Seminonlinear boundary value problems for nondegenerate differential-algebraic system
Abstract
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.
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References
Yu.E. Boyarintsev, V.F. Chistyakov. Algebro-differentsialnyye sistemy. Metody resheniya i issledovaniya. 1998. Nauka, Novosibirsk, 224 p.
A.M. Samoilenko, M.I. Shkil, V.P. Yakovets. Linijni systemy dyferentsialnykh rivnian z vyrodzhenniam. 2000. Vyshcha shkola, Kyiv, 296 p.
S.L. Campbell. Singular Systems of differential equations. 1980. Pitman Advanced Publishing Program, San Francisco - London - Melbourne, 178 p.
E. Khayrer, G. Vanner. Resheniye obyknovennykh differentsialnykh uravneniy. Zhestkiye i differentsialno-algebraicheskiye zadachi. 1999. Myr, M., 686 р.
S.M. Chuiko. Lineynyye neterovy krayevyye zadachi dlya differentsialno-algebraicheskikh sistem, Komp. issledov. i modelirovaniye.
- 2013. - 5. V.5. - P. 769-783.
S.M. Chuiko. A generalized matrix differential-algebraic equation, Journal of Mathematical Sciences (N.Y.). - 2015. - 210. V.1. - P. 9-21.
S.M. Chuiko. On a Reduction of the Order in a Differential-Algebraic System, Journal of Mathematical Sciences. - 2018. - 235, V.1. - P. 2-14.
A.A. Boichuk, A.A. Pokutnyi, V.F. Chistyakov. Application of perturbation theory to the solvability analysis of differential algebraic equations, Computational Mathematics and Mathematical Physics., - 2013. - 53, V.6. - P. 958-969.
A.M. Samoylenko, N.A. Perestyuk. Differentsialnyye uravneniya s impulsnym vozdeystviyem. 1987. Vishcha shkola, Kyiv, - 287p.
S.M. Chuiko. A Generalized Green operator for a boundary value problem with impulse action, Differential Equations., - 2001. - 37, V.8. - P. 1189-1193.
E.A. Grebenikov, Yu.A. Ryabov, Konstruktivnyye metody analiza nelineynykh sistem. 1979. Nauka, M., 432 p.
A.A. Boichuk, A.M. Samoilenko. Generalized inverse operators and Fredholm boundary-value problems; 2-th edition. 2016. Berlin; Boston, De Gruyter, 298 p.
A.S. Chuiko. Domain of Convergence of an Iteration Procedure for a Weakly Nonlinear Boundary-Value Problem, Nonlinear oscillation., - 2005. - 8, V.2. - P. 277-287.
S.M. Chuiko. Generalized Green Operator of Noetherian boundary-value problem for matrix differential equation, Russian Mathematics., - 2016. - 60, V.8. - P. 64-73.
S.М. Chuiko, О.V. Starkova. About an approximate solution of autonomous boundary-value problem with a least-squares methods, Nonlinear oscillation., - 2009. - 12, V.4. - P. 556-573.
Citations
On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case
(2019) V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics
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