Algorithms for the numerical solution of fractional differential equations with interval parameters

A. Yu. Morozov ; D. L. Reviznikov

Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 4, pp. 93-108.

Résumé

The paper deals with the numerical solution of fractional differential equations with interval parameters in terms of derivatives describing anomalous diffusion processes. Computational algorithms for solving initial-boundary value problems as well as the corresponding inverse problems for equations containing interval fractional derivatives with respect to time and space are presented. The algorithms are based on the previously developed and theoretically substantiated adaptive interpolation algorithm tested on a number of applied problems for modeling dynamical systems with interval parameters; this makes it possible to explicitly obtain parametric sets of states of dynamical systems. The efficiency and workability of the proposed algorithms are demonstrated in several problems.

Détail

Keywords: fractional derivative, difference scheme, inverse problem, parametric identification, interval parameter, dynamical system
Mots-clés : anomalous diffusion, adaptive interpolation algorithm.

Bibliographie

[1] Samko S.G., Kilbas A.A., Marichev O.I., Fractional integrals and derivatives and some of their applications, Nauka i tekhnika, Minsk, 1987 (in Russian)

[2] Nahushev A.M., Fractional calculus and its application, Fizmatlit, M., 2003 (in Russian)

[3] Goloviznin V.M., Kiselev V.P., Korotkin I.A., YUrkov YU.I., Some Features of Computational Algorithms for Fractional Diffusion Equations, IBRAE-2002-01, Nuclear Safety Institute, Russia Academy of Sciences, M., 2002

[4] Meerschaert M. M., Tadjeran C., “Finite difference approximations for two-sided space-fractional partial differential equations”, Appl. Numer. Math., 56:1 (2006), 80–90 | DOI | MR | Zbl

[5] Zhang Y., Benson D. A., Meerschaert M. M., LaBolle E. M., “Space-fractional advection-dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the MADE-site data”, Water Resour. Res., 43 (2007), W05439

[6] Moroz L.I., Maslovskaya A.G., “Fractional-Differential Model of the Process of Thermal Conduction of Ferroelectric Materials under Conditions of Intense Heating”, Mat. Mat. Model., 2 (2019), 29–47 (in Russian)

[7] Maslovskaya A. G., Moroz L. I., Chebotarev A. Yu., Kovtanyuk A. E., “Theoretical and numerical analysis of the Landau-Khalatnikov model of ferroelectric hysteresis”, Commun. Nonlinear Sci. Numer. Simul., 93 (2021), 105524 | DOI | MR | Zbl

[8] Moroz L.I., Maslovskaya A.G., “Numerical simulation of an anomalous diffusion process based on the higher-order accurate scheme”, Mathematical Models and Computer Simulations, 13:3 (2021), 492–501 | DOI | DOI | MR | Zbl

[9] Reviznikov D.L., Slastushenskij YU.V., “Numerical simulation of anomalous diffusion of billiard gas in a polygonal channel”, Mat. Model., 25:5 (2013), 3–14 (in Russian) | Zbl

[10] Tverdyi D., Parovik R., “Application of the Fractional Riccati Equation for Mathematical Modeling of Dynamic Processes with Saturation and Memory Effect”, Fractal Fract., 6:3 (2022), 163 | DOI

[11] Zaky M. A., Van Bockstal K., Taha T. R., Suragan D., Hendy A. S., “An L1 type difference/Galerkin spectral scheme for variable-order time-fractional nonlinear diffusion-reaction equations with fixed delay”, J. Comput. Appl. Math., 420 (2023), 114832 | DOI | MR | Zbl

[12] Moore R., Interval analysis, Prentice-Hall, N. Y., 1966 | MR | Zbl

[13] Moore R. E., Kearfott R. B., Cloud M. J., Introduction to Interval Analysis, SIAM, Philadelphia, 2009 | MR | Zbl

[14] SHaryj S.P., Finite-Dimensional Interval Analysis, XYZ, Novosibirsk, 2019 (in Russian)

[15] Dobronec B.S., Interval mathematics, Krasnoyarskij gosudarstvennyj universitet, Krasnoyarsk, 2007 (in Russian)

[16] Morozov A.Y., Reviznikov D.L., “Adaptive Interpolation Algorithm Based on a kd-Tree for Numerical Integration of Systems of Ordinary Differential Equations with Interval Initial Conditions”, Differential Equations, 54:7 (2018), 945–956 | DOI | DOI | MR | MR | Zbl

[17] Morozov A.Y., Zhuravlev A.A., Reviznikov D.L., “Analysis and Optimization of an Adaptive Interpolation Algorithm for the Numerical Solution of a System of Ordinary Differential Equations with Interval Parameters”, Differential Equations, 56:7 (2020), 935–949 | DOI | DOI | MR | MR | Zbl

[18] Morozov A. Yu., Zhuravlev A. A., Reviznikov D. L., “Sparse Grid Adaptive Interpolation in Problems of Modeling Dynamic Systems with Interval Parameters”, Mathematics, 9 (2021), 298 | DOI | MR

[19] Gidaspov V.Yu., Morozov A.Yu., Reviznikov D.L., “Adaptive Interpolation Algorithm Using TT-Decomposition for Modeling Dynamical Systems with Interval Parameters”, Computational Mathematics and Mathematical Physics, 61:9 (2021), 1387–1400 | DOI | DOI | MR | Zbl

[20] Morozov A.Yu., Reviznikov D.L., “Adaptive Interpolation Algorithm on Sparse Meshes for Numerical Integration of Systems of Ordinary Differential Equations with Interval Uncertainties”, Differential Equations, 57:7 (2021), 947–958 | DOI | DOI | MR | MR | Zbl

[21] Morozov A. Yu., Reviznikov D. L., “Interval approach to solving parametric identification problems for dynamical systems”, Differential Equations, 58:7 (2022), 952–965 | DOI | DOI | MR | Zbl

[22] A. Yu. Morozov and D. L. Reviznikov, “Sliding window algorithm for parametric identification of dynamical systems with rectangular and ellipsoid parameter uncertainty domains”, Differ. Equations, 59:6 (2023), 833–846 | DOI | DOI | MR | Zbl

[23] Eijgenraam P., The Solution of Initial Value Problems Using Interval Arithmetic, Mathematical Centre Tracts, 144, Mathematisch Centrum, Amsterdam, 1981 | MR | Zbl

[24] Lohner R. J., “Enclosing the solutions of ordinary initial and boundary value problems”, Comput. Arith. Sci. Comput. Program. Lang, 1987, 255–286 | MR

[25] CHernous'ko F.L., Estimation of phase states of dynamical systems. Ellipsoid method, Nauka, M., 1988 (in Russian)

[26] Makino K., Berz M., “Models and Their Applications”, Numer. Softw. Verificat. 2017: Conference (Heidelberg, Germany, July 22-23, 2017), Springer International Publishing AG, 2017, 3–13 | DOI

[27] Nataraj P. S. V., Sondur S., “The Extrapolated Taylor Model”, Reliab. Comput., 15:3 (2011), 251–278 | MR

[28] Rogalev A.N., “Guaranteed methods for solving systems of ordinary differential equations based on the transformation of symbolic formulas”, Vychisl. Tekhnol., 8:5 (2003), 102–116 (in Russian) | Zbl

[29] Fu C., Ren X., Yang Y.-F., Lu K., Qin W., “Steady-state response analysis of cracked rotors with uncertain but bounded parameters using a polynomial surrogate method”, Commun. Nonlinear Sci. Numer. Simul., 68 (2019), 240–256 | DOI | MR | Zbl

[30] Fu C., Xu Y., Yang Y., Lu K., Gu F., Ball A., “Response analysis of an accelerating unbalanced rotating system with both random and interval variables”, J. Sound Vib., 466 (2020), 115047 | DOI

[31] Xiao N., Fedele F., Muhanna R. L., “Inverse Problems Under Uncertainties — An Interval Solution for the Beam Finite Element”, Proc. 11th Internat. Conf. Struct. Safety Reliab., 2013 | DOI

[32] Petrikevich YA.I., Structural-parametric identification of dynamical objects by interval initial data, Cand. Sci. (Eng.) Dissertation, Kemerov. State Univ., Kemerovo, 2006 (in Russian)

[33] Diligenskaya A.N., Samokish A.V., “Parametric identification in inverse problems of heat conduction under conditions of interval uncertainty based on neural networks”, Vestn. Samarsk. Gos. Tekh. Univ., 28:4(68) (2020), 6–18 (in Russian)

[34] Petuhov A.A., Reviznikov D.L., “Algorithms for numerical solutions of fractional differential equations”, Vestn. MAI, 16:6 (2009), 228–243 (in Russian)

[35] Gill Ph., Murray W., Wright M., Practical Optimization, Academic Press, London-New York, 1981 | MR | Zbl

[36] Shary S. P., “Randomized algorithms in interval global optimization”, Numer. Anal. Appl., 1 (2008), 376–389 | DOI

[37] Shary S. P., “A Surprising Approach in Interval Global Optimization”, Reliab. Comput., 7:6 (2001), 497–505 | DOI | MR | Zbl