Geodesic
Using Partial Differential Algebraic Equations in Modelling
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 1, pp. 98-111 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider evolutionary systems of partial differential equations depending on a single space variable. It is assumed that the matrices multiplying the derivatives of the desired vector-function are singular in the domain. Such systems are commonly called partial differential algebraic equations (PDAEs). Properties of PDEAs are essentially different to the properties of non-singular systems. In particular, it is impossible to define a type of a system judging by roots of characteristic polynomials. In this paper, we introduce a notion of splittable systems by which we mean systems allowing existence of non-singular transformations that lead to splitting of the original system to the subsystem with a unique solution and the non-singular subsystem of partial differential equations. Such an approach makes it possible to investigate the structure of general solutions to differential algebraic equations and, in some cases, to establish solvability of initial-boundary value problems.
Keywords: partial derivative, differential-algebraic equations, hyperbolic, singular systems, index, canonical form, modelling. 
@article{VYURU_2013_6_1_a8,
     author = {Nguyen Khac Diep and V. F. Chistyakov},
     title = {Using {Partial} {Differential} {Algebraic} {Equations} in {Modelling}},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {98--111},
     year = {2013},
     volume = {6},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2013_6_1_a8/}
}
TY  - JOUR
AU  - Nguyen Khac Diep
AU  - V. F. Chistyakov
TI  - Using Partial Differential Algebraic Equations in Modelling
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
PY  - 2013
SP  - 98
EP  - 111
VL  - 6
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VYURU_2013_6_1_a8/
LA  - ru
ID  - VYURU_2013_6_1_a8
ER  - 
%0 Journal Article
%A Nguyen Khac Diep
%A V. F. Chistyakov
%T Using Partial Differential Algebraic Equations in Modelling
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
%D 2013
%P 98-111
%V 6
%N 1
%U http://geodesic.mathdoc.fr/item/VYURU_2013_6_1_a8/
%G ru
%F VYURU_2013_6_1_a8
Nguyen Khac Diep; V. F. Chistyakov. Using Partial Differential Algebraic Equations in Modelling. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 1, pp. 98-111. http://geodesic.mathdoc.fr/item/VYURU_2013_6_1_a8/

[1] S. M. Wade, I. B. Paul, “A differentiation index for partial differential-algebraic equations”, SIAM J. Sci. Comp., 21:6 (2000), 2295–2316 | DOI | MR

[2] Sobolev S. L., “On a New Problem of Mathematical Physics”, Math of the USSR Izvestiya, 18 (1954), 3–50 | MR | Zbl

[3] Sviridyuk G. A., Fedorov V. E., Linear Sobolev type equations and degenerate semigroups of operators, VSP, Utrecht–Boston–Köln, 2003 | MR | Zbl

[4] Demidenko G. V., Uspenskiy S. V., Equations and Systems not Solved for the Highest Derivative, Nauchnaya kniga, Novosibirsk, 1998 | MR | Zbl

[5] Tairov E. A., Zapov V. V., “Integrated Model of the Nonlinear Dynamics of the Steam-Generating Channel Based on Analytical Solutions”, Of Nuclear Science and Technology Series \flqq Physics of Nuclear Reactors\frqq, 1991, no. 3, 14–20

[6] M. Gunther, P. Rentrop, PDAE-Netzwerkmodelle in der elektrischen schaltungssimulation, Preprint 99/3, IWRMMM, Karlsruhe, 1999

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

[8] Sidorov N. A., Falaleev M. V., “Generalized Solutions of Differential Equations with the Fredholm Operator in the Derivative”, Differential Equations, 23:4 (1987), 726–728 | MR | Zbl

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

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

[11] Boyarintsev Yu. E., “Application of Generalized Inverse Matrix Solutions and Research Systems of Partial Differential Equations of First Order”, Methods of Optimization and Operations Research, SEI SO AN USSR, Irkutsk, 1984, 123–141

[12] Bormotova O. V., Chistyakov V. F., “Methods of Numerical Solutions and Research Systems Are not Cauchy–Kovalevskaya”, Computational Mathematics and Mathematical Physics, 44:8 (2004), 1380–1387 | MR | Zbl

[13] Gaidomak S. V., Chistyakov V. F., “On Systems not of Cauchy–Kovalevskaya Index $(1, k)$”, Computer Applications, 10:2 (2005), 45–59 | Zbl

[14] Boyarintsev Y. E., Chistyakov V. F., Differential Algebraic Equations. Methods of Numerical Solutions and Research, Nauka, Novosibirsk, 1998 | MR | Zbl

[15] Petrovskiy I. G., Lectures on Partial Differential Equations, Leningrad, M., 1950 | MR

[16] Chistyakov V. F., Pjescic M. R., “On the Continuous Dependence of Solutions of Linear Systems of Differential-Algebraic Equations”, Differential Equations, 45:3 (2009), 374–384 | DOI | MR | Zbl

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

[18] Petrovskiy I. G., Lectures on the Theory of Ordinary Differential Equations, Nauka, M., 1964 | MR

[19] Loginov A. A., Tairov E. A., Chistyakov V. F., “Algebraic-Differential Mathematical Model of the System Power Units”, Optimization Methods and Their Applications, Proceedings of the XI Intern. Baikal. sch-seminar (Irkutsk, Baikal, 5–12 July 1998), v. 4, Numerical Analysis, Inverse and Ill-Posed Problems, Irkutsk, 1998, 119–122