Volume 10, no. 2Pages 5 - 23

On the Concept of Index for Partial Differential Algebraic Equations Arising in Modelling of Processes in Power Plants

V.F. Chistyakov, E.V. Chistyakova
This paper addresses some classes of linear and quasi-linear partial differential algebraic equations (PDAEs), i.e. systems of partial differential equations with singular matrices multiplying the higher derivatives of the desired vector-function. Such systems do not belong to the class of the Cauchy - Kovalevskaya equations, and therefore do not not comply with known existence theorems. The current research focuses on the first order evolutionary systems with one variable and investigates PDAEs depending on the parameter. The concept of index for PDAEs is introduced and various statements of initial boundary problems are considered. The results obtained are used to simulate and analyze the heat and mass exchange processes in power plants.
Full text
partial differential algebraic equation; partial derivatives; integral differential equations; solution space; index.
1. Sobolev S.L. [On a New Problem of Mathematical Physics]. Izvestiya AN SSSR. Seriya matematicheskaya, 1954, vol. 18, pp. 3-50. (in Russian)
2. Sviridyuk G.A. [Quasi-Stationary Trajectories of the Sobolev Semi-Linear Dynamic Equations]. Russian Academy of Sciences. Izvestiya Mathematics, 1993, vol. 57, no. 3, pp. 192-207. (in Russian)
3. Sviridyuk G.A. On the General Theory of Operator Semigroups. Russian Mathematical Surveys, 1994, vol. 49, no. 4, pp. 45-74. DOI: /10.1070/RM1994v049n04ABEH002390
4. Sviridyuk G.A. [Sobolev-Type Linear Equations and Strongly Continuous Semigroups of Resolving Operators with Kernels]. Doklady Akademii Nauk, 1994, vol. 337, no. 5, pp. 581-584. (in Russian)
5. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, VSP, 2003. DOI: 10.1515/9783110915501
6. Sviridyuk G.A., Fedorov V.E. Lineynye uravneniya sobolevskogo tipa [Linear Sobolev Type Equations]. Chelyabinsk, Chelyabinskiy gosudarstvennyy universitet, 2003.
7. Zamyshlyaeva A.A., Sviridyuk G.A. Nonclassical Equations of Mathematical Physics. Linear Sobolev Type Equations of Higher Order. Bulletin of the South Ural State University. Series: Mathematics, Mechanics, Physics, 2016, vol. 8, no. 4, pp. 5-16.
8. Tairov E.A., Zapov V.V. [An Analytical Solutions Based Integral Model of the Steam Generator Channel]. Problems of Atomic Science and Engineering. Series: Physics of Nuclear Reactors, 1991, issue 3, pp. 14-20. (in Russian)
9. Sidorov N.A., Falaleyev M.V. [Generalized Solutions of Differential Equations with the Fredholm Operator at the Derivative]. Differential Equations, 1987, vol. 23, no. 4, pp. 726-728. (in Russian)
10. Sidorov N.A., Romanova O.A., Blagodatskaya Ye.B. [Partial Differential Equations with an Operator of Finite Index at the Principal Part]. Differential Equations, 1994, vol. 30, no. 4, pp. 729-731. (in Russian)
11. Falaleev M.V., Orlov S.S. Degenerate Integro-Differential Operators in Banach Spaces and Their Applications. Russian Mathematics, 2011, vol. 44, issue 10, pp. 59-69. DOI: 10.3103/S1066369X11100082
12. Krein S.G., Chernyshov K.I. Singulyarno vozmushchennye differentsial'nye uravneniya v banakhovom prostranstve [Singularly Perturbed Differential Equations in Banach Space]. Novosibirsk, Sobolev Institute of Mathematics, 1979.
13. Favini A., Yagi A. Multivalued Linear Operators and Degenerate Evolutions. Annali di Matematica Pura ed Applicata, 1993, vol. 163, issue 1, pp. 353-384. DOI: 10.1007/BF01759029
14. Ivanov V.K., Melnikova I.V., Filinkov A.I. Differencial'no-operatornye uravneniya i nekorrektnye zadachi [Differential Operator Equations and Ill-Posed Problems]. Moscow, Fizmatlit, 1995.
15. Pyatkov S.G. Operator Theory. Nonclassical Problems. Utrecht, VSP, 2002. DOI: 10.1515/9783110900163
16. Kozhanov A.I. Composite Type Equations and Inverse Problems. Utrecht, VSP, 1999. DOI: 10.1515/9783110943276
17. Palamodov V.P. Lineynye differentsial'nye operatory s postoyannymi koeffitsientami [Linear Differential Equations with Constant Coefficients]. Moscow, Nauka, 1967.
18. Demidenko G.V., Uspensky S.V. Uravneniya i sistemy ne razreshennye otnositel'no starshey proizvodnoy [Equations and Systems Unresolved with Respect to the Higher Derivative]. Novosibirsk, Nauchnaya kniga, 1998.
19. Boyarintsev Yu.E. [Application of Generalized Inverse Matrices to Solving and Investigating First Order Partial Differential Equations]. Metody optimizacii i issledovanie operatsiy, Irkutsk, 1984, pp. 123-141. (in Russian)
20. Martinson W.S., Barton P.I. A Differentiation Index for Partial Differential-Algebraic Equations. SIAM Journal on Scientific Computing, 2000, vol. 21, no. 6, pp. 2295-2316. DOI: 10.1137/S1064827598332229
21. Campbell S.L., Marzalek W. The Index of an Infinite Dimensional Implicit System. Mathematical and Computer Modelling of Systems, 1999, vol. 5, no. 1, pp. 18-42.
22. Bormotova O.V., Chistyakov V.F. Numerical Solutions and Studies of the Non-Cauchy-Kovalevskaya Type Systems. Computational Mathematics and Mathematical Physics, 2004, vol. 44, no. 8, pp. 1306-1313.
23. Gaidomak S.V., Chistyakov V.F. [On the Non-Cauchy-Kovalevskaya Type of Systems of Index $(1,k)$]. Computational Technologies, 2005, vol. 10, no. 2, pp. 45-59. (in Russian)
24. Bormotova O.V., Gaidomak S.V., Chistyakov V.F. [On the Solvability of Degenerate Systems of Partial Differential Equations]. Russian Mathematics, 2005, vol. 49, no. 4, pp. 15-26. (in Russian)
25. Lamour R., Marz R., Tischendorf C. Differential-Algebraic Equations: A Projector Based Analysis. Berlin, Springer, 2013. DOI: 10.1007/978-3-642-27555-5
26. Godunov S.K. Uravneniya matematicheskoy fiziki [Equations of Mathematical Physics]. Moscow, Nauka, 1971.
27. Moore E.H. On the Reciprocal of the General Algebraic Matrix. Bulletin of the American Mathematical Society, 1920, vol. 26, no. 9, pp. 394-395.
28. Chistyakov V.F. Algebro-differentsial'nye operatory s konechnomernym yadrom [Algebraic Differential Operators with a Finite-Dimensional Kernel]. Novosibirsk, Nauka, 1996.
29. Boyarintsev Yu.E., Chistyakov V.F. Algebro-differencial'nye sistemy. Metody resheniya i issledovaniya [Algebraic Differential Systems. Methods for Investigation and Solution]. Novosibirsk, Nauka, 1998.
30. Bulatov M.V., Lee M.-G. Application of Matrix Polynomials to the Analysis of Linear Differential-Algebraic Equations. Differential Equations, 2008, vol. 44, no. 10, pp. 1353-1360. DOI: 10.1134/S0012266108100017
31. Bulatov M.V. [On a Family of Matrix Triples]. Lyapunovskie chteniya i prezentatsiya informatsionnykh tekhnologiy [Lyapunov Readings and Presentation of Information Technologies], Irkutsk, 2002, p. 10. (in Russian)
32. Boyarintsev Yu.E., Boyarintseva T.P. [A Note on an Implicit Difference Scheme Approximating the Navier - Stokes System]. Chislennye metody analiza i ih prilozheniya, Irkutsk, 1983, pp. 127-131. (in Russian)
33. Chistyakova E.V., Nguyen Duc Bang. Investigation of the Unsteady-State Hydraulic Networks by Means of Singular Systems of Integral Differential Equations. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2016, vol. 9, no. 1, pp. 59-72. DOI: 10.14529/mmp160105
34. Boyarintsev Yu.E. Regulyarnye i singulyarnye sistemy lineynykh obyknovennykh differentsial'nykh uravneniy [Regular and Singular Systems of Ordinary Differential Equations]. Novosibirsk, Nauka, 1980.