Geodesic
Numerical method for a~system of linear equations of second order with a~small parameter on a~semi-infinite interval
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 2, pp. 103-114.

Voir la notice de l'article provenant de la source Math-Net.Ru

A boundary value problem for a linear system of ordinary second order differential equations with a small parameter at higher derivatives on a semi-infinite interval is considered. Systems of reaction-diffusion and convection-diffusion equations are considered. The method of reduction of a problem to a finite interval problem, based on the extraction of a set of solutions satisfying the limit conditions on infinity, is investigated. Auxiliary singular Cauchy problems for the differential matrix Riccati equations are solved with the use of a series in powers of a small parameter and an independent variable. Accuracy of the method proposed is estimated. The Shishkin mesh is proposed for solving a problem after its reduction to a finite interval. The results of numerical experiments are presented. 
@article{SJVM_2004_7_2_a1,
     author = {A. I. Zadorin and O. V. Kharina},
     title = {Numerical method for a~system of linear equations of second order with a~small parameter on a~semi-infinite interval},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {103--114},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2004_7_2_a1/}
}
TY  - JOUR
AU  - A. I. Zadorin
AU  - O. V. Kharina
TI  - Numerical method for a~system of linear equations of second order with a~small parameter on a~semi-infinite interval
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2004
SP  - 103
EP  - 114
VL  - 7
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2004_7_2_a1/
LA  - ru
ID  - SJVM_2004_7_2_a1
ER  - 
%0 Journal Article
%A A. I. Zadorin
%A O. V. Kharina
%T Numerical method for a~system of linear equations of second order with a~small parameter on a~semi-infinite interval
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2004
%P 103-114
%V 7
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2004_7_2_a1/
%G ru
%F SJVM_2004_7_2_a1
A. I. Zadorin; O. V. Kharina. Numerical method for a~system of linear equations of second order with a~small parameter on a~semi-infinite interval. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 2, pp. 103-114. http://geodesic.mathdoc.fr/item/SJVM_2004_7_2_a1/

[1] Abramov A. A., “O perenose usloviya ogranichennosti dlya nekotorykh sistem obyknovennykh lineinykh differentsialnykh uravnenii”, Zhurn. vychisl. matem. i mat. fiz., 1:4 (1961), 733–737 | MR | Zbl

[2] Abramov A. A., Balla K., Konyukhova N. B., “Perenos granichnykh uslovii iz osobykh tochek dlya sistem lineinykh obyknovennykh differentsialnykh uravnenii”, Soobsch. po vychisl. matem., VTs AN SSSR, M., 1981 | MR

[3] Miller J. J. H., O'Riordan E., Shishkin G. I., Fitted numerical methods for singular perturbation problems. Error estimates in the maximum norm for linear problems in one and two dimensions, World Scientific, Singapore, 1996 | MR

[4] Harina O. V., Zadorin A. I., “Numerical solution of a boundary value problem for a linear system of equations with a small parameter on a half-infinite interval”, Proceedings of the International Conference on Computational Mathematics, Part 2, Novosibirsk, 2002, 449–453 | MR

[5] Bellman R., “Some Inequalities for the Square Root of a Positive Definite Matrix”, Linear Algebra and its Applications, 1:3 (1968), 321–324 | DOI | MR

[6] Freiling G., Jank G., Abou-Kandil H., “Generalized Riccati Difference and Differential Equations”, Linear Algebra and Its Applications, 241–243 (1996), 291–303 | DOI | MR | Zbl

[7] Royden H. L., “Comparison theorems for the matrix Riccati equation”, Comm. Pure and Appl. Math., 41 (1988), 739–746 | DOI | MR | Zbl

[8] Pulay P., “An Iterative Method for the Determination of the Square Root of a Positive Definite Matrix”, Zamm., 46 (1966), 151–152 | DOI | MR

[9] Ikramov Kh. D., Chislennoe reshenie matrichnykh uravnenii, Nauka, M., 1984 | MR | Zbl

[10] Zadorin A. I., “Reduktsiya nelineinoi kraevoi zadachi dlya sistemy uravnenii vtorogo poryadka s malym parametrom s polubeskonechnogo intervala k konechnomu”, Sibirskii matem. zhurn., 42:5 (2001), 1057–1066 | MR | Zbl

[11] Abramov A. A., Balla K., “O priblizhennykh resheniyakh, osnovannykh na teoremakh sravneniya, skalyarnykh i matrichnykh uravnenii Rikkati na beskonechnom intervale”, Zhurn. vychisl. matem. i mat. fiz., 33:1 (1993), 35–51 | MR | Zbl