Existence and Uniqueness of Abstract Stochastic Fractional-Order Differential Equation

  • Mahmoud Mohammed Mostafa El-Borai Alexandria University
  • A. Tarek S.A. Alexandria University
Keywords: Stochastic Differential Equations, Fractional Order, Existance and uniqueness theorems, Abstract Differential Equations

Abstract

In this paper, the existence and uniqueness about the solution for a class of abstract stochastic fractional-order differential equations

                                         Capture_new1.PNG

where Capture_new2.PNG in and Capture_new31.PNG are given functions, are investigated, where the fractional derivative is described in Caputo sense. The fractional calculus, stochastic analysis techniques and the standard $Picard's$ iteration method are used to obtain the required.

Author Biographies

Mahmoud Mohammed Mostafa El-Borai, Alexandria University

Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt

A. Tarek S.A., Alexandria University

Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt

References

Hilfer R, Application of fractional calculus in physics, New Jersey: World Scientic, (2001).

Sabatier J, Agrawal O P, Machado J A T, Advances in fractional calculus,

Omar K. Jaradat, Ahmad Al-Omari, Shaher Momani, Existence of the mild solution for fractional semi-linear initial value problems, Nonlinear Analysis, 69(2008), 3153-3159.

J. Barrett, Dierential equations of non-integer order, Canad.J.Math.,

I. Podlubny, Fractional Dierential Equations, Acadimic Press, New York, (1999).

Mahmoud M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons and fractals, 14(2002), 433- 440.

El-Borai, M.M., On some fractional differential equations in the Hilbert space, Journal of Discrete and Continuous Dynamical systems, Series A (2005), 233-241.

El-Borai, M.M., The fundamental solutions for fractional evolution equations of parabolic type, International Journal of Stochastic Analysis, 3(2004), 197- 211.

Ahmed, H.M., El-Borai, M.M., Hilfer fractional stochastic integro- differential equations, Applied Mathematics and Computation, (2018), 182- 189.

El-Borai, M.M., On the correct formulation of the cauchy problem, Vestnik Moskov Univ. Scr. I Mat. Mch 23 (1968), 15-21.

El-Borai, M.M., On some stochastic fractional integro-dierential equa- tions, Advances in Dynamical Systems and Applications, 1(2006), 49-57.

El-Borai, M.M., El-Nadi, K.E-S., Ahmed, H.M., El-Owaidy, H.M., Ghanem, A.S., Sakthivel R., Existence and stability for fractional parabolic

integro-partial dierential equations with fractional brownian motion and non- local condition, Cogent Mathematics statistics, 1(2018), 146-166.

El-Borai, M.M., El-said, K., A parabolic transform and some stochastic ill-posed problems, British Journal of Mathematics and Computer, (2015), 418-426.

El-Nadi, K.E., On some stochastic parabolic dierential equations in a Hilbert space, International Journal of Stochastic Analysis, 2(2005), 167-173.

Hille E, Philips RS, Functional analysis and semigroup, American Mathematical Society colloquium Publication, Vol.31. Providence(RI): American Mathematical Society; (1957).

Mainradi F., The fundamental solutions for the fractional diusion-wave equation, Appl Math lett, 9(6)(1996).

W. Feller, An Introduction to Probability Theory and Its Applications, Vol.II, John WileySons, New York, (1971).

A. Friedman, Parial Dierental Equations, Holt, Rinehart and Winston, New York, (1969).

I. M.Gelfand and G. E.Shilov, Generalized Functions, Vol.I: Properties and Operations, Nauka, Moscow, (1959).

S. Kawatsu, Cauchy Problem for abstract evolution equations of Parabolic type, J. Math. Kyoto Univ.30(1990), no.1, 59-91.

W. Wyss, The fractional diffusion equation, J. Math. Phys., 27(1986), no.11, 2782-2785.

Bashir Ahmad, Ahmed Alsaedi, Sotiris K. Ntouyas, Jessada Tariboon, Hadamard-Type Fractional Dierential Equations, Inclusions and Inequali- ties, Springer International Publishing, (2017).

Khan N A, Ara A, Mahmood A., Approximate solution of time-fractional

chemical engineering equations: a comparative study, Int. J. Chem. Reactor Eng., 8(2010) Article A19.

Oldham K B., Fractional differential equations in electrochemistry, Adv. Eng. Softw., 41(1) (2010), 9-12.

D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204(1996), 609-625.

J.H. He, Some applications of nonlinear fractional dierential equations and their approximations, Bull. Sci. Technol., 15(2)(1999), 86-90.

Oksendal B., Stochastic differential equations: an introduction with appli- cations, Springer Science Business Media, (2013).

Mao X., Stochastic differential equations and applications, Elsevier, (2007).

Anatoly A. Kilbas, Hari M. Srivastava, Juan J. Trujillo, Theory and Ap- plications of Fractional Dierential Equations, Elsevier, (2006).

Guang-an Zoua, Bo Wanga, On the study of stochastic fractional-order differential equation systems, arXiv:1611.07618v1, (2016).

Published
2019-01-31
How to Cite
El-Borai, M. M., & S.A., A. T. (2019). Existence and Uniqueness of Abstract Stochastic Fractional-Order Differential Equation. JOURNAL OF ADVANCES IN MATHEMATICS, 16, 8280-8287. Retrieved from https://cirworld.com/index.php/jam/article/view/8097
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