Weighted accuracy estimates of the Cayley transform method for abstract boundary-value problems in a Banach space

TitleWeighted accuracy estimates of the Cayley transform method for abstract boundary-value problems in a Banach space
Publication TypeJournal Article
Year of Publication2020
AuthorsMakarov, VL, Mayko, NV
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published5/2020

We study the first BVP for linear second-order differential equations with a strongly positive operator coefficient in a Banach space. The exact solutions of these BVPs are represented in the form of infinite series by means of the Cayley transform of the operator coefficient, the Meixner-type polynomials in the independent variable, and the Fourier series representation of the right-hand side of the equation. The approximate solution of each problem is a partial sum of the corresponding series (with the discretization parameter N). We prove the weighted accuracy estimates taking the boundary effect into account. These estimates demonstrate that the proposed methods have the power rate of convergence or the exponential rate of convergence in accordance with the smoothness properties of the input data.

KeywordsBanach space, boundary effect, boundary-value problem (BVP), Cayley transform, exponential rate of convergence, method without saturation of accuracy

1. Makarov, V. (1989). On a priori estimates of differential schemes giving an account of the boundary effect. C. R. Acad. Bulg. Sci., 42, No. 5, pp. 41-44.
2. Babenko, K. I. (1986). Fundamentals of numerical analysis. Moscow: Nauka (in Russian).
3. Gavrilyuk, I. P., Makarov, V. L. & Mayko, N. V. (2020). Weighted estimates of the Cayley transform method for abstract differential equations. Comput. Methods Appl. Math. Doi: https://doi.org/10.1515/cmam-2019-0120
4. Pazy, А. (1983). Semigroups of linear operators and applications to partial differential equation. New York: Springer. Doi: https://doi.org/10.1007/978-1-4612-5561-1
5. Gavrilyuk, I. P. & Makarov, V. L. (1999). Explicit and approximate solutions of second-order elliptic differential equations in Hilbert and Banach spaces. Numer. Func. Anal. Opt., 20, No. 7-8, pp. 695-717. Doi: https://doi.org/10.1090/S0025-5718-03-01590-4
6. Makarov, V. L. (2019). Meixner polynomials and their properties. Dopov. Nac. akad. nauk. Ukr., No. 7, pp. 3-8 (in Ukrainian). Doi: https://doi.org/10.15407/dopovidi2019.07.003
7. Gavrilyuk, I. P. & Makarov, V. L. (2004). Strongly positive operators and numerical algorithms without saturation of accuracy. Kyiv: Institute of Mathematics of the NAS of Ukraine (in Russian).
8. Gorbachuk, V. I. & Knyazyuk, A. V. (1989). Boundary values of solutions of operator-differential equations. Russ. Math. Surv., 44, Iss. 3, pp. 67-111. Doi: https://doi.org/10.1070/RM1989v044n03ABEH002115
9. Radyno, Ya. V. (1985). Vectors of exponential type in the operator calculus, and differential equations. Differ. Uravn., 21, No. 9, pp. 1559-1569 (in Russian).