V.G. Pimenov, A.B. Lozhnikov. Asymptotic expansion of the error of a numerical method for solving a superdiffusion equation with functional delay ... P. 138-151

An equation with Riesz fractional space derivatives and a functional delay effect is considered. The problem is discretized. Constructions of an analog of the Crank—Nicolson difference method with piecewise linear interpolation and extrapolation by continuation are presented. The method has the second order of smallness with respect to the time and space sampling steps $\Delta$ and $h$. The basic Crank—Nicolson method with piecewise parabolic interpolation and extrapolation by continuation is constructed. The order of the residual without interpolation of the basic method is studied. The expansion coefficients of the residual with respect to $\Delta$ and $h$ are written. An equation is derived for the main term of the asymptotic expansion of the global error. Under certain assumptions, the legality of using the Richardson extrapolation procedure is substantiated and the corresponding method is constructed. The main of these assumptions is the consistency of the orders of smallness $\Delta$ and $h$. It is proved that the method has order $O(\Delta^3+h^3)$.

Keywords: Riesz fractional derivatives, superdiffusion equation, functional delay, Crank–Nicolson method, piecewise parabolic interpolation, extrapolation by continuation, Richardson method

Received April 5, 2024

Revised May 3, 2024

Accepted May 6, 2024

Vladimir Germanovich Pimenov, Dr. Phys.-Math. Sci., Prof., Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: v.g.pimenov@urfu.ru

Andrey Borisovich Lozhnikov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: ablozhnikov@yandex.ru

REFERENCES

1.   Wu J. Theory and application of partial functional differential equations. NY, Springer-Verlag, 1996, 438 p. doi: 10.1007/978-1-4612-4050-1

2.   Polyanin A., Sorokin V., Zhurov A. Delay ordinary and partial differential equations. CRC Press: Taylor & Francis Group Publ., 2023, 434 p. doi: 10.1201/9781003042310

3.   Kamont Z., Kropielnicka K. Implicit difference methods for evolution functional differential equations. Num. Anal. Appl., 2011, Vol. 4, no. 4, pp. 294–308. doi: 10.1134/S1995423911040033

4.   Pimenov V.G. Raznostnyye metody resheniya uravneniy v chastnykh proizvodnykh s nasledstvennost’yu [Difference methods for solving partial differential equations with heredity]. Yekaterinburg, Ural University Publ., 2014, 134 p. ISBN: 978-5-7996-1364-8 .

5.   Meerschaert M.M., Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math., 2006, vol. 56, no. 1, pp. 80–90. doi: 10.1016/j.apnum.2005.02.008

6.   Tadjeran C., Meerschaert M.M., Scheffler H.P. A second-order accurate numerical approximation for the fractional diffusion equation. J. Comp. Phys., 2006, vol. 213, no. 1, pp. 205–213. doi: 10.1016/j.jcp.2005.08.008

7.   Tian W., Zhou H., Deng W. A class of second order difference approximation for solving space fractional diffusion equations. Math. Comp., 2015, vol. 84, no. 294, pp. 1703–1727. doi: 10.1090/S0025-5718-2015-02917-2

8.   Pimenov V.G., Hendy A.S. A fractional analog of Crank — Nicholson method for the two sided space fractional partial equation with functional delay. Ural Math. J., 2016, vol. 2, no. 1, pp. 48–57. doi: 10.15826/umj.2016.1.005

9.   Ibrahim M., Pimenov V.G. Crank — Nikolson scheme for two-dimensional in space fractional equations with functional delay. Izv. IMI UdGU, 2021, vol. 57, pp. 128–141. doi: 10.35634/2226-3594-2021-57-05

10.   Marchuk G.I., Povysheniye tochnosti resheniy raznostnykh skhem [Improving the accuracy of solutions of difference schemes]. Moscow, Nauka Publ., 1979, 320 p.

11.   Hairer E., Nersett S., Wanner G. Solving ordinary differential equations. Nonstiff problems. Berlin, Heidelberg, Springer-Verlag Publ., 1993, 528 p. doi: 10.1007/978-3-540-78862-1

12.   Deng D., Chen J. Explicit Richardson extrapolation methods and their analyses for solving two-dimensional nonlinear wave equation with delays. Networks and Heterogeneous Media, 2023, vol. 18, no. 1, pp. 412–443. doi: 10.3934/nhm.2023017

13.   Zhang C., Tan Z. Linearized compact difference methods combined with Richardson extrapolation for nonlinear delay Sobolev equations. Communications in nonlinear science and numerical simulation, 2020, vol. 1, art. no. 105461, 18 p. doi: 10.1016/j.cnsns.2020.105461

14.   Pimenov  V.G., Lozhnikov A.B. Richardson Method for a Diffusion Equation with Functional Delay. Proc. Steklov Inst. Math. (Suppl.), 2023, vol. 321, suppl. 1, pp. S204–S215. doi: 10.1134/S0081543823030173

15.   Pimenov V.G., Tashirova E.E. Asymptotic expansion of the error of the numerical method for solving wave equation with functional delay. Izv. IMI UdGU, 2023, vol. 62, pp. 71–86 (in Russian). doi: 10.35634/2226-3594-2023-62-06

16.   Li C.P., Zeng F.H. Numerical methods for fractional calculus, London, NY, Boca Raton. CRC Press, 2015, 294 p. doi: 10.1201/b18503

17.   Kim A.V., Pimenov V.G. i-Gladkii analiz i chislennye metody resheniya funktsionalno-differentsialnykh uravnenii, Regulyarnaya i khaoticheskaya dinamika [i-Smooth analysis and numerical methods for solving functional-differential equations]. Izhevsk, Regulyarnaya i Khaoticheskaya Dinamika Publ., 2004, 256 p. ISBN: 5-93972-379-9 .

18.   Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal Control. NY, Plenum Press, 1987, 309 p. doi: 10.1007/978-1-4615-7551-1 . Original Russian text published in Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal’noe upravlenie, Moscow, Nauka Publ., 1979, 432 p.

Cite this article as: V.G. Pimenov, A.B. Lozhnikov. Asymptotic expansion of the error of a numerical method for solving a superdiffusion equation with functional delay. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 2, pp. 138–151.