Mild solutions of a fractional partial differential equation with noise
Noureddine Bouteraa
Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Oran, Algeria
Search for more papers by this authorCorresponding Author
Mustafa Inc
Department of Mathematics, Science Faculty, Firat University, Elazig, Turkey
Department of Medical Research, China Medical University, Taichung, Taiwan
Correspondence
Mustafa Inc, Science Faculty, Department of Mathematics, Firat University, Elazig 23119 , Turkey.
Email: [email protected]
Communicated by: M. Kirane
Search for more papers by this authorMehmet Ali Akinlar
Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey
Search for more papers by this authorBandar Bin-Mohsin
Department of Mathematics, College of Science, Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia
Search for more papers by this authorNoureddine Bouteraa
Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Oran, Algeria
Search for more papers by this authorCorresponding Author
Mustafa Inc
Department of Mathematics, Science Faculty, Firat University, Elazig, Turkey
Department of Medical Research, China Medical University, Taichung, Taiwan
Correspondence
Mustafa Inc, Science Faculty, Department of Mathematics, Firat University, Elazig 23119 , Turkey.
Email: [email protected]
Communicated by: M. Kirane
Search for more papers by this authorMehmet Ali Akinlar
Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey
Search for more papers by this authorBandar Bin-Mohsin
Department of Mathematics, College of Science, Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia
Search for more papers by this authorAbstract
This article focuses on studying mild solutions of an original fractional partial differential equation disturbed by multiplicative white noise. We employ techniques of semigroup theory, Hausdorff measure, and Darbo fixed point theorem.
REFERENCES
- 1Gomez-Aguilar JF, Atangana Abdon. New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications. Eur Phys J Plus. 2017; 132(1): 1-13.
- 2Gomez-Aguilar JF, Yepez-Martínez H, Torres-Jimenez J, Cordova-Fraga T, Escobar-Jimenez RF, Olivares-Peregrino VH. Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel. Adv Differ Equa. 2017; 2017(1): 68.
- 3Jiang Y, Wei T, Zhou X. Stochastic generalized Burgers equations driven by fractional noises. J Differential Equations. 2012; 252(2): 1934-1961.
- 4Momani S. Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos Soliton Fract. 2006; 28: 930-937.
- 5Morales-Delgado 1 VF, Gomez-Aguilar JF, Yepez-Martínez H, Baleanu D, Escobar-Jimenez RF, Olivares-Peregrino VH. Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular. Adv Differ Equa. 2016; 1: 17.
- 6Saad KM, Khader MM, Gomez-Aguilar JF, Baleanu D. Numerical solutions of the fractional Fisher's type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods. Chaos: Interdiscipl J Nonlin Sci. 2019; 29(2): 1-13.
- 7Yang XJ. Advanced Local Fractional Calculus and Its Applications. New York: World Science; 2012.
- 8Yepez-Martineza H, Gomez-Aguilarb JF, Sosaa IO, Reyesa JM, Torres-Jimenez J. The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation. Rev Mex Fís. 2016; 62(4): 310-316.
- 9Yepez-Martnez H, Gomez-Aguilar JF, Atangana Abdon. First integral method for non-linear differential equations with conformable derivative. Math Modell Nat Phenom. 2018; 13(1): 14.
- 10Cui J, Yan L. Existence result for fractional neutral stochastic integro-differential equation with infinite delay. J Phys A Math Theor. 2011; 44: 335201.
- 11Djourdem H, Bouteraa N. Mild solution for a stochastic partial differential equation with noise. WSEAS Trans Syst. 2020; 19: 246-256.
10.37394/23202.2020.19.29 Google Scholar
- 12Erraoui M, Nualart D, Ouknine Y. Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet. Stoch Dyn. 2003; 3: 121-139.
10.1142/S0219493703000681 Google Scholar
- 13Hosokawa I, Yamamoto K. Turbulence in the randomly forced one dimensional Burgers flow. J Stat Phys. 1975; 13: 245-272.
- 14Hu Y. Heat equation with fractional white noise potential. Appl Math Optim. 2001; 43: 221-243.
- 15Hu Y, Nualart D. Stochastic heat equation driven by fractional noise and local time. Probab Theory Related Fields. 2009; 143: 285-328.
- 16Rogovchenko YV. Nonlinear impulse evolution systems and applications to population models. J Math Anal Appl. 1997; 207: 300-315.
- 17Chemin JY, Gallagher I, Paicu M. Global regularity for some classes of large solutions to the Navier-Stokes equations. Ann Math. 2011; 173(2): 983-1012.
- 18Jeng DT. Forced model equation for turbulence. Phys Fluids. 1969; 12(10): 2006-2010.
- 19Inc M. The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. J Math Anal Appl. 2008; 345(1): 476-484.
- 20Kruse R. Strong and Weak Approximation of Semilinear Stochastic Evolution Equations. Switzerland: Springer; 2014.
10.1007/978-3-319-02231-4 Google Scholar
- 21Zou G, Wang B. Stochastic Burgers equation with fractional derivative driven by multiplicative noise. Comput Math Appl. 2017; 74(12): 3195-3208. https://doi.org/10.1016/j.camwa.2017.08.023
- 22Banas J, Goebel K. Measure of Noncompactness in Banach Space. Lecture Notes in Pure and Applied Mathematics. Vol 60. New York: Marcel Dekker, Inc.; 1980.
- 23Agarwal R, Meehan M, O'Regan D. Fixed point theory and applications. Cambridge Tracts in Mathematics. Cambridge, United Kingdom: Cambridge University Press; 2001: 78-94.
10.1017/CBO9780511543005 Google Scholar
