Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations

Mikhail Bulatov; Pedro Lima; Ewa Weinmüller

Open Mathematics (2014)

  • Volume: 12, Issue: 2, page 308-321
  • ISSN: 2391-5455

Abstract

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We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.

How to cite

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Mikhail Bulatov, Pedro Lima, and Ewa Weinmüller. "Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations." Open Mathematics 12.2 (2014): 308-321. <http://eudml.org/doc/269736>.

@article{MikhailBulatov2014,
abstract = {We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.},
author = {Mikhail Bulatov, Pedro Lima, Ewa Weinmüller},
journal = {Open Mathematics},
keywords = {Weakly singular; Two-dimensional Volterra integral-algebraic equations; Integro-differential equations; weakly singular kernel; two-dimensional Volterra integral-algebraic equations; integro-differential equations; system},
language = {eng},
number = {2},
pages = {308-321},
title = {Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations},
url = {http://eudml.org/doc/269736},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Mikhail Bulatov
AU - Pedro Lima
AU - Ewa Weinmüller
TI - Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations
JO - Open Mathematics
PY - 2014
VL - 12
IS - 2
SP - 308
EP - 321
AB - We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.
LA - eng
KW - Weakly singular; Two-dimensional Volterra integral-algebraic equations; Integro-differential equations; weakly singular kernel; two-dimensional Volterra integral-algebraic equations; integro-differential equations; system
UR - http://eudml.org/doc/269736
ER -

References

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