- PII
- S0044466925010016-1
- DOI
- 10.31857/S0044466925010016
- Publication type
- Article
- Status
- Published
- Authors
- Volume/ Edition
- Volume 65 / Issue number 1
- Pages
- 3-9
- Abstract
- Linear Volterra equations of the first kind are considered. A class of problems that have a single solution is identified, and collocation-variational methods are proposed to solve them numerically. The essence of these algorithms is that the approximate solution is found at the nodes of a uniform grid (the collocation condition) that yield an underdetermined system of linear algebraic equations. The system thus obtained is supplemented by the condition of minimum of the objective function, which approximates the squared norm of the approximate solution. As a result, a quadratic programming problem is obtained, viz. the objective function (the squared norm of the approximate solution) is quadratic, and the constraints (the collocation conditions) are equalities. This problem is solved by the method of Lagrange multipliers. Sufficiently simple third-order methods are considered in detail. The calculation results for test problems are given. Further development of this approach to solve other classes of integral equations numerically is discussed.
- Keywords
- интегральные уравнения Вольтерра квадратурные формулы коллокация метод множителей Лагранжа
- Date of publication
- 17.09.2025
- Year of publication
- 2025
- Number of purchasers
- 0
- Views
- 24
References
- 1. Краснов М.Л. Интегральные уравнения. М.: Наука, 1975.
- 2. Тихонов А.Н., Арсенин В.Я. Методы решения некорректных задач. М.: Наука, 1986.
- 3. Апарцин А.С. Неклассические уравнения Вольтерра I рода: теория и численные методы. Новосибирск: Наука, 1999.
- 4. Brunner H. Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge: Cambridge Univer. Press, 2017. 402 p.
- 5. Brunner H. Collocation methods for Volterra integral and related functional equations. Cambridge: Univer. Press, 2004.
- 6. Brunner H., van der Houwen P.J. The numerical solution of Volterra equations, CWI Monographs 3. Amsterdam: North Holland, 1986.
- 7. Linz P. Analytical and numerical methads for Volterra equations. SIAM, Philadelphia, 1985.
- 8. Тен Мен Ян. Приближенное решение линейных интегральных уравнений Вольтерра 1 рода. Дисс... канд. физ.-матем. наук, Иркутск, 1985.
- 9. Верлань А.Ф., Сизиков В.С. Интегральные уравнения: методы, алгоритмы, программы. Справочное пособие. Киев: Наук. думка, 1978.
- 10. Bulatov M., Solovarova L. Collocation-variation difference schemes with several collocation points for differentialalgebraic equations // Appl. Numer. Math. 2020. V. 149. P. 153–163. DOI: 10.1016/j.apnum.2019.06.014.
- 11. Булатов М.В., Маркова Е.В. Коллокационно-вариационные подходы к решению интегральных уравнений Вольтерра I рода // Ж. вычисл. матем. и матем. физ. 2022. Т. 62.№1. С. 105–112.
- 12. Бахвалов Н.С. Численные методы. М.: Наука, 1975.
