are run on two examples originating in optimal control problems.     

An efficient method for solving non-linear singularly perturbed two points boundary-value problems of fractional order

In this paper, we discuss a numerical solution of a class of non-linear fractional singularly perturbed two points boundary-value problem. The method of solution consists of solving reduced problem and boundary layer correction problem. A series method is used to solve the boundary layer correction problem, and then the series solutions is approximated by the Pade’ approximant of order [m, m]. Some theoretical results are established and proved. Two numerical examples are discussed to illustrate the efficiency of the present scheme.     

Averaging of boundary-value problems of control over a standard system with delay

For the terminal problem of optimal control over systems of standard form with constant delay, according to the Pontryagin maximum principle, we study a boundary-value problem with deviating arguments with delay and anticipation. We justify an averaging method for an asymptotic solution of the boundary-value problem obtained.     

Cubic spline method for solving two-point boundary-value problems

In this paper, we use uniform cubic spline polynomials to derive some new consistency relations. These relations are then used to develop a numerical method for computing smooth approximations to the solution and its first, second as well as third derivatives for a second order boundary value problem. The present method outperforms other collocations, finite-difference and splines methods of the same order. Numerical illustrations are provided to demonstrate the practical use of our method.     

Sinc-Galerkin method for solving linear sixth-order boundary-value problems

There are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

     

Back-and-forth shooting method for solving two-point boundary-value problems

The paper proposes a special iterative method for a nonlinear TPBVP of the form (t)=f(t, x(t),p(t)), (t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.     

Interval length continuation method for solving two-point boundary-value problems

We consider the solution of the following two-point boundary-value problem: $$\begin{gathered} \dot x(t) = f(t,x(t),p(t)), \dot p(t) = g(t,x(t),p(t)), t \in [0,T], \hfill \\ h(x(0),p(0)) = 0, p(T) = q. \hfill \\ \end{gathered} $$ We propose a combination technique consisting of the interval length continuation method and the back-and-forth shooting method. Certain alternative ways of employing continuation are discussed, and some of them are well suited for the problem under consideration. As a test for the method, a numerical example of a problem originating in optimal control is given.     

Convergence of the back-and-forth shooting method for solving two-point boundary-value problems

The back-and-forth shooting method of Orava and Lautala (Ref. 1) is considered. The method transforms a given boundary-value problem to a sequence of initial-value problems. The present paper studies the convergence properties of this sequence. A local convergence theorem is given, and the rate of convergence is found to be quadratic in sufficiently smooth cases. The necessary tools for this analysis concerning the Fréchet differentiability of certain mappings are given in the Appendix.     

A certain spline method for solving a boundary-value problem

Translated from Ukrainskii Matematichskii Zhurnal, Vol. 41, No. 5, pp. 703–707, May, 1989.     

A global convergent derivative-free method for solving a system of non-linear equations

Finding all zeros of a system of \(m \in \mathbb {N}\) real non-linear equations in \(n \in \mathbb {N}\) variables often arises in engineering problems. Using Newtons’ iterative method is one way to solve the problem; however, the convergence order is at most two, it depends on the starting point, there must be as many equations as variables and the function F, which defines the system of nonlinear equations F(x)=0 must be at least continuously differentiable. In other words, finding all zeros under weaker conditions is in general an impossible task. In this paper, we present a global convergent derivative-free method that is capable to calculate all zeros using an appropriate Schauder base. The component functions of F are only assumed to be Lipschitz-continuous. Therefore, our method outperforms the classical counterparts.     

High-order adaptive Gegenbauer integral spectral element method for solving non-linear optimal control problems

In this work, we propose an adaptive spectral element algorithm for solving non-linear optimal control problems. The method employs orthogonal collocation at the shifted Gegenbauer–Gauss points combined with very accurate and stable numerical quadratures to fully discretize the multiple-phase integral form of the optimal control problem. The proposed algorithm relies on exploiting the underlying smoothness properties of the solutions for computing approximate solutions efficiently. In particular, the method brackets discontinuities and ‘points of nonsmoothness’ through a novel local adaptive algorithm, which achieves a desired accuracy on the discrete dynamical system equations by adjusting both the mesh size and the degree of the approximating polynomials. A rigorous error analysis of the developed numerical quadratures is presented. Finally, the efficiency of the proposed method is demonstrated on three test examples from the open literature.     

Numerical method of solving boundary-value problems for integrodifferential equations with deviating argument

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 6, pp. 854–860, June, 1989.     

非光滑最优控制问题的一种数值解法

针对非光滑最优控制问题提出一种分段数值解法.首先对问题进行全局拟谱离散,然后选取分点,将时间区域进行剖分,在每段区域上对问题进行离散,离散过程采用Chebyshev-Legendre拟谱方法,可以有效借助快速Legendre变换提高算法的运算效率,比现有算法在很大程度上节省了计算时间.给出了相关的理论分析,数值结果表明方法的高精度和有效性.     

Parallel algorithms for solving nonlinear two-point boundary-value problems which arise in optimal control

This paper describes a collection of parallel optimal control algorithms which are suitable for implementation on an advanced computer with the facility for large-scale parallel processing. Specifically, a parallel nongradient algorithm and a parallel variablemetric algorithm are used to search for the initial costate vector that defines the solution to the optimal control problem. To avoid the computational problems sometimes associated with simultaneous forward integration of both the state and costate equations, a parallel shooting procedure based upon partitioning of the integration interval is considered. To further speed computations, parallel integration methods are proposed. Application of this all-parallel procedure to a forced Van der Pol system indicates that convergence time is significantly less than that required by highly efficient serial procedures.This research was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-77-3418.     

Numerical method for solving a boundary-value problem of thermoelasticity

We consider the problem of determining the stress-strain state of an elastoplastic layer under impulse heating. The theory of small elastoplastic strains with linear hardening is used. A boundary-value problem is obtained for the equations of thermoelasticity whose coefficients at any time are functionals of strain history. A method is developed for solving this problem, based on discretization by space and time variables and application of an appropriate difference scheme. This scheme constructs a recursive evolution process for the state column at the nodes of the space grid. Numerical implementation of the method has demonstrated its reliability and efficiency.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 66–71, 1986.