Numerical solution of multipoint problems in statics of shells

We purpose an approach to solving multipoint boundary-value problems for a system of ordinary differential equations in the theory of shells. The technique is based on reduction of the original problem to several two-point boundary-value problems, which are solved by a stable numerical method. Examples of calculation of variable-thickness cylindrical shells are given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, 58–65, 1988.     

Multipoint Boundary-Value Solution of Two-Point Boundary-Value Problems

An iterative scheme, in which two-point boundary-value problems (TPBVP) are solved as multipoint boundary-value problems (MPBVP), which are independent TPBVPs in each iteration and on each subdomain, is derived for second-order ordinary differential equations. Several equations are solved for illustration. In particular, the algorithm is described in detail for the first boundary-value problem (FBVP) and second boundary-value problem (SBVP). A possible extension to higher-order BVPs is discussed briefly. The procedure may be used when the original TPBVP cannot be solved (does not converge) in a single long domain. It is suitable for implementation on computers with parallel processing. However, that issue is beyond the scope of this paper. The long domain is cut into a large number of subdomains and, based on assumed boundary conditions at the interface points, the resulting local BVPs are solved by any convenient conventional method. The local solutions are then patched by using simple matching formulas, which are derived below, rather than solving large systems of algebraic equations, as it is done in similar existing methods. Assuming that the local solutions are obtained by the most efficient methods, the overall convergence speed depends on the speed of matching. The proposed matching algorithm is based on a fixed-point iteration and has only a linear convergence rate. The rate can be made quadratic by applying standard accelerating schemes, which is beyond the scope of this article.     

Numerical solution of singular perturbation problems by a terminal boundary-value technique

In this paper, we present a new approach for numerically solving linear singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in implicit form is introduced. Then, the outer region problem is solved as a two-point boundary-value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, the inner region problem is modified and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Three numerical examples have been solved to demonstrate the applicability of the method.     

An approach to numerical solution of multipoint boundary-value problems

An approach is proposed to solving multipoint boundary-value problems for linear differential equation of w-th order, based on reduction to two-point boundary-value problems. The two-point problems are solved by the stable discrete orthogonalization method. Some numerical examples are considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 36–45, 1986.     

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.     

Boundary-value problems for systems of ordinary differential equations

This article contains an exposition of fundamental results of the theory of boundary-value problems for systems of linear and nonlinear ordinary differential equations. In particular, criteria are given for problems with functional, many-point, and two-point boundary conditions to be solvable and well-posed, as well as methods of finding approximate solutions. We also examine questions of existence, uniqueness, and stability of periodic and bounded solutions of nonautonomous differential systems.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 3–103, 1987.     

Boundary and initial-value methods for solving Fredholm equations with semidegenerate kernels

We develop a new approach to the theory and numerical solution of a class of linear and nonlinear Fredholm equations. These equations, which have semidegenerate kernels, are shown to be equivalent to two-point boundary-value problems for a system of ordinary differential equations. Applications of numerical methods for this class of problems allows us to develop a new class of numerical algorithms for the original integral equation. The scope of the paper is primarily theoretical; developing the necessary Fredholm theory and giving comparisons with related methods. For convolution equations, the theory is related to that of boundary-value problems in an appropriate Hilbert space. We believe that the results here have independent interest. In the last section, our methods are extended to certain classes of integrodifferential equations.     

Helly’s Principle and Its Application to an Infinite-Horizon Optimal Control Problem

In this paper, a numerical method is presented to solve singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a discontinuous source term. First, an asymptotic expansion approximation of the solution of the boundary-value problem is constructed using the basic ideas of the well-known WKB perturbation method. Then, some initial-value problems and terminal-value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial-value problems and terminal-value problems are singularly-perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples are provided to illustrate the method.     

Multipoint Boundary Conditions for Differential Operators

We establish differential properties of generalized solutions of multipoint boundary-value problems for ordinary differential equations.     

Optimal finite-dimensional solution for a class of nonlinear observation problems

The filtering problem in a differential system with linear dynamics and observations described by an implicit equation linear in the state is solved in finite-dimensional recursive form. The original problem is posed as a deterministic fixed-interval optimization problem (FIOP) on a finite time interval. No stochastic concepts are used. Via Pontryagin's principle, the FIOP is converted into a linear, two-point boundary-value problem. The boundary-value problem is separated by using a linear Riccati transformation into two initial-value problems which give the equations for the optimal filter and filter gain. The optimal filter is linear in the state, but nonlinear with respect to the observation. Stability of the filter is considered on the basis of a related properly linear system. Three filtering examples are given.     

On the unique solvability of a family of two-point boundary-value problems for systems of ordinary differential equations

We consider a family of two-point boundary-value problems for systems of ordinary differential equations with functional parameters. This family is the result of the reduction of a boundary-value problem with nonlocal condition for a system of second-order, quasilinear hyperbolic equations by the introduction of additional functions. Using the parametrization method, we establish necessary and sufficient conditions of the unique solvability of the family of two-point boundary-value problems for a linear system in terms of the initial data. We also prove sufficient conditions of the unique solvability of the problem considered and propose an algorithm for its solution. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 21–39, 2006.     

Initial-Value Technique for Singularly Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations Arising in Chemical Reactor Theory

An initial-value technique is presented for solving singularly perturbed two-point boundary-value problems for linear and semilinear second-order ordinary differential equations arising in chemical reactor theory. In this technique, the required approximate solution is obtained by combining solutions of two terminal-value problems and one initial-value problem which are obtained from the original boundary-value problem through asymptotic expansion procedures. Error estimates for approximate solutions are obtained. Numerical examples are presented to illustrate the present technique.     

Fundamental matrix and two-point boundary-value problems

Theoretically, the solution of all linear ordinary differential equation problems, whether initial-value or two-point boundary-value problems, can be expressed in terms of the fundamental matrix. The examination of well-known two-point boundary-value methods discloses, however, the absence of the fundamental matrix in the development of the techniques and in their applications. This paper reveals that the fundamental matrix is indeed present in these techniques, although its presence is latent and appears in various guises.     

Resolvent kernels of Green's function kernels and other finite-rank modifications of Fredholm and Volterra kernels

Many important Fredholm integral equations have separable kernels which are finite-rank modifications of Volterra kernels. This class includes Green's functions for Sturm-Liouville and other two-point boundary-value problems for linear ordinary differential operators. It is shown how to construct the Fredholm determinant, resolvent kernel, and eigenfunctions of kernels of this class by solving related Volterra integral equations and finite, linear algebraic systems. Applications to boundary-value problems are discussed, and explicit formulas are given for a simple example. Analytic and numerical approximation procedures for more general problems are indicated.This research was sponsored by the United States Army under Contract No. DAA29-75-C-0024.     

Numerical integration of a class of singular perturbation problems

In this paper, we discuss an approximate method for the numerical integration of a class of linear, singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This method requires a minimum of problem preparation and can be implemented easily on a computer. We replace the original singular perturbation problem by an approximate first-order differential equation with a small deviating argument. Then, we use the trapezoidal formula to obtain the three-term recurrence relationship. Discrete invariant imbedding algorithm is used to solve a tridiagonal algebraic system. The stability of this algorithm is investigated. The proposed method is iterative on the deviating argument. Several numerical experiments have been included to demonstrate the efficiency of the method.The authors wish to express their sincere thanks to Dr. S. M. Roberts for his comments and valuable suggestions.     

Some experiments with interval methods for two-point boundary-value problems in ordinary differential equations

Methods of interval mathematics are used to find upper and lower bounds for the solution of two-point boundary-value problems at discrete mesh points. They include interval versions of shooting and of finite-difference techniques for linear and non-linear differential equations of second order, and of finite-difference methods for Sturm-Liouville eigenvalue problems.Good results are obtained whenever the difficulties of dependency-width can be avoided, and particularly for the finite-difference method when the associated matrix is anM matrix.     

Fredholm One-Dimensional Boundary-Value Problems in Sobolev Spaces

Ukrainian Mathematical Journal - For systems of ordinary differential equations on a compact interval, we study the character of solvability of the most general linear boundary-value problems in...     

二阶线性常微分方程的两点边值问题的泰勒展开式解法

本文用泰勒展开公式求解二阶线性常微分方程的两点边值问题.首先将两点边值问题化为一个F redho lm积分方程,进一步通过泰勒展开公式化F redho lm积分方程为线性方程组,利用G ramm er法则可求得问题的近似解.     

A noniterative algebraic solution for Riccati equations satisfying two-point boundary-value problems

A noniterative algebraic method is presented for solving differential Riccati equations which satisfy two-point boundary-value problems. This class of numerical problems arises in quadratic optimization problems where the cost functionals are composed of both continuous and discrete state penalties, leading to piecewise periodic feedback gains. The necessary condition defining the solution for the two-point boundary value problem is cast in the form of a discrete-time algebraic Riccati equation, by using a formal representation for the solution of the differential Riccati equation. A numerical example is presented which demonstrates the validity of the approach.The authors would like to thank Dr. Fernando Incertis, IBM Madrid Scientific Center, who reviewed this paper and pointed out that the two-point boundary-value necessary condition could be manipulated into the form of a discrete-time Riccati equation. His novel approach proved to be superior to the authors' previously proposed iterative continuation method.     

Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations

A numerical method is suggested for solving systems of nonautonomous loaded linear ordinary differential equations with nonseparated multipoint and integral conditions. The method is based on the convolution of integral conditions into local ones. As a result, the original problem is reduced to an initial value (Cauchy) problem for systems of ordinary differential equations and linear algebraic equations. The approach proposed is used in combination with the linearization method to solve systems of loaded nonlinear ordinary differential equations with nonlocal conditions. An example of a loaded parabolic equation with nonlocal initial and boundary conditions is used to show that the approach can be applied to partial differential equations. Numerous numerical experiments on test problems were performed with the use of the numerical formulas and schemes proposed.