OPTIMAL QUADRATURE FORMULA FOR THE APPROXIMATION OF THE RIGHT RIEMANN-LIOUVILLE INTEGRAL
Keywords:
optimal quadrature formula, the extremal function, the error functional, optimal coefficient, Lagrange function.Abstract
In the present article, the problem of construction of the optimal quadrature formula in the sense of Sard is discussed for numerical integration of the right Riemann-Liouville integral in the Hilbert space of real-valued functions. Initially, the norm of the error functional is found using the extremal function of the error functional of the quadrature formula. Since the error functional is defined on the Hilbert space, the quadrature formula that we are constructing is exact for zeros of this space, that is, we have the conditions that the influence of the error functional on these functions is equal to zero. Then, the Lagrange function is constructed to find the conditional extremum of the error functional. Thereby, a system of linear equations is obtained for the coefficients of the optimal quadrature formula. The existence and uniqueness of the solution of the obtained system are studied.
References
Kilbas A.A., Srivastava H.M., and Trujillo J.J. 2006. Theory and Applications of Fractional Differential Equations. // in: North–Holland Mathematics Studies, Elsevier, Amersterdam. – vol. 204..
Miller K.S. and Ross B. 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. // A Wiley–Interscience Publication Wiley, New York.
Podlubny I. 1999. Fractional Differential Equations. // Academic Press, New York.
Samko S.G., Kilbas A.A., and Marichev O.I. 1993. Fractional Integrals and Derivatives. //Theory and Applications. Gordon and Breach Science Publishers, Switzer-land.
Sobolev S.L. 2006. The coefficients of optimal quadrature formulas. // Selected Works of S.L. Sobolev, Springer US, – P. 561–566.
Sobolev S.L. 1974. Introduction to the theory of cubature formulas. // (Russian), Nauka, Moscow. – 808 p.
Changpin Li, and Zeng Fanhai 2015. Numerical Methods for Fractional Calculus. CRC Press, Taylor and Francis Group.
A.O.Mamanazarov. Nonlocal problems for a fractional order mixed parabolic equation. Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences. Vol. 4, № 1. 2021.
Маманазаров А.О. Краевые задачи для смешанно-параболического уравнения дробного порядка в неограниченных областях. Бюллетень Института математики. Т.4, № 6, с. 37-48.
I.U. Khaydarov. A problem with non-local conditions for a mixed parabolic-hyperbolic equation with two lines of changing type. International Journal of Research in Commerce, IT, Engineering and Social Sciences. V. 16, Issue 10. 2022
A.K.Urinov, Q.S.Khalilov, A Nonlocal Problem for a Third Order Parabolic-Hyperbolic Equation with a Singular Coefficient, J. Sib. Fed. Univ. Math. & Phys., 2022. 15(4), 467-481. DOI: 10.17516/1997-1397-2022-15-4-467-481.
А.К. Уринов, И.У. Хайдаров. Об одной задаче с нелокальными условиями для параболо-гиперболического уравнения. Доклады Адыгской (Черкесской) Международной академии наук. Т.12, № 2, 2010. С.80-87.
Уринов А.К., Халилов К.С. Об одной нелокальной задаче для параболо-гиперболических уравнений, Доклады АМАН. – Нальчик. 2013. Т.15. №1. – С. 24-30.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 GEJournals
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
