A Sequential Random Airy Type Problem of Fractional Order: Existence, Uniqueness and ß-Differential Dependance

Year 2021, , 277 - 286, 30.09.2021

Zoubir Dahmani , Yfrah Hafssa

https://doi.org/10.31197/atnaa.891115

Abstract

In this work, a new class of sequential random differential equations of Airy type is introduced. An existence and uniqueness criteria for stochastic process solutions for the introduced class is discussed. Some notions on β−differential dependance are also introduced. Then, new results on the β−dependance are discussed. At the end, some illustrative examples are discussed.

Keywords

Airy equation, mean square calculus, random differential equation

References

• [1] S. Abbas, N. Al Arifi, M. Benchohra, J. Graef, Random coupled systems of implicit Caputo-Hadamard fractional differential equations with multi-point boundary conditions in generalized Banach spaces, Dynamics Systems and Applications. 28(2) (2019) 329-350.
• [2] S. S. Alshehri, Properties of Airy functions and application to the V-Shape potential, MECSJ. 15 (2018).
• [3] C. Burgos, J. C. Cort¨s, A. Debbouche, L. Villafuerte and R.J. Villanueva, Random fractional generalized Airy differential equations/ a probabilistic analysis using mean square calculus, Applied Mathematics and Computation.352 (2019) 15-29.
• [4] C. Burgos, J. C. Cort¨s, M.D. Rosello and R.J. Villanueva, Some tools to study random fractional differential equations and applications, Springer. (2020)
• [5] Z. Dahmani and M.A. Abdellaouil, On a three point boundary value problem of arbitrary order, Journal of Interdisciplinary Mathematics.19(5-6) (2016) 893-906.
• [6] Z. Dahmani, M.A. Abdelaoui and M. Houas, Polynomial solutions for a class of fractional differential equations and systems, Journal of Interdisciplinary Mathematics. 21(3) (2018) 669-680.
• [7] Z. Dahmani and L. Marouf, Numerical study of differential equation governing speech gestures with Caputo derivative, Journal of Interdisciplinary Mathematics. 16(4-5) (2013) 287-296.
• [8] D. B. Dhaigude, S. G. Jadhav, L.J . Mahmood, Solution of space time fractional partial differential equations by Adomian decomposition method, Bulletin of the Marathwada Mathematical Society.15(1) (2014) 26-37.
• [9] A. M. A. El-Sayed, The mean square Riemann-Liouville stochastic fractional derivative and stochastic fractional order differential equation, Math. Sei. Res. J.
• [10] A.M.A. El-Sayed, On the stochastic fractional calculus operators, Journal of Fractional Calculus and Applications.6(1) (2015) 101-109.
• [11] 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.7(1) (2017) 135-149.
• [12] R. Gorenflo, F. Mainardi, Essentials of fractional calculus, Maphysto Center. (2000). [13] F.M. Hafiz, The fractional calculus for some stochastic processes, Stoch. Anal. Appl. 22 (2004) 507-523.
• [14] 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.
• [15] N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional diffrential equations with Reimann-Liouville fractional derivatives, Rheologica Acta. 45(5) (2006) 765-771.
• [16] J. S. Jacob, J. H. Priya, A. Karthika, Applications of fractional calculus in science and engineering, JCR. 7(13) (2020) 4385-4394.
• [17] V. Ho, Random fractional functional differential equations, International journal of nonlinear analysis and applications.7(2) (2016) 253-267.
• [18] H. M. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North Holland Math. Stud. Elsivier, Amsterdam.204 (2006).
• [19] V. Kiryakova,The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Computers and Mathematics with applications. 59 (2010). 1128-1141.
• [20] V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of fractional dynamic systems, Cambridge Scientific Publishers, Cambridge. (2009).
• [21] V. Lakshminarayanan, L. S. Varadharajan, Special functions for optical science and engineering, SPIE. (2015).
• [22] L. D. Landau, E. M. Lifshitz, Quantum mechanics: non-relativistic theory, Institute of physical problems, U.S.S.R. Academy of sciences. (1965).
• [23] A. Loverro, Fractional calculus: History, definitions and applications for the engineer, Institute of physical problems, U.S.S.R. Academy of sciences. (2004).
• [24] F. Mainardi, A. Mura and G. Pagnini, The M-Wright function in time-fractional diffusion processes: a tutorial survey, International Journal of Differential Equations. 2010.
• [25] M. D Ovidio, E. Orsingher and B. Toaldo, Time-chenged processes governed by spacetime fractional telegraph equations, Math.PR. (2013).
• [26] S. Pitts, Mean-square fractional calculus and some applications, Scool of Mathematics, Statistics and Computer Science University of Kwazulu-Natal. (2012).
• [27] V. E. Tarasov, Mathematical economics: application of fractional calculus, Mathematics. 8(5) (2020) 660.
• [28] O. Vallee, M. Soares, Airy functions and applications to physics, Imperial College Press, London. (2010).
• [29] A. Vinodkumar, K. Malar, M. Gowrisankar, P. Mohankumar, Existence, uniqueness and stability of random impulsive fractional differential equations, Acta Mathematica Scientia.36(2) (2016) 428-442.
• [30] 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.
• [31] H. Yfrah, Z. Dahmani, L. Tabharit and A. Abdelenbi, High order random fractional differential equations: existence, uniqueness and data dependence, Journal of Interdisciplinary Mathematics. (2020) Accepted.