A finite-difference interpretation of a nonlocalproblem with a parameterised first-order differential equation

Authors

  • Dovlet M. Dovletov Independent researcher, Ashgabat, Turkmenistan

Keywords:

nonlocal initial value problem, uniform stability, uniform convergence, multipoint condition, exponentially fitted parameter

Abstract

In this work the nonlocal initial value problem and corresponding finite-difference interpretation for the first-order linear ordinary differential equation with a parameter at the derivative is studied. The nonlocal initial value condition is given by terms of multipoint linear combination. The difference scheme with exponential fit is proposed on a uniform mesh. The article identifies the requirements on the location of nonlocal data carriers in the multipoint condition, on the values of corresponding coefficients and on the parameter variation interval, under which a uniform on parameter stability of classical and difference solutions, as well as a uniform on parameter convergence of difference solution to classical solution are proved. The identification and proof of such conditions, which provide a uniform on parameter approximation of the nonlocal initial value problem classical solution by the solution of exponentially fitted difference scheme, define the novelty of the current work.

Author Biography

  • Dovlet M. Dovletov, Independent researcher, Ashgabat, Turkmenistan

    PhD (physics and mathematics); independent researcher

References

  1. Ильин ВА, Моисеев ЕИ. Нелокальная краевая задача первого рода для оператора Штурма – Лиувилля в дифференциальной и разностной трактовках. Дифференциальные уравнения. 1987;23(7):1198–1207.
  2. Ильин ВА, Моисеев ЕИ. Нелокальная краевая задача второго рода для оператора Штурма – Лиувилля. Дифференциальные уравнения. 1987;23(8):1422–1431.
  3. Ильин ВА, Моисеев ЕИ. Априорная оценка решения задачи, сопряженной к нелокальной краевой задаче первого рода. Дифференциальные уравнения. 1988;24(5):795–804.
  4. Ciegis R. Numerical solution of a problem with small parameter for the highest derivative and a nonlocal condition. Lithuanian Mathematical Journal. 1988;28(1):90–96. DOI: 10.1007/BF00972255.
  5. Довлетов ДМ. О нелокальной краевой задаче первого рода в дифференциальной и разностной трактовках. Дифференциальные уравнения. 1989;25(8):1297–1307.
  6. Dovletov DM. Nonlocal boundary value problem in terms of flow for Sturm – Liouville operator in differential and difference statements. e-Journal of Analysis and Applied Mathematics. 2018;2018:37–55. DOI: 10.2478/ejaam-2018-0004.
  7. Ažić N. Spectral approximation and nonlocal boundary value problems. Novi Sad Journal of Mathematics. 2000;30(3):1–10.
  8. Amiraliyev GM, Cakir M. Numerical solution of the singularly perturbed problem with nonlocal boundary condition. Applied Mathematics and Mechanics. 2002;23(7):755–764. DOI: 10.1007/BF02456971.
  9. Cakir M, Amiraliyev GM. A finite difference method for the singularly perturbed problem with nonlocal boundary condition. Applied Mathematics and Computation. 2005;160(2):539–549. DOI: 10.1016/j.amc.2003.11.035.
  10. Arslan D, Cakir M. A numerical solution study on singularly perturbed convection-diffusion nonlocal boundary problem. Communications Faculty of Sciences University of Ankara. Series A1, Mathematics and Statistics. 2019;68(2):1482–1491. DOI: 10.31801/cfsuasmas.540631.
  11. Dovletov DM. A uniformly stable solvability of NLBVP for parameterized ODE. Proceedings of International Mathematical Sciences. 2021;3(2):50–69. DOI: 10.47086/pims.975424.
  12. Dovletov DM. On a multipoint nonlocal initial value problem for a singularly-perturbed first-order ODE. e-Journal of Analysis and Applied Mathematics. 2019;2019:84–103. DOI: 10.2478/ejaam-2019-0006.
  13. Дулан Э, Миллер Дж, Шилдерс У. Равномерные численные методы решения задач с пограничным слоем. Демидова ГВ, переводчик; Яненко НН, редактор. Москва: Мир; 1983. 200 с.
  14. Ильин ВА, Моисеев ЕИ. Двумерная нелокальная краевая задача для оператора Пуассона в дифференциальной и разностной трактовках. Математическое моделирование. 1990;2(8):139–156.
  15. Dovletov DM. New solvability condition of 2-d nonlocal boundary value problem for Poisson’s operator on rectangle. e-Journal
  16. of Analysis and Applied Mathematics. 2021;2021:12–28. DOI: 10.2478/ejaam-2021-0002.

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Published

2025-10-13

Issue

No. 2 (2025): Journal of the Belarusian State University. Mathematics and Informatics

Section

Computational Mathematics

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How to Cite

[1]
Dovletov, D.M. 2025. A finite-difference interpretation of a nonlocalproblem with a parameterised first-order differential equation. Journal of the Belarusian State University. Mathematics and Informatics. 2 (Oct. 2025), 89–104. DOI:https://doi.org/10.33581/2520-6508-2025-2-89-104.