Geodesic
The factorization method for linear and quasilinear singularly perturbed boundary problems for ordinary differential equations
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 12 (2009) no. 1, pp. 1-15 
A. F. Voevodin. The factorization method for linear and quasilinear singularly perturbed boundary problems for ordinary differential equations. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 12 (2009) no. 1, pp. 1-15. http://geodesic.mathdoc.fr/item/SJVM_2009_12_1_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

For linear singularly perturbed boundary value problems we offer the method that reduces solving a differential problem to a discrete (difference) problem. The difference equations are constructed by the factorization method and are an exact analogy of differential equations. The coefficients of difference equations are calculated by solving the Cauchy problems for first order differential equations. In this case, the nonlinear Ricatti equations with a small parameter are solved by the asymptotic method, and linear equations are solved by the numerical methods. Solution to the quasilinear singularly perturbed equations is obtained by the implicit relaxation method. The solution to a linearized problem is calculated by analogy with a linear problem at each iterative steP. The method is tested with the known Lagestrome-Cole problem.

[1] Shishkin G.I., “Metod povyshennoi tochnosti dlya kvazilineinogo singulyarno vozmuschennogo ellipticheskogo uravneniya konvektsii-diffuzii”, Sib. zhurn. vychisl. matematiki / RAN. Sib. otd-nie, 9:1 (2006), 81–108 | Zbl

[2] Dulan E., Miller Dzh., Shilders U., Ravnomernye chislennye metody resheniya zadach s pogranichnym sloem, Mir, M., 1983

[3] Chang K., Khaues F., Nelineinye singulyarno vozmuschennye kraevye zadachi. Teoriya i prilozheniya, Mir, M., 1988

[4] Samarskii A.A., Vvedenie v teoriyu raznostnykh skhem, Nauka, M., 1971

[5] Gaevoi V.P., “Ob odnom metode postroeniya raznostnykh uravnenii dlya dvukhtochechnykh kraevykh zadach”, Metody splain-funktsii (vychislitelnye sistemy), 75, IM, Novosibirsk, 1978, 96–110

[6] Krasnoselskii M.A., Burd V.Sh., Kolesov Yu.S., Nelineinye pochti periodicheskie kolebaniya, Nauka, M., 1970

[7] Voevodin A.F., Shugrin S.M., Metody resheniya odnomernykh evolyutsionnykh sistem, Nauka, Novosibirsk, 1993

[8] Mamedov Ya.D., Amirov S., Atdaev S., Teoremy o neravenstvakh, Ylym, Ashkhabad, 1980

[9] Voevodin V.V., Kuznetsov Yu.A., Matritsy i vychisleniya, Nauka, M., 1984

[10] Samarskii A.A., Nikolaev E.S., Metody reshenii setochnykh uravnenii, Nauka, M., 1978