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Transformation of the second order boundary value problem into integral form : different approaches and a numerical solution

Siedlecki J., Ciesielski M., Błaszczyk T.

Journal of Applied Mathematics and Computational Mechanics

2015

Vol. 14, nr 3

103--108

In this paper we present different approaches to the transformation of the second order ordinary differential equation, with respect to adequate boundary conditions, into integral equations. The obtained equations are Fredholm integral equations of the second kind. Next, a numerical method based on quadrature methods has been proposed to get an approximate solution of these equations.

A hybrid parallel approach to one-parameter nonlinear boundary value problems

Domokos G., Szeberényi I.

Computer Assisted Mechanics and Engineering Sciences

2004

Vol. 11, No. 1

15-34

This paper presents a global algorithm for parallel computers, suitable to solve nonlinear boundary value problems depending on one parameter. Our method offers a mixture of path continuation and scanning. The former is well-known, the latter is a novel approach introduced a few years ago, capable to find all equilibria in a given domain. The hybrid method combines the speed of path continuation with the robustness and generality of scanning, offering a transition between the two methods which depends on the choice of some characteristic control parameters. We introduce the algorithms on a small example and test it on large-scale problems.

Oblique wave scattering by undulations on the bed of an ice-covered ocean

Mandal B. N., Maiti P.

Archives of Mechanics

2004

Vol. 56, nr 6

485-493

The problem of oblique wave scattering by cylindrical undulations on the bed of an ice-covered ocean is investigated by using a simplified perturbation analysis. The first-order potential function satisfies a boundary value problem (BVP) which is solved by employing the Green integral theorem after constructing an appropriate Green function. Analytical expressions for the first-order reflection and transmission coefficients are then obtained from the solution of this BVP, in terms of the integrals involving the shape function describing undulations. Three particular forms of the shape function are considered for which the reflection and transmission coefficients up to the first-order are obtained exactly.

Partial material replacement without stress redistribution

Matuszak A.

Archives of Mechanics

2002

Vol. 54, nr 1

35-54

A partial material replacement causes stress redistribution in comparison to the original structure made of a homogeneous material. The article presents a possibility to design a geometry of the replacement which keeps the state of stress unchanged. It is shown that for a class of two-dimensional configurations, it is possible to find a solution of this problem. Conditions for the existence of an appropriate geometry are given. A method to obtain the shape of the replaced part is proposed.