Some applications of the generalized Laplace transform and the representation of a solution to Sobolev-type evolution equations with the generalized Caputo derivative

• Autorzy: Aydin, Mustafa , Mahmudov, Nazim I.
• Języki publikacji: EN

Abstrakty

EN

We introduce the Sobolev-type multi-term μ-fractional evolution with generalized fractional orders with respect to another function. We make some applications of the generalized Laplace transform. In the sequel, we propose a novel type of Mittag-Leffler function generated by noncommutative linear bounded operators with respect to the given function and give a few of its properties. We look for the mild solution formula of the Sobolev-type evolution equation by building on the aforementioned Mittag-Leffler-type function with the aid of two different approaches. We share new special cases of the obtained findings.

Słowa kluczowe

PL

Sobolew; pochodna ułamkowa względem innej funkcji; funkcja typu Mittaga-Lefflera; uogólniona transformata Laplace'a; równanie ewolucji zdegenerowane

EN

degenerate evolution equation; Sobolev; generalized Laplace transform; fractional derivative with respect to another function; Mittag-Leffler type function

• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2024
• Tom: Vol. 72, nr 2
• Strony: art. no. e149170
• Opis fizyczny: Bibliogr 28 poz.

Twórcy

autor

Aydin, Mustafa

• Department of Medical Services and Techniques, Muradiye Vocational School, Van Yuzuncu Yil University, Tuşba 65080 Van, Turkey,

autor

Mahmudov, Nazim I.

• Department of Mathematics, Eastern Mediterranean University, Famagusta 99628 T.R. North Cyprus, Turkey
• Research Center of Econophysics, Azerbaijan State University of Economics (UNEC), Istiqlaliyyat Str. 6, Baku 1001, Azerbaijan

Bibliografia

• [1] R. Kamocki, “Existence of optimal control for multi-order fractional optimal control problems,” Arch. Control Sci., vol. 32, no. 2, pp. 279–303, 2022, doi: 10.24425/acs.2022.141713.
• [2] M. Aydin and N. Mahmudov, “𝜓-Caputo type time-delay Langevin equations with two general fractional orders,” Math. Meth. Appl. Sci., vol. 46, no. 8, pp. 9187–9204, 2023, doi: 10.1002/mma.9047.
• [3] M. Aydin and N. Mahmudov, “On a study for the neutral Caputo fractional multi-delayed differential equations with noncommutative coefficient matrices,” Chaos Solitons Fractals, vol. 161, no. 112372, pp. 1–11, 2022, doi: 10.1016/j.chaos.2022.112372.
• [4] B. Sikora, “Results on the controllability of Caputo’s fractional descriptor systems with constant delays,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 71, no. 4, p. e146287, 2023, doi: 10.24425/bpasts.2023.146287.
• [5] J. Lightbourne and S. Rankin, “A partial functional differential equation of Sobolev type,” J. Math. Anal. Appl., vol. 93, no. 2, pp. 328–337, 1983.
• [6] A. Debbouche and D. Torres, “Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional non-local conditions,” Fract. Calc. Appl. Anal., vol. 18, pp. 95–121, 2015.
• [7] K. Balachandran, S. Kiruthika, and J. Trujillo, “On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces,” Comput. Math. Appl., vol. 62, pp. 1157–1165, 2011.
• [8] K. Balachandran and J. Dauer, “Controllability of functional differential systems of Sobolev type in Banach spaces,” Kybernetika, vol. 34, no. 3, pp. 349–357, 1998.
• [9] J. Wang, M. Feckan, and Y. Zhou, “Controllability of Sobolev type fractional evolution systems,” Dyn. Partial Differ. Equ., vol. 11, no. 1, pp. 71–87, 2014.
• [10] M. Feckan, J. Wang, and J. Zhou, “Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators,” J. Optim. Theory Appl., vol. 156, pp. 79–95, 2013.
• [11] N. Mahmudov, “Existence and approximate controllability of Sobolev type fractional stochastic evolution equations,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 62, no. 2, pp. 1–11, 2014, doi: 10.2478/bpasts-2014-0020.
• [12] N. Mahmudov, “Approximate controllability of fractional Sobolev-type evolution equations in Banach spaces,” Abstract Appl. Anal., vol. 2013, 2013, doi: 10.1155/2013/502839.
• [13] Y. Chang, R. Ponce, and S. Rueda, “Fractional differential equations of Sobolev type with sectorial operators,” Semigroup Forum, vol. 99, pp. 591–606, 2019.
• [14] J. Wang and X. Li, “A uniform method to ulam–hyers stability for some linear fractional equations,” Mediterr. J. Math., vol. 13, pp. 625–635, 2016.
• [15] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science, 2006.
• [16] R. Almeida, “A Caputo fractional derivative of a function with respect to another function,” Commun. Nonlinear Sci. Numer. Simul., vol. 44, pp. 460–481, 2017. doi: 10.1016/j.cnsns.2016.09.006.
• [17] M. Aydin, N. Mahmudov, H. Aktuğlu, E. Baytunç, and M. Atamert, “On a study of the representation of solutions of a 𝜓-Caputo fractional differential equations with a single delay,” Electron. Res. Arch., vol. 30, no. 3, pp. 625–635, 2022, doi: 10.3934/era.2022053.
• [18] N. Mahmudov, A. Ahmadova, and I. Huseynov, “A novel technique for solving Sobolev-type fractional multi-order evolution equations,” Comput. Appl. Math., vol. 41, no. 71, 2022, doi: 10.1007/s40314-022-01781-x.
• [19] Y. Gambo, F. Jarad, D. Baleanu, and A. T, “On Caputo modification of the hadamard fractional derivatives,” Adv. Differ. Equ., vol. 10, no. 2014, 2014, doi: 10.1186/1687-1847-2014-10.
• [20] F. Jarad, T. Abdeljawad, and D. Baleanu, “Caputo-type modification of the hadamard fractional derivatives,” Adv. Differ. Equ., vol. 142, no. 2012, 2012, doi: 10.1186/1687-1847-2012-142.
• [21] Y. Luchko and J. Trujillo, “Caputo-type modification of the erdélyi–kober fractional derivative,” Fract. Calc. Appl. Anal., vol. 10, no. 3, pp. 249–267, 2016.
• [22] R. Almeida, A. Malinowska, and M. Monteiro, “Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications,” Math. Meth. Appl. Sci., vol. 41, no. 1, pp. 336–352, 2018.
• [23] F. Jarad and T. Abdeljawad, “Generalized fractional derivatives and laplace transform,” Discret. Contin. Dyn. Syst.-Ser. S, vol. 13, no. 3, pp. 709–722, 2020, doi: 10.3934/dcdss.2020039.
• [24] A. Ansari, “The generalized laplace transform and fractional differential equations of distributed orders,” J. Differ. Equ. Control Process., vol. 2012, no. 3, pp. 1–11, 2012.
• [25] H. Fahad, M. Rehman, and A. Fernandez, “On laplace transforms with respect to functions and their applications to fractional differential equations,” Math. Meth. Appl. Sci., vol. 46, no. 7, pp. 8304–8323, 2021, doi: 10.1002/mma.7772.
• [26] T. Prabhakar, “A singular integral equation with a generalized mittag-leffler function in the kernel,” Yokohama Math. J., vol. 19, pp. 7–15, 1971.
• [27] A. Fernandez, C. Kürt, and M. Özarslan, “A naturally emerging bivariate mittag- leffler function and associated fractionalcalculus operators,” Comput. Appl. Math., vol. 39, 2020, doi: 10.1007/s40314-020-01224-5.
• [28] E. Kreyszig, Introductory Functional Analysis with Applications. New York: Jon Wiley & Sons, 1978.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).

Identyfikatory

DOI

10.24425/bpasts.2024.149170