Abstract
In this paper, we aim to prove the existence and uniqueness of the mean square solution to the Caputo fractional random boundary value problem by using the fixed point theorems. Moreover, we introduce the Ulam–Hyers stability and generalized Ulam–Hyers stability for this problem. Finally, we give an example to illustrate our results.
Similar content being viewed by others
References
Burgos, C., Cortes, J.C., Villafuerte, L., Villanueva, R.J.: Mean square calculus and random linear fractional differential equations: theory and applications. Appl. Math. Nonlinear Sci. 2(2), 317–328 (2017). https://doi.org/10.21042/AMNS.2017.2.00026
Burgos, C., Cortes, J.C., Villafuerte, L., Villanueva, R.J.: Solving random mean square fractional linear differential equations by generalized power series : analysis and computing. J. Comput. Appl. Math. 2018(339), 94–110 (2018). https://doi.org/10.1016/j.cam.2017.12.042
Burton, T.A.: A fixed-point theorem of Krasnoselskii. Appl. Math. Lett. 11(1), 85–88 (1998). https://doi.org/10.1016/S0893-9659(97)00138-9
Dong, L.S., Hoa, N.V., Ho, V.: Existence and Ulam stability for random fractional integro-differential equation. Afrika Matematika 31(7), 1283–1294 (2020). https://doi.org/10.1007/s13370-020-00795-0
El-Sayed, A.M.A.: The mean square Riemann–Liouville stochastic fractional derivative and stochastic fractional order differential equation. Math. Sci. Res. J. 9(6), 142–150 (2005)
El-Sayed, A.M.A.: On the stochastic fractional calculus operators. J. Fract. Calc. Appl. 6(2), 101–109 (2015)
El-Sayed, A.M.A., Fouad, H.A.: On a coupled system of stochastic ito-differential and the arbitrary (fractional) order differential equations with nonlocal random and stochastic integral conditions. Mathematics 9(20), 2571 (2021). https://doi.org/10.3390/math9202571
El-Sayed, A.M.A., Fouad, H.A.: On a coupled system of random and stochastic nonlinear differential equations with coupled nonlocal random and stochastic nonlinear integral conditions. Mathematics 9(17), 2111 (2021). https://doi.org/10.3390/math9172111
Hafiz, F.M.: The fractional calculus for some stochastic processes. Stoch. Anal. Appl. 22(2), 507–523 (2004). https://doi.org/10.1081/SAP-120028609
Hafiz, F.M., El-Sayed, A.M.A., El-Tawil, M.A.: On a stochastic fractional calculus. Fracti. Calc. Appl. Anal. 4(1), 81–90 (2001)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin, Heidelberg (1981)
Hilfer, R.: Applications of fractional calculus in physics. World Scientific Publishing Company (2000)
Ho, V., Ngo, V.H.: On initial value problem of random fractional differential equation with impulses. Hacettepe J. Math. Stat. 49(1), 282–293 (2020). https://doi.org/10.15672/hujms.546989
Ionescu, C., Lopes, A., Copot, D., Machado, J.A.T., Bates, J.H.T.: The role of fractional calculus in modeling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul 2017(51), 141–159 (2017). https://doi.org/10.1016/j.cnsns.2017.04.001
Khursheed, J. A., Asma, Fatima I., Kamal, S., Aziz, K., Thabet, A.: On new updated concept for delay differential equations with piecewise Caputo fractional-order derivative. Waves Rand. Complex Media (2023). https://doi.org/10.1080/17455030.2023.2187241
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2004)
Mahmoud, M.B., Khairia, E.N., Fouad, H.A.: On some fractional stochastic delay differential equations. Comput. Math. Appl. 59(3), 1165–1170 (2010). https://doi.org/10.1016/j.camwa.2009.05.004
Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterran. J. Math. 12(3), 1123–1140 (2015). https://doi.org/10.1007/s11075-020-00967-w
Shen, X.: Applications of Fractional Calculus in Chemical Engineering. University of Ottawa, Ottawa (2018)
Slimane, I., Dahmani, Z.: A continuous and fractional derivative dependance of random differential equations with nonlocal conditions. J. Interdiscip. Math. 24(5), 1457–1470 (2021). https://doi.org/10.1080/09720502.2020.1868661
Soong, T.T.: Random differential equations in science and engineering. Academic Press, New York City (1973)
Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.Q.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 2018(64), 213–231 (2018). https://doi.org/10.1016/j.cnsns.2018.04.019
Traore, A., Sene, N.: Model of economic growth in the context of fractional derivative. Alexand. Eng. J. 59(6), 4843–4850 (2020). https://doi.org/10.1016/j.aej.2020.08.047
Yfrah, H., Dahmani, Z., Tabharit, L., Abdelnebi, A.: High order random fractional differential equations: Existence, uniqueness and data dependence. J. Interdiscip. Math. (2021). https://doi.org/10.1080/09720502.2020.1860291
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Vu, H. On the Caputo fractional random boundary value problem. Afr. Mat. 34, 79 (2023). https://doi.org/10.1007/s13370-023-01121-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13370-023-01121-0