On boundary value problems in differential-algebraic equations

1989 ◽  
Author(s):  
R. Marz
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Differential Algebraic Equations ◽  
Algebraic Equations

scholarly journals Seminonlinear boundary value problems for nondegenerate differential-algebraic system

2019 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Differential Algebraic Equations ◽  
Problem Solution ◽  
Algebraic Equations ◽  
Arbitrary Continuous Function ◽  
Differential Algebraic System

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

Singularities in Differential-Algebraic Boundary-Value Problems Governing the Excitation Response of Beam Structures

2014 ◽  
Vol 10 (1) ◽  
Author(s):  
Mehdi Saghafi ◽  
Harry Dankowicz
Keyword(s):  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Convergence Rates ◽  
Boundary Value ◽  
Differential Algebraic Equations ◽  
Group Symmetry ◽  
Algebraic Equations ◽  
Beam Structures ◽  
Group Symmetries

The objective of this paper2 is to identify and, where possible, resolve singularities that may arise in the discretization of spatiotemporal boundary-value problems governing the steady-state behavior of nonlinear beam structures. Of particular interest is the formulation of nondegenerate continuation problems of a geometrically-nonlinear model of a slender beam, subject to a uniform harmonic excitation, which may be analyzed numerically in order to explore the parameter-dependence of the excitation response. In the instances of degeneracy investigated here, the source is either found (i) directly in a differential-algebraic system of equations obtained from a finite-element-based spatial discretization of the governing partial differential boundary-value problem(s) together with constraints on the trial functions or (ii) in the further collocation-based discretization of the time-periodic boundary-value problem. It is shown that several candidate spatial finite-element discretizations of a mixed weak formulation of the governing boundary-value problem either result in (i) spatial group symmetries corresponding to equivariant vector fields and one-parameter families of periodic orbits along the group symmetry orbit or (ii) temporal group symmetries corresponding to ghost solutions and indeterminacy in a subset of the field variables. The paper demonstrates several methods for breaking the spatial equivariance, including projection onto a symmetry-reduced state space or the introduction of an artificial continuation parameter. Similarly, the temporal indeterminacy is resolved by an asymmetric discretization of the governing differential-algebraic equations. Finally, in the absence of theoretical bounds, computation is used to estimate convergence rates of the different discretization schemes, in the case of numerical calibration experiments performed on equilibrium and periodic responses for a linear beam, as well as for the full nonlinear models.

Sign in / Sign up

Export Citation Format

Share Document