A Sequential Random Airy Type Problem of Fractional Order: Existence, Uniqueness and ß-Differential Dependance
Yıl 2021,
Cilt: 5 Sayı: 3, 277 - 286, 30.09.2021
Zoubir Dahmani
,
Yfrah Hafssa
Öz
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.
Kaynakça
- [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.
Yıl 2021,
Cilt: 5 Sayı: 3, 277 - 286, 30.09.2021
Zoubir Dahmani
,
Yfrah Hafssa
Kaynakça
- [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.