Volume 10, no. 2Pages 63 - 73
Solution of Irregular Systems of Partial Differential Equations Using Skeleton Decomposition of Linear OperatorsD.N. Sidorov, N.A. Sidorov
The linear system of partial differential equations is considered. It is assumed that there is an irreversible linear operator in the main part of the system. The operator is assumed to enjoy the skeletal decomposition. The differential operators of such system are assumed to have sufficiently smooth coefficients. In the concrete situations the domains of such differential operators are linear manifolds of smooth enough functions with values in Banach space. Such functions are assumed to satisfy additional boundary conditions. The concept of a skeleton chain of linear operator is introduced. It is assumed that the operator generates a skeleton chain of the finite length. In this case, the problem of solution of a given system is reduced to a regular split system of equations. The system is resolved with respect to the highest differential expressions taking into account certain initial and boundary conditions. The proposed approach can be generalized and applied to the boundary value problems in the nonlinear case. Presented results develop the theory of degenerate differential equations summarized in the monographs MR 87a:58036, Zbl 1027.47001. Full text
- ill-posed problems; Cauchy problems; irreversible operator; skeleton decomposition; skeleton chain; boundary value problems.
- 1. Petrowsky I.G., Oleinik O.A. Selected Works. Part I: Systems of Partial Differential Equations and Algebraic Geometry. Amsterdam, Gordon and Breach Publishers, 1996.
2. Voropai N.I., Kurbatsky V.G., Tomin N.V., Panasetsky D.A., Sidorov D.N. Complex intellektualnih sredstv dlya predotvrashenia krupnih avarii v elektroenergenicheskih sistemah [Intellectual Algorithms for Major Accidents Prevention in Electric Power Systems]. Novosibirsk, Nauka, 2016.
3. Sidorov N.A. Obshchie voprosy regulyarizatsii v zadachakh teorii vetvleniya [General Regularization Questions in Problems of Bifurcation Theory]. Irkutsk, Izdatel'stvo Irkutskogo Universiteta, 1982.
4. Sidorov N., Sidorov D., Li Y. Skeleton Decomposition of Linear Operators in the Theory of Degenerate Differential Equations. 2015. arXiv:1511.08976. 4 p.
5. Sidorov N., Loginov B., Sinitsyn A., Falaleev M. Lyapunov - Schmidt Methods in Nonlinear Analysis and Applications. Springer Netherlands, 2013. DOI 10.1007/978-94-017-2122-6
6. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, VSP, 2003. DOI: 10.1515/9783110915501
7. Zamyshlyaeva A.A., Tsyplenkova O.N. Optimal Control of Solutions of the Showalter - Sidorov - Dirichlet Problem for the Boussinesq - L'ove Equation. Differential Equations, 2013, vol. 49, issue 11, pp. 1356-1365. DOI: 10.1134/S0012266113110049
8. Keller A.V., Shestakov A.L., Sviridyuk G.A., Khudyakov Yu.V. The Numerical Algorithms for the Measurement of the Deterministic and Stochastic Signals. Semigroups of Operators - Theory and Applications, Springer International Publishing, 2015, vol. 113, pp. 183-195.
9. Sidorov D. Integral Dynamical Models: Singularities, Signals and Control. Singapore, London, World Scientific, 2015.
10. Li Y., Han J., Cao Y., Li Yu., Xiong J., Sidorov D., Panasetsky D. A Modular Multilevel Converter Type Solid State Transformer with Internal Model Control Method. International Journal of Electrical Power & Energy Systems, 2017, vol. 85, pp. 153-163. DOI: 10.1016/j.ijepes.2016.09.001
11. Gantmacher F.R. The Theory of Matrices. N.-Y., Chelsea, 1959.
12. Tikhonov A.N., Samarskii A.A. Equations of Mathematical Physics. Courier Corporation, 2013.
13. Sobolev S.L. [The Cauchy Problem for a Special Case of System That Are not of Kovalevskaya Type]. Doklady akademii nauk SSSR, 1952, vol. 82, no. 2, pp. 1007-1009. (in Russian)
14. Loginov B.V., Rousak Yu.B., Kim-Tyan L.R. Differential Equations with Degenerate, Depending on the Unknown Function Operator at the Derivative. Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations, 2016, pp. 119-147.