Geodesic
The special difference schemes for numerical solution of ordinary differential equation with the alternating-sign coefficient
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2010), pp. 5-17 Cet article a éte moissonné depuis la source Math-Net.Ru

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The special difference first and second order schemes for the through numerical calculation of the Cauchy problem for an ordinary differential equation with the alternating-sign coefficient at the unknown function are proposed. The construction of schemes is based on the use of the exact integral solution of the equation and approximations of the grid functions invariant to the sign of the equation's coefficient. Properties, asymptotics, and the rational approximation of the schemes are considered. An opportunity of application of the schemes at rough steps of integration and convergence of numerical results to exact solutions is shown on test examples.
Keywords: ordinary differential equation, first order, alternating-sign coefficient, special difference schemes, through numerical calculation. 
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V. G. Zverev. The special difference schemes for numerical solution of ordinary differential equation with the alternating-sign coefficient. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2010), pp. 5-17. http://geodesic.mathdoc.fr/item/VTGU_2010_4_a0/

[1] Samarskii A. A., Vvedenie v chislennye metody, Nauka, M., 1982, 272 pp. | MR

[2] Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M., Chislennye metody, Nauka, M., 1987 | MR | Zbl

[3] Kalitkin N. N., Chislennye metody, Nauka, M., 1978 | MR

[4] Kakhaner D., Mouler K., Nesh S., Chislennye metody i matematicheskoe obespechenie, per. s angl., Mir, M., 1998

[5] Khairer E., Nërsett S., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Nezhestkie zadachi, Mir, M., 1990 | MR

[6] Khairer E., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Zhestkie i differentsialno-algebraicheskie zadachi, Mir, M., 1999

[7] Zverev V. G., “Raznostnye skhemy povyshennogo poryadka tochnosti dlya chislennogo resheniya zhestkogo obyknovennogo differentsialnogo uravneniya s lineinymi koeffitsientami”, Matematicheskoe modelirovanie, 19:9 (2007), 94–104 | MR | Zbl

[8] Zverev V. G., Goldin V. D., “Ob odnoi spetsialnoi raznostnoi skheme dlya resheniya zhestkogo obyknovennogo differentsialnogo uravneniya”, Vychislitelnye tekhnologii, 13:3 (2008), 54–64 | Zbl

[9] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1976, 576 pp. | MR

[10] Titov V. A., Shishkin G. I., “Chislennoe reshenie zadachi Koshi dlya obyknovennogo differentsialnogo uravneniya s malym parametrom pri proizvodnoi”, Chislennye metody mekhaniki sploshnoi sredy (Novosibirsk), 9:7 (1978), 112–121 | MR

[11] Boglaev I. P., “O chislennom integrirovanii singulyarno vozmuschennoi zadachi Koshi dlya obyknovennogo differentsialnogo uravneniya”, Zhurn. vychisl. matem. i matem. fiziki, 25:7 (1985), 1009–1022 | MR | Zbl

[12] Naife A., Vvedenie v metody vozmuschenii, Mir, M., 1984, 535 pp. | MR

[13] De Brein N. G., Asimptoticheskie metody v analize, IL, M., 1961, 247 pp.

[14] Fedoryuk M. V., Metod perevala, Nauka, M., 1977, 369 pp. | MR | Zbl

[15] Abramovits M., Stigan I., Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979, 832 pp. | MR

[16] Vasenin I. M., Arkhipov V. A., Butov V. G. i dr., Gazovaya dinamika dvukhfaznykh techenii v soplakh, Izd-vo Tom. un-ta, Tomsk, 1986

[17] Rychkov A. D., Matematicheskoe modelirovanie gazodinamicheskikh protsessov v kanalakh i soplakh, Nauka, Novosibirsk, 1988 | MR | Zbl