Abstract:
A direct method for finding exact solutions of differential or Fredholm integro-differential
equations with nonlocal boundary conditions is proposed. We investigate the abstract equations of
the form Bu=Au−gF(Au)=f and B1u=A2u−qF(Au)−gF(A2u)=f with abstract nonlocal
boundary conditions Φ(u)=NΨ(Au) and Φ(u)=NΨ(Au), Φ(Au)=DF(Au)+NΨ(A2u),
respectively, where q, g are vectors, D, N are matrices, F, Φ, Ψ are vector-functions. In this paper:
we investigate the correctness of the equation Bu=f and find its exact solution,
we investigate the correctness of the equation B1u=f and find its exact solution,
we find the conditions under which the operator B1 has the decomposition B1=B2, i.e. B1
is a quadratic operator, and then we investigate the correctness of the equation B2u=f1 and find its exact solution.
Keywords and phrases:
differential and Fredholm integro-differential equations, nonlocal integral boundary conditions, decomposition of operators, correct operators, exact solutions.
Citation:
I. N. Parasidis, “Extension and decomposition method for differential and integro-differential equations”, Eurasian Math. J., 10:3 (2019), 48–67
\Bibitem{Par19}
\by I.~N.~Parasidis
\paper Extension and decomposition method for differential and integro-differential equations
\jour Eurasian Math. J.
\yr 2019
\vol 10
\issue 3
\pages 48--67
\mathnet{http://mi.mathnet.ru/emj338}
\crossref{https://doi.org/10.32523/2077-9879-2019-10-3-48-67}
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Linking options:
https://www.mathnet.ru/eng/emj338
https://www.mathnet.ru/eng/emj/v10/i3/p48
This publication is cited in the following 4 articles:
A. T. Asanova, E. A. Bakirova, A. E. Imanchiev, “Kraevaya zadacha dlya integro-differentsialnogo uravneniya smeshannogo tipa”, Geometriya, mekhanika i differentsialnye uravneniya, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 211, VINITI RAN, M., 2022, 3–13
E. Providas, L. S. Pulkina, I. N. Parasidis, “Factorization of ordinary and hyperbolic integro-differential equations with integral boundary conditions in a Banach space”, Vestn. SamU. Estestvennonauchn. ser., 27:1 (2021), 29–43
A. T. Assanova, A. P. Sabalakhova, Z. M. Toleukhanova, “On the unique solvability of a family of boundary value problems for integro-differential equations of mixed type”, Lobachevskii J. Math., 42:6, SI (2021), 1228–1238
I. N. Parasidis, E. Providas, “Factorization method for solving nonlocal boundary value problems in Banach space”, Bull. Karaganda Univ-Math., 103:3 (2021), 76–86
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