Geodesic
Asymptotic solutions of Fredholm integro-differential equations with rapidly changing kernels and irreversible limit operator
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2015), pp. 3-18

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We consider a system of singularly perturbed integro-differential Fredholm equations with rapidly changing kernel in the case of irreversible operator of differential part. We develop an algorithm for constructing regularized asymptotic solutions. It is shown that in the presence of rapidly decreasing multiplier in the kernel the original problem is not on the spectrum (i.e., it is solvable for any right-hand side). We study the limit transition (with small parameter tending to zero), and solve the problem of initialization, i.e., the problem of extracting of the source data for which an exact solution to the system tends to the limit at all duration (including a zone of boundary layer).
Mots-clés : singular perturbation
Keywords: boundary layer, integro-differential equation, initialization. 
@article{IVM_2015_10_a0,
     author = {A. A. Bobodzhanov and V. F. Safonov},
     title = {Asymptotic solutions of {Fredholm} integro-differential equations with rapidly changing kernels and irreversible limit operator},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--18},
     publisher = {mathdoc},
     number = {10},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2015_10_a0/}
}
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A. A. Bobodzhanov; V. F. Safonov. Asymptotic solutions of Fredholm integro-differential equations with rapidly changing kernels and irreversible limit operator. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2015), pp. 3-18. http://geodesic.mathdoc.fr/item/IVM_2015_10_a0/