DOI
10.34229/KCA2522-9664.26.2.11
UDC 519.6
V.L. Makarov
Institute of Mathematics, National Academy of Sciences of Ukraine,
Kyiv, Ukraine,
makarovimath@gmail.com
N.V. Mayko
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine,
mayko@knu.ua
V.L. Ryabichev
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine,
ryabichev@knu.ua
AN EXPONENTIALLY CONVERGENT NUMERICAL METHOD
FOR A FRACTIONAL-ORDER EVOLUTION EQUATION IN A BANACH SPACE
Abstract. We consider an initial-value problem for a fractional-order evolution equation with a strongly positive operator in a Banach space. To solve it approximately, we propose and study a numerical method for both homogeneous and inhomogeneous equations based on applying the Cayley transform of the operator coefficient, using the Laguerre–Cayley polynomials, and approximating the solution operator by a partial sum of its defining series. A detailed analysis indicates an exponential rate of convergence, which is supported by numerical results.
Keywords: initial-value problem, evolution equation, fractional derivative, Banach space, sectorial operator, strongly positive operator, Mittag-Leffler function, Cayley transformation, exponentially convergent algorithm.
full text
REFERENCES
- 1. Sun H., Zhang Y., Baleanu D., Chen W., Chen Y. A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation. 2018. Vol. 64. P. 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019.
- 2. Hilfer R. (ed.) Applications of fractional calculus in physics. Singapore: World Scientific, 2000. 472 p. https://doi.org/10.1142/3779.
- 3. Mainardi F. Fractional calculus and waves in linear viscoelasticity. London: Imperial College Press, 2010. 368 p. https://doi.org/10.1142/p614.
- 4. Magin R.L. Fractional calculus in bioengineering (2nd ed.). Redding, CT: Begell House, 2020. 704 p. https://doi.org/10.1615/critrevbiomedeng.v32.i1.10.
- 5. Comte F., Renault E. Long memory in continuous-time stochastic volatility models. Mathematical Finance. 2002. Vol. 8, Iss. 4. P. 291–323. https://doi.org/10.1111/1467-9965.00057.
- 6. Cartea A., del-Castillo-Negrete D. Fractional diffusion models of option prices in markets with jumps. Physica A: Statistical Mechanics and its Applications. 2007. Vol. 374, Iss. 2. P. 749–763. https://doi.org/10.1016/j.physa.2006.08.071.
- 7. Monje C.A., Chen Y., Vinagre B.M., Xue D., Feliu V. Fractional-order systems and controls: Fundamentals and Applications. London: Springer, 2010. XIV+499 p. https://doi.org/10.1007/978-1-84996-335-0.
- 8. Bohaienko V.O., Bulavatskyi V.M., Khimich O.M. Mathematical and computer modeling in problems of hydrogeomigration dynamics. Kyiv: Naukova dumka, 2022. 249 p.
- 9. Sun Z.-Z., Gao G.-H. Fractional differential equations: Finite difference methods. Berlin: De Gruyter, 2020. 396 p.
- 10. Chakraverty S., Jena R.M., Jena S.K. Computational fractional dynamical systems: Fractional differential equations and applications. Hoboken, NJ: Wiley, 2022. http://doi.org/10.1002/9781119697060.
- 11. Xue D., Bai L. Fractional calculus: High-precision algorithms and numerical iImplementations. Singapore: Springer, 2024. XI+406 p. https://doi.org/10.1007/978-981-99-2070-9.
- 12. Milici C., Drgnescu G., Tenreiro Machado J. Introduction to fractional differential equations (Nonlinear Systems and Complexity, Vol. 25). Cham; Switzerland: Springer, 2019. https:// doi.org/10.1007/978-3-030-00895-6.
- 13. Gavrilyuk I.P., Makarov V.L., Mayko N.V. Weighted estimates for boundary value problems with fractional derivatives. Computational Methods in Applied Mathematics. 2020. Vol. 20, N 4. P. 609–630. https://doi.org/10.1515/cmam-2018-0305.
- 14. Makarov V., Mayko N. Traditional functional-discrete methods for the problems of mathematical physics: New Aspects. Wiley-ISTE, 2024. XIX+352 p. http://doi.org/10.1002/9781394276660.
- 15. Mclean W., Thome V. Numerical solution via Laplace transforms of a fractional order evolution equation. J. Integral Equations Applications. 2010. Vol. 22, N 1. P. 57–94. https://doi.org/10.1216/JIE-2010-22-1-57.
- 16. Makarov V.L., Mayko N.V. Weighted estimates of the Cayley transform method for boundary value problems in a Banach space. Numerical Functional Analysis and Optimization. 2021. Vol. 42, N 2. P. 211–233. https://doi.org/10.1080/01630563.2020.1871010.
- 17. Gavrilyuk I.P., Makarov V.L. The Cayley transform and the solution of an initial value problem for a first order differential equation with an unbounded operator coefficient in Hilbert space. Numerical Functional Analysis and Optimization. 1994. Vol. 15, N 5–6. P. 583–598. https://doi.org/10.1080/01630569408816582.
- 18. Makarov V., Mayko N., Ryabichev V. A numerical method without accuracy saturation for a fractional-order equation in a Banach space. J. Applied Numer. Analysis. 2025. Vol. 3. P. 101–112. http://doi:10.30970/ana.2025.3.102.
- 19. Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V. Mittag-Leffler functions. Related Topics and Applications. Berlin; Heidelberg: Springer-Verlag, 2014. XIV+443 p. https://doi.org/10.1007/978-3-662-43930-2.
- 20. Makarov V.L., Mayko N.V., Ryabichev V.L. Finding the recurrence relation for the system of polynomials used in the fractional differential problem. Cybern. Syst. Anal. 2025. Vol. 61. P. 53–65. https://doi.org/10.1007/s10559-025-00746-2.
- 21. Gorbachuk V.I., Gorbachuk M.L. Boundary value problems for differential-operator equations [in Russian]. Kyiv: Nauk. Dumka, 1984. 283 p.
- 22. Makarov V.L., Makarov S.V. Functions and Cayley polynomials. Dopov. NAN Ukrayiny,, 2022, No. 5. pp. 3–9. https://doi.org/10.15407/dopovidi2022.05.003.