Geodesic
Hereditary Riccati Equation with Fractional Derivative of Variable Order
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Tome 154 (2018), pp. 105-112.

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The Riccati differential equation with a fractional derivative of variable order is considered. A derivative of variable fractional order in the original equation implies the hereditary property of the medium, i.e., the dependence of the current state of a dynamic system on its previous states. A software called Numerical Solution of a Fractional-Differential Riccati Equation (briefly NSFDRE) is created; it allows one to compute a numerical solution of the Cauchy problem for the Riccati differential equation with a derivative of variable fractional order. The numerical algorithm implemented in the software is based on the approximation of the variable-order derivative by finite differences and the subsequent solution of the corresponding nonlinear algebraic system. New distribution modes depending on the specific type of variable order of the fractional derivative were obtained. We also show that some distribution curves are specific for other hereditary dynamic systems.
Mots-clés : Riccati equation
Keywords: fractional derivative, heredity, numerical methods, differential equation. 
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     author = {D. A. Tvyordyj},
     title = {Hereditary {Riccati} {Equation} with {Fractional} {Derivative} of {Variable} {Order}},
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D. A. Tvyordyj. Hereditary Riccati Equation with Fractional Derivative of Variable Order. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Tome 154 (2018), pp. 105-112. http://geodesic.mathdoc.fr/item/INTO_2018_154_a12/

[1] Baranov S. V., Kuznetsov S. P., Ponomarenko V. I., “Khaos v fazovoi dinamike ostsillyatora Van der Polya s modulirovannoi dobrotnostyu i dopolnitelnoi zapazdyvayuschei obratnoi svyazyu”, Izv. vuzov. Prikl. nelin. dinamika, 18:1 (2010), 12–23 | MR

[2] Berezin I. S., Zhidkov N. P., Metody vychislenii, M.-L., 1962

[3] Kumakshev S. A., “Issledovanie regulyarnykh i relaksatsionnykh kolebanii ostsillyatorov Releya i Van der Polya”, Vestn. Nizhegorod. un-ta im. N. I. Lobachevskogo, 2011, no. 4-2, 203–205

[4] Makarov D. V., Parovik R. I., “Obobschennaya matematicheskaya model Dubovskogo dlya prognozirovaniya ekonomicheskikh krizisov”, Nauch.-tekhn. vestn. Povolzhya, 2016, no. 1, 74–77

[5] Nakhushev A. M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003

[6] Novikova E. R., “Ostsillyator Van der Polya—Duffinga s effektom ereditarnosti”, Vestn. KRAUNTs. Fiz.-mat. nauki, 2017, no. 2 (18), 65–75

[7] Parovik R. I., Matematicheskoe modelirovanie lineinykh ereditarnykh ostsillyatorov, KamGU imeni Vitusa Beringa, Petropavlovsk-Kamchatskii, 2015

[8] Parovik R. I., “O chislennom reshenii uravneniya fraktalnogo ostsillyatora s proizvodnoi drobnogo peremennogo poryadka ot vremeni”, Vestn. KRAUNTs. Fiz.-mat. nauki, 2014, no. 1 (8), 60–-65 | Zbl

[9] Tverdyi D. A., “Uravnenie Rikkati s proizvodnoi drobnogo peremennogo poryadka”, Mezhdunar. stud. nauch. vestn., 2017, no. 2, 42

[10] Tverdyi D. A., “Uravnenie Rikkati s peremennoi ereditarnostyu”, Vestn. KRAUNTs. Fiz.-mat. nauki, 2017, no. 1 (17), 44–-53 | MR

[11] Uchaikin V. V., Metod drobnykh proizvodnykh, Artishok, Ulyanovsk, 2008

[12] Feder E., Fraktaly, Mir, M., 1991

[13] Makarov D. V., Parovik R. I., “Modeling of the economic cycles using the theory of fractional calculus”, J. Internet Banking Commerce, 21:S6 (2016), 1

[14] Parovik R. I., “Explicit finite-difference scheme for the numerical solution of the model equation of nonlinear hereditary oscillator with variable-order fractional derivatives”, Arch. Control Sci., 26:3 (2016), 429–435 | DOI | MR

[15] Petras I., Fractional-order nonlinear systems. Modeling, analysis and simulation, Springer, 2011 | MR | Zbl

[16] Sweilam N. H., Khader M. M., Mahdy A. M. S., “Numerical studies for solving fractional Riccati differential equation”, Appl. Appl. Math., 7:2 (2012), 595–-608 | MR | Zbl