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

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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. 
@article{ZVMMF_2009_49_9_a4,
     author = {E. V. Chistyakova},
     title = {Properties of finite-difference schemes for singular integrodifferential equations of index~1},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1579--1588},
     publisher = {mathdoc},
     volume = {49},
     number = {9},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_9_a4/}
}
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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/