Nonlinear Random Differential Equations with n Sequential Fractional Derivatives

Hafssa, Yfrah; Dahmani, Zoubir

doi:10.2478/mjpaa-2022-0001

Abstract

This article deals with a solvability for a problem of random fractional differential equations with n sequential derivatives and nonlocal conditions. The existence and uniqueness of solutions for the problem is obtained by using Banach contraction principle. New random data concepts for the considered problem are introduced and some related definitions are given. Also, some results related to the dependance on the introduced data are established for both random and deterministic cases.

References

[1] S. Abbas, M. Benchohra, J.R. Greaf and J. Henderson, Implicit fractional differential and integral equations: Existence and Stability. De Gryter Series in Nonlinear Analysis and Applications, (2018)10.1515/9783110553819
Search in Google Scholar Back to article
[2] G.B. Airy, On the intensity of light in the neighbourhood of a caustic, Trans. Cambridge Philos. Soc., 6 (1838), 379-402
Search in Google Scholar Back to article
[3] A. Benzidane and Z. Dahmani, A class of nonlinear singular differential equations, Journal of Interdisciplinary Mathematics, 6 (2019), 991-100710.1080/09720502.2019.1698805
Search in Google Scholar Back to article
[4] S.P. Bhairat, D.B. Dhaigude, Existence of solutions of generalized fractional differential equation with nonlocal initial condition, Mathematica Bohemica, 2 (2019), 203-22010.21136/MB.2018.0135-17
Search in Google Scholar Back to article
[5] C. Burgos, J.C. Cortes, A. Debbouche, L. Villafuerte, R.J. Villanueva, Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus, Applied Mathematics and Computation, 352 (2019), 15-2910.1016/j.amc.2019.01.039
Search in Google Scholar Back to article
[6] A. Chadha, R. Sakthivel, S.N. Bora, Solvability of control problem for fractional nonlinear differential inclusions with nonlocal conditions, Nonlinear Analysis: Modelling and Control, 4 (2019)10.15388/NA.2019.4.2
Search in Google Scholar Back to article
[7] Z. Dahmani, M.A. Abdelaoui and M. Houas, Polynomial solutions for a class of fractional differential equations and systems, Journal of Interdisciplinary Mathematics, 3 (2018), 669-68010.1080/09720502.2016.1142723
Search in Google Scholar Back to article
[8] M. Dahmani, A. Anber and Z. Dahmani, Speech movements on vocal tract : Fractional nonlinear dynamics, Journal of Information and Optimization Sciences, 6 (2019), 1307-131510.1080/02522667.2018.1499600
Search in Google Scholar Back to article
[9] Z. Dahmani, M.M. Belhamiti, Integral inequalities and differential equations via fractional calculus, IntechOpen, (2020)10.5772/intechopen.91140
Search in Google Scholar Back to article
[10] M.Di Paola, J. N. Reddy and E. Ruocco, On the applications of fractional calculus for the formulation of viscoelastic Reddy beam, Mathmatics, 5 (2020), 1365-137810.1007/s11012-020-01177-3
Search in Google Scholar Back to article
[11] A.M.A. El-Sayed, The mean square Riemann-Liouville stochastic fractional derivative and stochastic fractional order differential equation, Math. Sei. Res. J., 9 (2005), 142-150
Search in Google Scholar Back to article
[12] A.M.A. El-Sayed, On the stochastic fractional calculus operators, Journal of Fractional Calculus and Applications, 1 (2015), 101-109
Search in Google Scholar Back to article
[13] A.M.A. El- Sayed, F. Gaafar and M. El-Gendy, Continuous dependence of the solution of random fractional-order differential equation with nonlocal conditions, J. Fractional Differential Calculus, 1 (2017), 135-14910.7153/fdc-07-03
Search in Google Scholar Back to article
[14] F.M. Hafiz, The fractional calculus for some stochastic processes, Stoch. Anal. Appl, 22 (2004), 507-52310.1081/SAP-120028609
Search in Google Scholar Back to article
[15] F.M. Hafiz, A.M.A. El-Sayed, and M.A. El-Tawil, On a stochastic fractional calculus, Frac. Calc; Appl. Anal, 4 (2001), 81-90
Search in Google Scholar Back to article
[16] M. Houas and Z. Dahmani, New results for a system of two fractional differential equations involving n Caputo derivatives, Kragujevac Journal of Mathematics, 2 (2014), 283-30110.5937/KgJMath1402283H
Search in Google Scholar Back to article
[17] D.C. Labora, R.R. López, M. Belmekki, Existence of solutions to nonlocal boundary value problems for fractional differential equations with impulses, Electronic Journal of Differential Equations, 15 (2020), 1-1610.58997/ejde.2020.15
Search in Google Scholar Back to article
[18] V. Lupulescu, S.K. Ntouyas, Random fractional differential equations, International Electronic Journal of Pure and Applied Mathematics, 2 (2012), 119-136
Search in Google Scholar Back to article
[19] V. Lupulescu, D. O’Regan, and G. Rahman, Existence results for random fractional differential equations, Opuscula Mathematica, 4 (2014), 813-82510.7494/OpMath.2014.34.4.813
Search in Google Scholar Back to article
[20] M.Z. Sarikaya, M. Bezziou and Z. Dahmani, New operators for fractional integration theory with some applications, Journal of Mathematical Extension, 1 (2018), 87-100
Search in Google Scholar Back to article
[21] I. Slimane and Z. Dahmani, A continuous and fractional derivative dependence of random differential equations with nonlocal conditions, Journal of Interdisciplinary Mathematics, Accepted (2021)10.1080/09720502.2020.1868661
Search in Google Scholar Back to article
[22] V.E. Tarasov, Mathematical Economics : Applications of fractional calculus,Mathematics, 5 (2020), 66010.3390/math8050660
Search in Google Scholar Back to article
[23] S. Tate and H.T. Dinde, Existence and uniqueness results for nonlinear implicit fractional differential equations with non local conditions, Palestine journal of Mathematics, 1 (2020), 212-219
Search in Google Scholar Back to article
[24] H. Vu, H.V. Ngo, On initial value problem of random fractional differential equation with impulses, Hacettepe Journal of Mathematics and Statistics, 1 (2020), 282-29310.15672/hujms.546989
Search in Google Scholar Back to article
[25] H. Yfrah, Z. Dahmani, M.Z. Sarikaya and F.A. Gujar, A sequential nonlinear random fractional differential equation: existence, uniqueness and new data dependence, Submitted.
Search in Google Scholar Back to article
[26] H. Yfrah, Z. Dahmani, L. Tabharit and A. Abdelenbi, High order random fractional differential equations: existence, uniqueness and data dependence, Journal of Interdisciplinary of Mathematics, Accepted, (2020)10.22541/au.159031805.52880067
Search in Google Scholar Back to article

Nonlinear Random Differential Equations with n Sequential Fractional Derivatives

Abstract

Paradigm

My account