Email this article On Stability of an Approximate Solution of the Cauchy Problem for Some First-Order Integrodifferential Equations
- Authors: Vabishchevich P.N.1,2
-
Affiliations:
- Nuclear Safety Institute, Russian Academy of Sciences
- North-Caucasus Center of Mathematical Studies
- Issue: Vol 63, No 2 (2023)
- Pages: 328-335
- Section: МАТЕМАТИЧЕСКАЯ ФИЗИКА
- URL: https://journals.rcsi.science/0044-4669/article/view/136127
- DOI: https://doi.org/10.31857/S004446692302014X
- EDN: https://elibrary.ru/BRJQJC
- ID: 136127
Cite item
Open Access
Access granted
Subscription Access
Abstract
The Cauchy problem for a first-order evolutionary equation with memory with the time derivative of the Volterra integral term and difference kernel in the finite-dimensional Banach space is considered. The fundamental difficulties of the approximate solution of such problems are caused by nonlocality with respect to time when the solution at the current time depends on the entire history. Transformation of the first-order integrodifferential equation to a system of evolutionary local equations with the approximation of the difference kernel by a sum of exponential functions is used. For the weakly coupled system of local equations with additional ordinary differential equations, estimates of stability of solution with respect to initial data and right-hand side are obtained using the concept of logarithmic norm. Similar estimates are obtained for the approximate solution using two-level time approximations.
About the authors
P. N. Vabishchevich
Nuclear Safety Institute, Russian Academy of Sciences; North-Caucasus Center of Mathematical Studies
Author for correspondence.
Email: vabishchevich@gmail.com
115191, Moscow, Russia; 355017, Stavropol, Russia
References
- Gripenberg G., Londen S.-O., Staffans O. Volterra Integral and Functional Equations. Cambridge: Springer, 1990.
- Prüss J. Evolutionary Integral Equations and Applications. Basel: Springer, 1993.
- Kochubei A.N. General fractional calculus, evolution equations, and renewal processes // Integral Equations and Operator Theory. 2011. V. 71. № 4. P. 583–600.
- Chen C., Shih T. Finite Element Methods for Integrodifferential Equations. Singapore: World Scientific, 1998.
- Samarskii A.A. The Theory of Difference Schemes. New York: Marcel Dekker, 2001.
- Самарский А.А., Гулин А.В. Устойчивость разностных схем. М.: Наука, 1973.
- Luchko Y., Yamamoto Y. The general fractional derivative and related fractional differential equations // Mathematics. 2020. V. 8. № 2115. P. 1–20.
- Вабищевич П.Н. Численные методы решения нестационарных задач. М.: ЛЕНАНД, 2021.
- Вабищевич П.Н. Монотонные схемы для задач конвекции-диффузии с конвективным переносом в различной форме // Ж. вычисл. матем. и матем. физ. 2021. Т. 61. № 1. С. 95–107.
- Linz P. Analytical and Numerical Methods for Volterra Equations. Philadelphia: Springer, 1985.
- Vabishchevich P.N. Numerical solution of the Cauchy problem for Volterra integrodifferential equations with difference kernels // Applied Numerical Mathematics. 2022. V. 174. P. 177–190.
- Vabishchevich P.N. Numerical solution of the heat conduction problem with memory // Computers and Mathematics with Applications. 2022. № 2022.05.020 P. 1–7.
- Joseph D.D., Preziosi L. Heat waves // Reviews of Modern Physics. 1989. V. 61. № 1. P. 1–41.
- Straughan B. Heat Waves. Berlin: Springer, 2011.
- Gurtin M.E., Pipkin A.C. A general theory of heat conduction with finite wave speeds // Archive for Rational Mechanics and Analysis. 1968. V. 31. № 2. P. 113–126.
- Nunziato J.W. On heat conduction in materials with memory // Quarterly of Applied Mathematics. 1971. V. 29. № 2. P. 187–204.
- Лозинский С.М. Оценка погрешности численного интегрирования обыкновенных дифференциальных уравнений. I // Изв. вузов. Математика. 1958. № 5. С. 52–90.
- Dekker K., Verwer J.G. Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. Amsterdam: North-Holland, 1984.
- McLean W., Thomee V., Wahlbin L.B. Discretization with variable time steps of an evolution equation with a positive- type memory term // J. of Computational and Applied Mathematics. 1996. V. 69. № 1. P. 49–69.
- Halanay A. On the asymptotic behavior of the solutions of an integro-differential equation // J. of Mathematical Analysis and Applications. 1965. V. 10. № 2. P. 319–324.
- Söderlind G. The logarithmic norm. History and modern theory // BIT Numerical Mathematics. 2006. V. 46. № 3. P. 631–652.
- Pachpatte B.G. Inequalities for differential and integral equations. San Diego: Academic Press, 1998.
