Numerical solution of multipoint boundary-value problems in improved shell theory

For some classes of stratified shells subject to transverse shear (shells of revolution with parameters varying along the generatrix, noncircular cylindrical shells with characteristics varying along the directrix, rectangular shallow shells), we propose an approach that reduces the solution of multipoint boundary-value problems to a number of two-point problems. As an example, we consider the stressed state of an open noncircular cylindrical shell supported in some section along the generatrix.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 35–38, 1987.     

Limit theorems in the theory of multipoint boundary-value problems

We present a reduction of a countable system of differential equations with countably-point boundary conditions to the case of a finite-dimensional multipoint boundary-value problem. We separately consider the case of a linear system. Kamenets-Podol'sk Pedagogic University, Kamenets-Podol'sk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 519–531, April, 1999.     

Contrasting solutions of singularly perturbed multipoint boundary-value problems with singularities

Translated from Matematicheskie Zametki, Vol. 56, No. 4, pp. 95–101, October, 1994.     

Application of the collocation method to multipoint boundary-value problems with integral boundary conditions

An algebraic collocation method for approximating solutions of systems of nonlinear ordinary differential equations is shown to be applicable in the case of linear multipoint boundary conditions containing definite integrals.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1548–1555, November, 1992.     

An application of a multipoint differential-difference scheme to a boundary-value problem

For the boundary-value problem (2) we construct a scheme of the method of lines with a central-difference approximation of the derivative for any odd pattern. In particular cases we investigate the behavior at the net refinement of the direct solution of the boundary-value problem for the determination of the difference between the approximate solution obtained by the method of lines and the exact solution of the problem (1), (2). We also consider some modifications of the method of lines: the number of the lines of the net is taken to be equal to that of the pattern. We give an estimate for the norm of the difference between the approximate solution obtained by this method and the exact solution of the problem (1), (2).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 76–88, 1977.     

Discrete invariant imbedding for the numerical solution of boundary-value problems over infinite intervals

A numerical method to solve boundary-value problems posed on infinite intervals is given by reducing the infinite interval to a finite interval which is large, and impossing appropriate asymptotic boundary conditions at the far end. Then the two-point boundary-value problem is solved by using discrete invariant-imbedding method, which is also analyzed for its stability. The theory is illustrated by solving a test example.     

Multipoint solution of two-point boundary-value problems

The purpose of this paper is to report on the application of multipoint methods to the solution of two-point boundary-value problems with special reference to the continuation technique of Roberts and Shipman. The power of the multipoint approach to solve sensitive two-point boundary-value problems with linear and nonlinear ordinary differential equations is exhibited. Practical numerical experience with the method is given.Since employment of the multipoint method requires some judgment on the part of the user, several important questions are raised and resolved. These include the questions of how many multipoints to select, where to specify the multipoints in the interval, and how to assign initial values to the multipoints.Three sensitive numerical examples, which cannot be solved by conventional shooting methods, are solved by the multipoint method and continuation. The examples include (1) a system of two linear, ordinary differential equations with a boundary condition at infinity, (2) a system of five nonlinear ordinary differential equations, and (3) a system of four linear ordinary equations, which isstiff.The principal results are that multipoint methods applied to two-point boundary-value problems (a) permit continuation to be used over a larger interval than the two-point boundary-value technique, (b) permit continuation to be made with larger interval extensions, (c) converge in fewer iterations than the two-point boundary-value methods, and (d) solve problems that two-point boundary-value methods cannot solve.     

An approach for computing eigenvalues of discrete symplectic boundary-value problems

We propose a new method for calculating eigenvalues of discrete symplectic boundaryvalue problems. This method is based on the discrete oscillation theory and on a modification of the Abramov double sweep method for discrete self-adjoint boundary-value problems.     

An approximate solution for linear boundary-value problems with slowly varying coefficients

In this paper, an approximate closed-form solution for linear boundary-value problems with slowly varying coefficient matrices is obtained. The derivation of the approximate solution is based on the freezing technique, which is commonly used in analyzing the stability of slowly varying initial-value problems as well as solving them. The error between the approximate and the exact solutions is given, and an upper bound on the norm of the error is obtained. This upper bound is proportional to the rate of change of the coefficient matrix of the boundary-value problem. The proposed approximate solution is obtained for a two-point boundary-value problem and is compared to its solution obtained numerically. Good agreement is observed between the approximate and the numerical solutions, when the rate of change of the coefficient matrix is small.     

An effective approach for numerical solution of linear and nonlinear singular boundary value problems

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.     

An effective boundary-integral approach for the mixed boundary-value problems of linear elastostatics

A recent BIE formulation is applied to the mixed problem of two-dimensional elastostatics. The discretized form of the equations is presented in detail for easy numerical implementation. Some numerical examples are presented from which it is seen that this method is a significant improvement over a previous indirect approach.     

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.     

On a method for construction of successive approximations for investigation of multipoint boundary-value problems

We suggest a new scheme of successive approximations. This scheme allows one to study the problem of existence and approximate construction of solutions of nonlinear ordinary differential equations with multipoint linear boundary conditions. This method enables one to study problems both with singular and nonsingular matrices in boundary conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1243–1253, September, 1995.     

On the solution of boundary-value problems of Kolmogorov-Petrovskii-Piskunov type

We describe a number of properties of solutions of boundary-value problems for nonlinear ordinary differential equations of the same type as those studied by Kolmogorov, Petrovskii, and Piskunov in their well-known paper on waves described by the parabolic equation. We construct and justify the asymptotics of such solutions for large values of the modulus of the independent variable.     

An approach to the solution of recognition problems using neural networks

Conditions are determined under which, for pattern recognition problems with standard information (Ω-regular problems), a correct algorithm and a six-level spatial neural network reproducing the calculations performed by the correct algorithm can be constructed. The proposed approach to constructing the neural network is not related to the traditional approach based on minimizing a functional.     

On local uniqueness of the solution of boundary-value problems

In this paper we present conditions under which differentiability of the mappings F: ACⁿ(I) Lⁿ(I) and :ACⁿ(I) Rⁿ at x₀ ACⁿ(I) and the uniqueness of the solution of the boundaryvalue problem u = F(x₀)(u), (x₀)(u) = 0 imply local uniqueness of the solution x₀ of the boundary-value problem x = F(x), (x) = 0.Translated from Matematicheskie Zametki, Vol. 15, No. 6, pp. 891–895, June, 1974.The author thanks A. Ya. Lepin for the help he has given him in the preparation of this paper.