Differential-symbol method for solving the Nicoletti problem for a homogeneous equation with second-order time partial derivatives
DOI:
https://doi.org/10.3842/umzh.v78i3-4.9345Keywords:
диференціально-символьний метод, задача Ніколетті, двоточкові умови, квазіполіномні розв'язкиAbstract
UDC 517.95
The Nicoletti problem is investigated for a homogeneous differential equation with partial time derivatives of the second order and, in general, of the infinite order in the spatial variable. We select classes of quasipolynomial functions in which the solution to the problem exists and is unique. The case where the solution to the Nicoletti problem exists in the spaces of continuously differentiable functions is considered separately. A method for constructing solutions to the problem based on the differential-symbol method is described. We also present examples of solving the problems.
References
1. O. Nicoletti, Sulle condizioni iniziali che determiniano gli integrali delle equazioni differenziali ordinarie, Atti Accad. Sci. Torino, 33, 746–759 (1897–1898).
2. Б. Й. Пташник, Задача типу Валле–Пуссена для гіперболічних рівнянь із сталими коефіцієнтами, Допов. АН УРСР, № 10, 1254–1257 (1966).
3. A. T. Assanova, A. E. Imanchiev, A nonlocal problem with multipoint conditions for partial differential equations of higher order, Filomat, 38, № 1, 295–304 (2024).
4. V. Il'kiv, Incorrect nonlocal boundary value problem for partial differential equations, North-Holland Math. Stud., 197, 115–121 (2004).
5. B. I. Ptashnyk, M. M. Symotyuk, Multipoint problem for nonisotropic partial differential equations with constant coefficients, Ukr. Math. J., 55, No 2, 293–310 (2003).
6. V. S. Il'kiv, B. I. Ptashnyk, An ill-posed nonlocal two-point problem for systems of partial differential equations, Sib. Math. J., 46, № 1, 94–102 (2005).
7. Z. M. Nytrebych, The differential-symbol method of solving the problem with local two-point in time conditions for a partial differential equation, J. Math. Sci. (N.Y.), 254, № 3, 406–415 (2021).
8. І. Л. Віленць, Класи єдиності розв'язку загальної крайової задачі в шарі для систем лінійних диференціальних рівнянь у частинних похідних, Допов. АН УРСР, Сер. А, 3, 195–197 (1974).
9. L. V. Fardigola, Nonlocal two-point boundary-value problems in a layer with differential operators in a boundary condition, Ukr. Mat. Zh., 47, № 8, 1122–1128 (1995).
10. B. Pelloni, D. A. Smith, Nonlocal and multipoint boundary value problems for linear evolution equations, Stud. Appl. Math., 141, Issuе 1, 46–88 (2018).
11. A. S. Fokas, B. Pelloni, Two-point boundary value problems for linear evolution equations, Math. Proc. Cambridge Philos. Soc., 131, № 3, 521–543 (2001).
12. M. Denche, A. Kourta, Boundary value problem for second-order differential operators with mixed nonlocal boundary conditions, JIPAM. J. Inequal. Pure Appl. Math., 5, № 2, Article 38 (2004).
13. В. С. Ільків, М. М. Симотюк, Я. О. Слоньовський, Метричні оцінки характеристичного визначника задачі Ніколетті для рівняння типу Ейлера, Прикл. пробл. механіки та математики, 20, 31–38 (2022).
14. Ю. П. Матурін, М. М. Симотюк, Оцінки характеристичного визначника задачі Ніколетті для строго гіперболічного рівняння, Наук. вісн. Ужгород. нац. ун-ту, 2, № 33, 100–108 (2018).
15. В. Кирилич, А. Філімонов, Деякі зауваження до задачі Ніколетті з невідомими внутрішніми межами, Вісн. Львів. ун-ту. Сер. мех.-мат., 76, 7–20 (2012).
16. I. T. Kiguradze, On the singular problem o] Cauchy–Nicotetti, Ann. Mat. Pura Appl. (4), 104, 151–175 (1975).
17. J. Diblik, The singular Cauchy–Nicoletti problem for the system of two ordinary differential equations, Math. Bohem., 117, № 1, 55–67 (1992).
18. G. S. Ladde, S. Seikkala, On sample solutions of random initial value and Nicoletti boundary value problems, Math. Comput. Simulation, 29, 223–231 (1987).
19. A. Lasota, C. Olech, An optimal solution of Nicoletti’s boundary value problem, Ann. Polon. Math., 18, 131–139 (1966).
20. K. Marynets, On the Cauchy–Nicoletti type two-point boundary-value problem for fractional diferential systems, Differ. Equat. Dyn. Syst., 31, № 4, 847–867 (2020).
21. Z. M. Nytrebych, O. M. Malanchuk, The differential-symbol method of solving the problem two-point in time for nonhomogeneous partial differential equation, J. Math. Sci. (N.Y.), 227, № 1, 68–80 (2017).
22. Z. M. Nytrebych, O. M. Malanchuk, The conditions of existence of a solution of the two-point in time problem for nonhomogeneous PDE, Ital. J. Pure Appl. Math., № 41, 242–250 (2019).
23. Z. M. Nitrebich, An operator method of solving the Cauchy problem for a homogeneous system of partial differential equations, J. Math. Sci., 81, № 6, 3034–3038 (1996).
24. T. G. Anderson, M. Bonnet, L. M. Faria, C. Perez-Arancibia, Construction of polynomial particular solutions of linear constant-coefficient partial differential equations, Comput. Math. Appl., 162, 94–103 (2024).
25. G. N. Hile, A. Stanoyevitch, Heat polynomial analogous for equations with higher order time derivatives, J. Math. Anal. Appl., 295, 595–610 (2004).
26. Z. Nytrebych, O. Malanchuk, The differential-symbol method of constructing the quasi-polynomial solutions of two-point problem, Demonstr. Math., 52, Issue 1, 88–96 (2019).
27. T. Dangal, C. S. Chen, J. Lin, Polynomial particular solutions for solving elliptic partial differential equations, Comput. Math. Appl., 1, 60–70 (2017).
28. E. N. Petropoulou, P. D. Siafarikas, Polynomial solutions of linear partial differential equations, Commun. Pure Appl. Anal., 8, № 3, 1053–1065 (2009).
