# On Boundary Value Problems for Singular Systems of Linear Integro-Differential Equations Method of Least Squares

B.D. Nguyen, V.F. ChistyakovAt present, in the analysis of complex electrical and electronic circuits, the system often includes interconnected differential, integral and algebraic equations. Algebraic equations are responsible for the difference of balance relations in the models, in particular, the conservation laws or equations of state, the system of differential equations describing the dynamics of the process. If the process has afteraction, the mathematical model can include integral equation (IE). The systems of interconnected differential, algebraic and integral equations can be written in the form of vector integro-differential equations with a matrix at the highest derivative of a searched vector-function of not full rank in the domain. Numerical solution of boundary and initial problems for such systems conjugates with great difficulty. In this paper we discuss the least squares method and the results of numerical calculations.Full text

- Keywords
- degenerate system; general solution; integro-differential equations; boundary value problem; least squares method.
- References
- 1. Sidorov N.A. Study of Continuous Solutions of the Cauchy Problem in a Neighborhood of the Branch. Russian Mathematics (Izvestiya VUZ. Matematika), 1976, vol. 20, issue 9, pp. 77-87.

2. Boyarintsev Yu.E. Reguljarnye i singuljarnye sistemy linejnyh obyknovennyh differencial'nyh uravneniy [Regular and Singular Systems of Linear Ordinary Differential Equations]. Novosibirsk, Nauka, 1980. (in Russian)

3. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, Tokyo, VSP, 2003. DOI: 10.1515/9783110915501

4. Sviridyuk G.A., Keller A.V. Invariant Spaces and Dichotomies of Solutions of a Class of Linear Equations of the Sobolev Type. Russian Mathematics (Izvestiya VUZ. Matematika), 1997, vol. 41, issue 5, pp. 57-65.

5. Keller A.V. Numerical Solution of Optimal Control Problem Degenerate Linear System of Ordinary Differential Equations with Initial Showalter-Sidorov Conditions. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2008, no. 27 (127), issue 2, pp. 50-56. (in Russian)

6. Chistyakov V. F. Algebro-differentsial'nye operatory s konechnomernym yadrom [Algebraic-Differential Operators with Finite-Dimensional Kernel]. Novosibirsk, Nauka, Siberian Publishing Company of the RAS, 1996.(in Russian)

7. Chistyakova E.V. [Research Methods and Solving Degenerate Integro-Differential Equations and Their Applications. Candidate's Dissertation in Mathematics and Physics]. Irkutsk, 2006.(in Russian)

8. Chistyakov V.F. On the Solvability and Numerical Methods for Solution of Linear Integro-Algebraic Equations. Siberian Mathematical Journal, 2013, vol. 54, issue4, pp. 746-758. DOI: 10.1134/S0965542511090065

9. Bulatov M.V., Chistyakova E.V. On a Family of Singular Integro-Differential Equations. Computational Mathematics and Mathematical Physics, 2011, vol. 51, issue 9, pp. 1558-1566. DOI: 10.1134/S0965542511090065

10. Falaleev M.V., Orlov S.S. Degenerate Integro-Differential Operators in Banach Spaces and Their Applications. Russian Mathematics, 2011, vol. 55, issue 10, pp. 59-69.

11. Clark K.D., Petzold L.R. Numerical Solution of Boundary Value Problems in Differential-Algebraic Systems. SIAM Journal on Scientific and Statistical Computing archive, 1989, vol. 10, issue 5, pp. 915-930.

12. Marz R. On Difference and Shooting Methods for Boundary Value Problems in Differential-Algebraic Equations. ZAMM Journal of applied mathematics and mechanics: Zeitschrift fur angewandte Mathematik und Mechanik, 2006, vol. 64, issue 11, pp. 463-473.

13. Chistyakov V.F., Chistyakova E.V. Application of the Least Squares Method for Solving Linear Differential-Algebraic Equations. Numerical Analysis and Applications, 2013, vol. 6, issue 1, pp. 77-90. DOI: 10.1134/S1995423913010102

14. Ushakov E.I. Staticheskaya ustoychivost’ elektricheskikh sistem [Static Stability of Electrical Systems]. Novosibirsk, Nauka, 1988.

15. Maslov V.P. Operatornye metodi [Operator Methods]. Moscow, Nauka, 1973.

16. Bertsekas D.P. Conditional Optimization and Lagrange Multiplier Methods. N.Y., Academic Press Inc. 1987.(in Russian)

17. Bahvalov N.S., Zhidkov N.P., Kobelkov G.M. Chislennyye metody [Numeriсal Methods]. Moscow, Nauka, 1987. (in Russian)