On the concept of index for partial differential algebraic equations arising in modelling of processes in power plants

V. F. Chistyakov ; E. V. Chistyakova

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 10 (2017) no. 2, pp. 5-23

Résumé

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.

Détail

Keywords: partial differential algebraic equation; partial derivatives; integral differential equations; solution space; index.

Bibliographie

[1] Sobolev S. L., “On a New Problem of Mathematical Physics”, Izvestiya AN SSSR. Seriya matematicheskaya, 18 (1954), 3–50 (in Russian) | MR | Zbl

[2] Sviridyuk G. A., “Quasi-Stationary Trajectories of the Sobolev Semi-Linear Dynamic Equations”, Russian Academy of Sciences. Izvestiya Mathematics, 57:3 (1993), 192–207 (in Russian) | MR | Zbl

[3] Sviridyuk G. A., “On the General Theory of Operator Semigroups”, Russian Mathematical Surveys, 49:4 (1994), 45–74 | DOI | MR | Zbl

[4] Sviridyuk G. A., “Sobolev-Type Linear Equations and Strongly Continuous Semigroups of Resolving Operators with Kernels”, Doklady Akademii Nauk, 337:5 (1994), 581–584 (in Russian) | MR | Zbl

[5] Sviridyuk G. A., Fedorov V. E., Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht–Boston–Köln, 2003 | DOI | MR | Zbl

[6] Sviridyuk G. A., Fedorov V. E., Linear Sobolev Type Equations, Chelyabinskiy gosudarstvennyy universitet, Chelyabinsk, 2003 | MR | Zbl

[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, 8:4 (2016), 5–16 | DOI

[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, no. 3, 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, 23:4 (1987), 726–728 (in Russian) | Zbl

[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, 30:4 (1994), 729–731 (in Russian) | Zbl

[11] Falaleev M. V., Orlov S. S., “Degenerate Integro-Differential Operators in Banach Spaces and Their Applications”, Russian Mathematics, 44:10 (2011), 59–69 ; М. В. Фалалеев, С. С. Орлов, “Обобщенные решения вырожденных интегро-дифференциальных уравнений в банаховых пространствах и их приложения”, Труды ИММ УрО РАН, 18, No 4, 2012, 286–297 | DOI | MR

[12] Krein S. G., Chernyshov K. I., Singularly Perturbed Differential Equations in Banach Space, Sobolev Institute of Mathematics, Novosibirsk, 1979

[13] Favini A., Yagi A., “Multivalued Linear Operators and Degenerate Evolutions”, Annali di Matematica Pura ed Applicata, 163:1 (1993), 353–384 | DOI | MR | Zbl

[14] Ivanov V. K., Melnikova I. V., Filinkov A. I., Differential Operator Equations and Ill-Posed Problems, Fizmatlit, M., 1995 | MR | Zbl

[15] Pyatkov S. G., Operator Theory. Nonclassical Problems, VSP, Utrecht, 2002 | DOI | MR | Zbl

[16] Kozhanov A. I., Composite Type Equations and Inverse Problems, VSP, Utrecht, 1999 | DOI | MR | Zbl

[17] Palamodov V. P., Linear Differential Equations with Constant Coefficients, Nauka, M., 1967 | MR

[18] Demidenko G. V., Uspensky S. V., Equations and Systems Unresolved with Respect to the Higher Derivative, Nauchnaya kniga, Novosibirsk, 1998 | MR

[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, 123–141 (in Russian)

[20] Martinson W. S., Barton P. I., “A Differentiation Index for Partial Differential-Algebraic Equations”, SIAM Journal on Scientific Computing, 21:6 (2000), 2295–2316 | DOI | MR

[21] Campbell S. L., Marzalek W., “The Index of an Infinite Dimensional Implicit System”, Mathematical and Computer Modelling of Systems, 5:1 (1999), 18–42 | DOI | MR | Zbl

[22] Bormotova O. V., Chistyakov V. F., “Numerical Solutions and Studies of the Non-Cauchy–Kovalevskaya Type Systems”, Computational Mathematics and Mathematical Physics, 44:8 (2004), 1306–1313 | MR | Zbl

[23] Gaidomak S. V., Chistyakov V. F., “On the Non-Cauchy–Kovalevskaya Type of Systems of Index $(1,k)$”, Computational Technologies, 10:2 (2005), 45–59 (in Russian) | Zbl

[24] Bormotova O. V., Gaidomak S. V., Chistyakov V. F., “On the Solvability of Degenerate Systems of Partial Differential Equations”, Russian Mathematics, 49:4 (2005), 15–26 | MR | Zbl

[25] Lamour R., Marz R., Tischendorf C., Differential-Algebraic Equations: A Projector Based Analysis, Springer, Berlin, 2013 | DOI | MR | Zbl

[26] Godunov S. K., Equations of Mathematical Physics, Nauka, M., 1971 | MR

[27] Moore E. H., “On the Reciprocal of the General Algebraic Matrix”, Bulletin of the American Mathematical Society, 26:9 (1920), 394–395 | DOI

[28] Chistyakov V. F., Algebraic Differential Operators with a Finite-Dimensional Kernel, Nauka, Novosibirsk, 1996 | MR

[29] Boyarintsev Yu. E., Chistyakov V. F., Algebraic Differential Systems. Methods for Investigation and Solution, Nauka, Novosibirsk, 1998 | MR | Zbl

[30] Bulatov M. V., Lee M.-G., “Application of Matrix Polynomials to the Analysis of Linear Differential-Algebraic Equations”, Differential Equations, 44:10 (2008), 1353–1360 | DOI | MR | Zbl

[31] Bulatov M. V., “On a Family of Matrix Triples”, Lyapunov Readings and Presentation of Information Technologies, Irkutsk, 2002, 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, 127–131 (in Russian) | MR | Zbl

[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, 9:1 (2016), 59–72 | DOI | Zbl

[34] Boyarintsev Yu. E., Regular and Singular Systems of Ordinary Differential Equations, Nauka, Novosibirsk, 1980 | MR | Zbl