Linear equations with discretely distributed fractional derivative in Banach spaces

V. E. Fedorov   ; N. V. Filin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 2, pp. 264-280

Résumé

We study the unique solvability of linear equations in Banach spaces with discretely distributed Gerasimov–Caputo fractional derivative in terms of analytic resolving families of operators. Necessary and sufficient conditions for the existence of such a family of operators are obtained in terms of the resolvent of a closed operator from the right-hand side of the equation, and the properties of this family are studied. These results are used to prove the existence of a unique solution to the Cauchy problem for a linear equation of the corresponding class with inhomogeneity which is either continuous in the norm of the graph of the operator from the right-hand side of the equation or Hölderian. Based on the abstract results obtained, we investigate the unique solvability of initial–boundary value problems for a class of equations with discretely distributed fractional time derivative and with polynomials in an elliptic self-adjoint differential operator with respect to spatial variables.

Détail

Keywords: Gerasimov–Caputo fractional derivative, discretely distributed fractional derivative, Cauchy problem, resolving family of operators, initial–boundary value problem.

Bibliographie

[1] Nakhushev A. M., “O nepreryvnykh differentsialnykh uravneniyakh i ikh raznostnykh analogakh”, Dokl. AN SSSR, 300:4 (1988), 796–799 | MR | Zbl

[2] Caputo M., “Mean fractional order derivatives. Differential equations and filters”, Ann. Univ. Ferrara, 41:1 (1995), 73–84 | DOI | MR | Zbl

[3] Sokolov I. M., Chechkin A. V., Klafter J., “Distributed-order fractional kinetics”, Acta Physica Polonica B, 35 (2004), 1323–1341

[4] Diethelm K., Ford N., Freed A. D., Luchko Y., “Algorithms for the fractional calculus: A selection of numerical methods”, Computer Methods in Applied Mechanics and Engineering, 194:6–8 (2003), 743–773 | DOI | MR

[5] Pskhu A. V., Uravneniya v chastnykh proizvodnykh drobnogo poryadka, Nauka, M., 2005, 199 pp.

[6] Atanackovic T. M., Oparnica L., Pilipovic S., “On a nonlinear distributed order fractional differential equation”, J. Math. Anal. Appl., 328 (2007), 590–608 | DOI | MR | Zbl

[7] Kochubei A. N., “Distributed order calculus and equations of ultraslow diffusion”, J. Math. Anal. Appl., 340 (2008), 252–280 | DOI | MR

[8] Fedorov V. E., Abdrakhmanova A. A., “A class of initial value problems for distributed order equations with a bounded operator”, Stability, Control and Differential Games, eds. A. Tarasyev, V. Maksimov, T. Filippova, Springer Nature, Cham, 2020, 251–262 | DOI

[9] Fedorov V. E., Streletskaya E. M., “Initial-value problems for linear distributed-order differential equations in Banach spaces”, Electronic J. Diff. Eq., 2018:176 (2018), 1–17 | MR | Zbl

[10] Fedorov V. E., Fuong T. D., Kien B. T., Boiko K. V., Izhberdeeva E. M., “Odin klass polulineinykh uravnenii raspredelennogo poryadka v banakhovykh prostranstvakh”, Chelyab. fiz.-mat. zhurn., 5:3 (2020), 342–351 | DOI | MR

[11] Fedorov V. E., “O porozhdenii analiticheskogo v sektore razreshayuschego semeistva operatorov differentsialnogo uravneniya raspredelennogo poryadka”, Zap. nauch. seminarov POMI, 489, 2020, 113–129

[12] Fedorov V. E., “Generators of analytic resolving families for distributed order equations and perturbations”, Mathematics, 8 (2020), 1306, 1–15 | DOI

[13] Pskhu A. V., “Uravnenie drobnoi diffuzii s operatorom diskretno raspredelennogo differentsirovaniya”, Sib. elektron. mat. izv., 13 (2016), 1078–1098 | DOI | MR | Zbl

[14] Kostic M., “Degenerate multi-term fractional differential equations in locally convex spaces”, Publication de l'Institut Mathematique. Nouvelle serie, 100 (114) (2016), 49–75 | DOI | MR | Zbl

[15] Novozhenova O. G., “Life and science of Alexey Gerasimov, one of the pioneers of fractional calculus in Soviet Union”, Fractional Calculus and Applied Analysis, 20:3 (2017), 790–809 | DOI | MR | Zbl

[16] Podlubny I., Fractional Differential Equations, Academic Press, San Diego; Boston, 1999, 340 pp. | MR | Zbl

[17] Pruss J., Evolutionary integral equations and applications, Springer, Basel, 1993, 369 pp. | MR

[18] Arendt W., Batty C. J. K., Hieber M., Neubrander F., Vector-valued Laplace transforms and Cauchy problems, Springer, Basel, 2011, 539 pp. | MR

[19] Bajlekova E. G., Fractional evolution equations in Banach spaces, PhD thesis, Eindhoven University of Technology: University Press Facilities, Eindhoven, 2001 | MR | Zbl

[20] Iosida K., Funktsionalnyi analiz, Mir, M., 1967, 616 pp.

[21] Goldstein J. A., “Semigroups and second-order differential equations”, J. Functional Analysis, 4 (1969), 50–70 | DOI | MR | Zbl

[22] Tribel Kh., Teoriya interpolyatsii. Funktsionalnye prostranstva. Differentsialnye operatory, Mir, M., 1980, 664 pp.