Fredholm integral equation with a difference kernel containing a periodic multiplier

Authors

  • Maxim G. Muradyan Armenian State University of Economics, 128 Nalbandjana Street, Yerevan 0025, Armenia
  • Aram G. Muradyan Armenian State University of Economics, 128 Nalbandjana Street, Yerevan 0025, Armenia

Keywords:

Fredholm integral equation, periodic multiplier of the difference kernel, periodically layered medium, recurrent solution method

Abstract

A Fredholm integral equation with a kernel of the form λ(x)K(x–t) is considered on an interval [0; n] (n is a natural number), where λ(x) is a 1-periodic function, and K(x–t) represents an integral superposition of exponential functions. It is assumed that the solution of this equation is known for one period, and based on this, a recurrent method for the successive continuation of the solution over the entire interval under consideration is developed. An equation with such a kernel describes processes of transport, optimal filtering, wave propagation, and other phenomena in periodically layered media (for example, in crystals or systems whose properties repeat at regular intervals). The proposed method can serve as a basis for a stable computational algorithm for solving the given integral equation. The obtained a priori estimates make it possible to control the accuracy and convergence of the computational process.

Author Biographies

  • Maxim G. Muradyan, Armenian State University of Economics, 128 Nalbandjana Street, Yerevan 0025, Armenia

    doctor of science (physics and mathematics), full professor; professor at the department of higher mathematics

  • Aram G. Muradyan, Armenian State University of Economics, 128 Nalbandjana Street, Yerevan 0025, Armenia

    PhD (physics and mathematics), docent; associate professor at the department of higher mathematics

References

  1. Middleton D. Topics in communication theory. New York: McGraw-Hill; 1965. X, 126 p. (Bellman RE, Kalaba R, editors. McGraw-Hill monographs in modern engineering science).
  2. Helstrom CW. Statistical theory of signal detection. 2nd edition. Oxford: Pergamon Press; 1975. XIV, 470 p. (Fry DW, Costrell L, Kandiah K, editors. International series of monographs in electronics and instrumentation; volume 9).
  3. Busbridge IW. The mathematics of radiative transfer. Cambridge: Cambridge University Press; 1960. 143 p. (Hall P, Smithies F, editors. Cambridge tracts in mathematics and mathematical physics; number 50).
  4. Allen RC Jr. Functional relationships for Fredholm integral equations arising from pseudo-transport problems. Journal of Mathematical Analysis and Applications. 1970;30(1):48 –78. DOI: 10.1016/0022-247X(70)90183-6.
  5. Yeh P. Optical waves in layered media. Hoboken: John Wiley & Sons; 2005. X, 406 p. (Saleh BE, editor. Wiley series in pure and applied optics).
  6. Арабаджян ЛГ, Енгибарян НБ. Уравнения в свертках и нелинейные функциональные уравнения. Итоги науки и техники. Математический анализ. 1984;22:175 –244.
  7. Крейн МГ, Шмульян ЮД. Уравнения Винера – Хопфа, ядра которых допускают интегральное представление через экспоненты. Известия Академии наук Армянской ССР. Математика. 1982;17(4):307–327.
  8. Крейн МГ, Шмульян ЮД. Уравнения Винера – Хопфа, ядра которых допускают интегральное представление через экспо­ненты (продолжение). Известия Академии наук Армянской ССР. Математика. 1982;17(5):335 –375.
  9. Касти Дж, Калаба Р. Методы погружения в прикладной математике. Чеботарев СП, переводчик; Цукерман ГМ, редактор. Москва: Мир; 1976. 224 с.
  10. Соболев ВВ. Рассеяние света в атмосферах планет. Москва: Наука; 1972. 336 с.
  11. Kagiwada HH, Kalaba RE. Verification of the invariant imbedding method for certain Fredholm integral equations. Journal of Mathematical Analysis and Applications. 1968;23(3):540 –550. DOI: 10.1016/0022-247X(68)90135-2.
  12. Allen RC Jr, Boicourt GP. An algorithm for solving Fredholm integral equations with displacement kernels. Journal of Mathematical Analysis and Applications. 1981;80(1):245 –260. DOI: 10.1016/0022-247X(81)90103-7.
  13. Мурадян МГ, Мурадян АГ. Линейное интегральное уравнение на полуоси с ядром λ(x)K(x–t), где λ(x) – периодическая функция. Дифференциальные уравнения. 1988;24(6):1061–1064.

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Published

2026-09-16

Issue

No. 2 (2026): Journal of the Belarusian State University. Mathematics and Informatics

Section

Real, Complex and Functional Analysis

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How to Cite

[1]
Muradyan, M.G. and Muradyan, A.G. 2026. Fredholm integral equation with a difference kernel containing a periodic multiplier. Journal of the Belarusian State University. Mathematics and Informatics. 2 (Sep. 2026), 6–13.