Geodesic
Properties of finite-difference schemes for singular integrodifferential equations of index 1
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 9, pp. 1579-1588 
E. V. Chistyakova. Properties of finite-difference schemes for singular integrodifferential equations of index 1. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 9, pp. 1579-1588. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_9_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Systems of integrodifferential equations with a singular matrix multiplying the highest derivative of the unknown vector function are considered. An existence theorem is formulated, and a numerical solution method is proposed. The solutions to singular systems of integrodifferential equations are unstable with respect to small perturbations in the initial data. The influence of initial perturbations on the behavior of numerical processes is analyzed. It is shown that the finite-difference schemes proposed for the systems under study are self-regularizing.

[1] Ushakov E. I., Staticheskaya ustoichivost elektricheskikh sistem, Nauka, Novosibirsk, 1988 | Zbl

[2] Boyaryntsev Yu. E., Regulyarnye i singulyarnye sistemy lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, Novosibirsk, 1980

[3] Brenan K. E., Campbell S. L., Petzold L. R., Numerical solution of initial-value problems in differential-algebraic equations, SIAM, Philadelphia, 1996 | MR | Zbl

[4] Abramov A. A., Ulyanova V. I., Yukhno A. F., “Metod resheniya nelineinoi spektralnoi zadachi differentsialno- algebraicheskikh sistem uravnenii”, Zh. vychisl. matem. i matem. fiz., 47:5 (2007), 829–834 | MR

[5] Gorbunov V. K., “Development of the normal spilne method for linear intego-differential equations”, Proc. ICCS-2003, Part II, Springer, Berlin, 2003 | MR

[6] Dmitriev C. C., Kuznetsov E. B., “Chislennoe reshenie sistem integrodifferentsialno-algebraicheskikh uravnenii s zapazdyvayuschim argumentom”, Zh. vychisl. matem. i matem. fiz., 48:3 (2008), 430–444 | MR | Zbl

[7] Bulatov M. V., “Ob integrodifferentsialnykh sistemakh s vyrozhdennoi matritsei pered proizvodnoi”, Differents. ur-niya, 38:5 (2002), 692–697 | MR | Zbl

[8] Chistyakov V. F., Algebro-differentsialnye operatory s konechnomernym yadrom, Nauka, SO RAN, Novosibirsk, 1996 | MR

[9] Bulatov M. V., Chistyakova E. V., “Chislennoe reshenie integrodifferentsialnykh sistem s vyrozhdennoi matritsei pered proizvodnoi mnogoshagovymi metodami”, Differents. ur-niya, 42:9 (2006), 1248–1255 | MR | Zbl

[10] Bulatov M. V., Chistyakova E. V., “Numerical solution of singular systems of integral differential equations”, Proc. Internat. Conf. Comput. Math., V. 2, Novosibirsk, 2004, 813–817

[11] Ten M. Ya., Priblizhennoe reshenie lineinykh integralnykh uravnenii Volterra I roda, Dis. $\dots$ kand. fiz.-matem. nauk, SO RAN SSSR, Irkutsk, 1985

[12] Krasnov M. L., Integralnye uravneniya, Nauka, M., 1975 | MR | Zbl

[13] Apartsin A. C., Neklassicheskie uravneniya Volterra I roda: teoriya i chislennye metody, Nauka, SO RAN, Novosibirsk, 1999