On solvability of an initial value problem for Hilfer type fractional differential equation with nonlinear maxima

T. K. Yuldashev (Iuldashev) ; B. J. Kadirkulov

Daghestan Electronic Mathematical Reports, Tome 14 (2020), pp. 48-65.

Résumé

In this article we consider the questions of one-valued solvability of initial value problem for a nonlinear Hilfer type fractional differential equation with nonlinear maxima. By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation with nonlinear maxima. It is proved the theorem of existence and uniqueness of the solution of given initial value problem in an interval under consideration. It is proved also the stability of the desired solution with respect to given parameter.

Détail

Keywords: ordinary differential equation, initial value problem, nonlinear maxima, Hilfer operator, one-valued solvability.

Bibliographie

[1] Application of fractional calculus in physics, ed. Hilfer R., World Scientific Publishing Company, Singapore, 2000, 472 pp. | MR

[2] Handbook of fractional calculus with applications, v. 1–8, ed. Tenreiro Machado J.A., Walter de Gruyter GmbH, Berlin–Boston, 2019.

[3] Hilfer R., “Experimental evidence for fractional time evolution in glass forming materials”, Chem. Phys., 284:1–2 (2002), 399–408 | DOI | MR

[4] Hilfer R., “On fractional relaxation”, Part III: Scaling., Fractals, 1, Supp. 01 (2003), 251–257 | DOI | MR

[5] Hilfer R., Luchko Y., Tomovski Z., “Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives”, Fract. Calc. Appl. Anal., 12:3 (2009), 299–318 http://www.math.bas.bg/f̃caa/volume12/fcaa123 | MR | Zbl

[6] Kim M. Ha, Chol Ri. G., Chol O. H., “Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives”, Fract. Calc. Appl. Anal., 17:1 (2014), 79–95 | DOI | MR | Zbl

[7] Samko S.G., Kilbas A.A., Marichev O.I., Fractional integrals and derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993 | MR | Zbl

[8] Mainardi F., Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics, eds. Carpinteri A., Mainardi F., Springer, Wien, 1997 | MR

[9] Area I., Batarfi H., Losada J., Nieto J. J., Shammakh W., Torres A., “On a fractional order Ebola epidemic model”, ID 278, Adv. El. J. Differ. Equations., 1 (2015) | MR | Zbl

[10] Hussain A., Baleanu D., Adeel M., “Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model”, Adv. in Differ Equations, 384 (2020) | MR | Zbl

[11] Ullah S., Khan M. A., Farooq M., Hammouch Z., Baleanu D., “A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative”, Discrete Contin. Dyn. Syst., Ser. S, 13:3 (2020), 975-993 | MR | Zbl

[12] Kumar D., Baleanu D., “Editorial: Fractional calculus and its applications in physics”, Front. Phys., 7:6 (2019.)

[13] Sun H., Chang A., Zhang Y., Chen W., “A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications”, Fract. Calc. Appl. Anal., 22:1 (2019), 27–59 | DOI | MR | Zbl

[14] Saxena R.K., Garra R., Orsingher E., “Analytical solution of space-time fractional telegraph-type equations involving Hilfer and Hadamard derivatives”, Integral Transforms Spec. Funct., 27:1 (2015), 30–42 | DOI | MR

[15] Patnaik S., Hollkamp J. P., Semperlotti F., “Applications of variable-order fractional operators: a review”, Proc. A., 476:2234 (2020), 1–32 | MR

[16] Sandev T., Tomovski Z., Fractional equations and models: Theory and applications, v. 61, Dev. Math., Springer Nature Switzerland AG, Cham. – Switzerland, 2019, 345 pp.

[17] Yuldashev T.K., Kadirkulov B.J., “Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator”, ID 68, Axioms, 9:2 (2020), 1–19 | MR

[18] Yuldashev T.K., Kadirkulov B.J., “Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator”, Ural Math. J., 6:1 (2020), 153–167 | DOI | MR | Zbl

[19] Yuldashev T.K., Karimov E.T., “Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters”, ID 121, Axioms, 9:4 (2020), 24 pp. | DOI | MR | Zbl

[20] Yuldashev T.K., Ovsyanikov S.M., “Priblizhennoe reshenie sistemy nelineinykh integralnykh uravnenii s zapazdyvayuschim argumentom i priblizhennoe vychislenie funktsionala kachestva”, Zhurnal Srednevolzhskogo Mat. Obschestva, 17:2 (2015), 85–95 | Zbl

[21] Yuldashev T. K., “Predelnaya zadacha dlya sistemy integro-differentsialnykh uravnenii s dvukhtochechnymi smeshannymi maksimumami”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki., 1:16 (2008), 15–22 | Zbl

[22] Berdyshev A. S., Kadirkulov B. J. On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator, Differ. Equations, 52:1 (2016), 122–127 | DOI | MR | Zbl

[23] Malik S. A., Aziz S. An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions, Comput. Math. Appl., 73:12 (2017), 2548–2560 | DOI | MR | Zbl

[24] Erdelyi A., Higher transcendental functions (Bateman Manuscript Project), v. 1, California Institute of Technology, McGraw-Hill New York, 1953 | MR

[25] Chen S., Shen J., Wang L., “Generalized Jacobi functions and their applications to fractional differential equations”, Math. Comp., 85 (2016), 1603–1638 | DOI | MR | Zbl

[26] Filatov A.N., Sharova L.V., Integralnye neravenstva i teoriya nelineinykh kolebanii, Nauka, M., 1976, 152 pp.