Abstract
The this paper, we introduce a pair of Sobolev spaces with special Jacobi–Gegenbauer weights, in which the general boundary-value problem for a class of ordinary integro-differential equations characterized by the positivity of the difference of orders of the inner and outer differential operators is well-posed in the Hadamard sense. Based on this result, we justify the general polynomial projection method for solving the corresponding problem. An application of general results to the proof of the convergence of the polynomial Galerkin method for solving the Cauchy problem in the Sobolev weighted space is given. The convergence rate of the method is characterized in terms of the best polynomial approximations of an exact solution, which automatically responds to the smoothness properties of the coefficients of the equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu. R. Agachev, “Convergence of the general polynomial projection method for ill-posed integro-differential equations,” Izv. Vyssh. Ucheb. Zaved. Mat., No. 8, 3–14 (2007).
Yu. R. Agachev and M. Yu. Pershagin, “Well-posed statement of conditionally well-posed integro-differential equations in a new pair of weightless Sobolev spaces,” Izv. Vyssh. Ucheb. Zaved. Mat., No. 8, 80-85 (2017).
R. Z. Dautov and M. R. Timerbaev, “Exact estimates of polynomial approximations in weighted Sobolev spaces,” Differ. Uravn., 51, No. 7, 890–898 (2015).
B. G. Gabdulkhaev, “Somce problems of the theory of approximate methods, II,” Izv. Vyssh. Ucheb. Zaved. Mat., No. 10, 21–29 (1968).
B. G. Gabdulkhaev, Optimal Approximations of Solutions of Linear Problems [in Russian], Kazan (1980).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 175, Proceedings of the XVII All-Russian Youth School-Conference “Lobachevsky Readings-2018,” November 23-28, 2018, Kazan. Part 1, 2020.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Agachev, Y.R., Pershagin, M.Y. Well-Posedness of Boundary-Value Problems for Conditionally Well-Posed Integro-Differential Equations and Polynomial Approximations of Their Solutions. J Math Sci 272, 816–825 (2023). https://doi.org/10.1007/s10958-023-06475-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06475-1
Keywords and phrases
- Sobolev space
- Jacobi–Gegenbauer weight
- integro-differential equation
- general boundary-value problem
- well-posedness
- projection method
- polynomial approximation
- convergence