Acceleration of multiple solution of a boundary value problem involving a linear algebraic system

2016 ◽  
Author(s):  
Talgat R. Gazizov ◽  
Sergey P. Kuksenko ◽  
Roman S. Surovtsev
Keyword(s):  
Boundary Value Problem ◽  
Algebraic System ◽  
Boundary Value ◽  
Multiple Solution ◽  
Linear Algebraic System

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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.

Matrix boundary-value problems for differential equations with p-Laplacian

2020 ◽  
Vol 17 (1) ◽  
pp. 41-57
Author(s):  
Olga Nesmelova
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Arbitrary Function ◽  
Algebraic System ◽  
Boundary Value ◽  
System Of Differential Equations ◽  
Differential Algebraic System ◽  
Uniqueness Of The Solution

We consider the boundary-value problem for a linear system of differential equations with matrix p-Laplacian, which is reduced to the traditional differential-algebraic system with an unknown in the form of the vector function. A generalization of various boundary-value problems for differential equations with p-Laplacian, which preserves the features of the solution of such problems, namely, the lack of uniqueness of the solution and, in this case, the dependence of the desired solution on an arbitrary function, is given.

scholarly journals The Approximation Solution of a Nonlinear Parabolic Boundary Value Problem Via Galerkin Finite Elements Method with Crank-Nicolson

2018 ◽  
Vol 31 (3) ◽  
pp. 126
Author(s):  
Jamil Amir Al-Hawasy ◽  
Marwa Ahmed Jawad
Keyword(s):  
Boundary Value Problem ◽  
Algebraic System ◽  
Boundary Value ◽  
Approximation Solution ◽  
Parabolic Boundary ◽  
Nonlinear Parabolic ◽  
Nonlinear Algebraic System ◽  
Matlab Package ◽  
Cn Scheme ◽  
Crank Nicolson

    This paper deals with finding the approximation solution of a nonlinear parabolic boundary value problem (NLPBVP) by using the Galekin finite element method (GFEM) in space and Crank Nicolson (CN) scheme in time, the problem then reduce to solve a Galerkin nonlinear algebraic system(GNLAS). The predictor and the corrector technique (PCT) is applied here to solve the GNLAS, by transforms it to a Galerkin linear algebraic system (GLAS). This GLAS is solved once using the Cholesky method (CHM) as it appear in the matlab package and once again using the Cholesky reduction order technique (CHROT) which we employ it here to save a massive time. The results, for CHROT are given by tables and figures and show the efficiency of this method, from other sides we conclude that the both methods are given the same results, but the CHROT is very fast than the CHM.  

About the equilibrium positions of a matrix differential-algebraic boundary value problem

2019 ◽  
Vol 33 ◽  
pp. 218-231
Author(s):  
Sergei Chuiko ◽  
Olga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Green Operator ◽  
Solvability Conditions ◽  
Differential Algebraic System ◽  
The Matrix ◽  
Matrix Differential ◽  
Generalized Green Operator

In the article we found the solvability conditions and the construction of the generalized Green operator of the linear Noetherian matrix differential-algebraic boundary value problem. We obtained sufficient conditions of transformationsof the matrix differential-algebraic equation to a traditional differential-algebraic equation with an unknown in the form of a column vector. The problem that reviewed in the article continues the study of solvability conditions for the linear Noetherian boundary value problems given in the monographs of M.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko and A.A. Boichuk. We investigated the general case when the linear bounded operator corresponding to the homogeneous part of the linear Cauchy problem for the matrix differential-algebraic system does not have the reverse operator. We introduced the definition of the equilibrium positions of the matrix differential-algebraic system and the matrix differential-algebraic boundary-value problem to solve the matrix differential-algebraic boundary-value problem. We proposed sufficient conditions of existence and constructive schemes for finding the equilibrium positions of the matrix differential-algebraic system and the matrix differential-algebraic boundary value problem. The cases~of equilibrium positions of the matrix differential-algebraic system, which are constant matrices, and equilibrium positions depending on an independent variable are considered separately. To solve the matrix differential-algebraic boundary-value problem, we used the original solvability conditions and~the construction of the general solution of the Sylvester-type matrix equation, while the Moore-Penrose matrix pseudoinverse technique was essentially used. In the article we constructed the generalized Green operator of the linear Noetherian matrix differential-algebraic boundary value problem. The proposed solvability conditions and the construction of the generalized Green operator of the linear Noetherian matrix differential-algebraic boundary value problem, were illustrated in detail with examples.

On the solution of a linear Noetherian boundary-value problem for a differential-algebraic system with lumped delay by the method of least squares

2019 ◽  
Vol 16 (4) ◽  
pp. 503-513
Author(s):  
Sergei Chuiko
Keyword(s):  
Boundary Value Problem ◽  
Least Squares ◽  
Algebraic System ◽  
Boundary Value ◽  
Method Of Least Squares ◽  
Differential Algebraic System

The conditions of existence and the construction of pseudosolutions, being the best by the method of least squares, of a Noetherian differential-algebraic boundary-value problem with concentrated delay, respectively, are determined.

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