Stochastic sub-diffusion equation with conformable derivative driven by standard Brownian motion

Year 2021, Volume: 5 Issue: 3, 287 - 299, 30.09.2021

Ngo Hung Ho Binh Nguyen Luc , An Nguyen Thı Kıeu Le Dinh Long

https://doi.org/10.31197/atnaa.906952

Cited By: 3

Abstract

This article is concerned with a forward problem for the following sub-diffusion equation driven by standard Brownian motion
\begin{align*}
\left( ^{\mathcal C} \partial^\gamma_t + A \right) u(t) = f(t) + B(t) \dot{W}(t), \quad t\in J:=(0,T),
\end{align*}
where $^{\mathcal C} \partial^\gamma_t$ is the conformable derivative, $\gamma \in (\frac{1}{2},1].$ Under some flexible assumptions on $f,B$ and the initial data, we investigate the existence, regularity, continuity of the solution on two spaces $L^r(J;L^2(\Omega,\dot{H}^\sigma))$ and $C^\alpha(\overline{J};L^2(\Omega,H))$ separately.

Keywords

Diffusion equation, Standard Brownian motion, Fractional Brownian motion, Existence and regularity., Conformable derivative

References

• [1] T. Abdeljawad, On conformable fractional calculus, J Comput Appl Math, 279 (2015): 57-66.
• [2] R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Mathematical Methods in the Applied Sciences, 2020, https://doi.org/10.1002/mma.665
• [3] H. Afshari, E, Karapinar, A discussion on the existence of positive solutions of the boundary value problems via-Hilfer fractional derivative on b-metric spaces, Advances in Di?erence Equations, 1 (2020): 1-11.
• [4] H.Afshari, S. Kalantari, E. Karapinar; Solution of fractional differential equations via coupled fixed point, Electronic Journal of Differential Equations, 286 (2015): 2015.
• [5] B.Alqahtani, H. Aydi, E. Karapinar, V. Rakocevic, A Solution for Volterra Fractional Integral Equations by Hybrid Contractions, Mathematics, 7.8 (2019): 694.
• [6] E. Karapinar, A.Fulga, M. Rashid, L.Shahid, H. Aydi, Large Contractions on Quasi-Metric Spaces with an Application to Nonlinear Fractional Differential-Equations, Mathematics, 7.5 (2019): 444.
• [7] A.Salim, B. Benchohra, E. Karapinar, J. E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional di?erential equations, Adv Di?er Equ. 2020.1 (2020): 1-21.
• [8] E. Karapinar, T.Abdeljawad, F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations, Advances in Difference Equations, 1 (2019): 1-25.
• [9] A. Abdeljawad, R.P. Agarwal, E. Karapinar, P.S. Kumari, Solutions of he Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space, Symmetry, 11.5 (2019): 686.
• [10] N. Arfin, A.C. Yadav, H B. Bohidar, Sub-diffusion and trapped dynamics of neutral and charged probes in DNA-protein coacervates, AIP Advances, 3.11 (2013): 13192.
• [11] F.M. Alharbi, D. Baleanu, A. Ebaid, Physical properties of the projectile motion using the conformable derivative, Chin J Phys. 58 (2019): 18-28.
• [12] A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math, 13.1 (2015): 889?898.
• [13] D. Baleanu, F. Jarad, E. Ugurlu, Singular conformable sequential differential equations with distributional potentials, Quaestiones Mathematicae, 42.3 (2019): 277-287.
• [14] D. Baleanu, A. Mousalou, S. Rezapour (2017), A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative, Adv. Difference Equ., 2017.1 (2017): 1-12.
• [15] M. Bohner, V.F. Hatipoglu, Dynamic cobweb models with conformable fractional derivatives, Nonlinear Analysis: Hybrid Systems, 32 (2019): 157-167.
• [16] M. Bohner, V. F. Hatipoglu, Cobweb model with conformable fractional derivatives, Mathematical Methods in the Applied Sciences, 41.18 (2018): 9010-9017.
• [17] H. Brezis, Functional analysis, sobolev spaces and partial differential equations, New York: Springer, (2010).
• [18] Y. Cai, H.E. Wang, S.Z. Huang, J. Jin, C. Wang, B. L. Su (2015), Hierarchical nanotube-constructed porous TiO 2-B spheres for high performance lithium ion batteries, Scienti?c reports, 5.1 (2015): 1-8.
• [19] V. Capasso, D. Morale, Stochastic modelling of tumour-induced angiogenesis, Journal of Mathematical Biology, 58.1 (2009): 219-233.
• [20] A.N. Carvalho, J.W. Cholewa, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J Math Anal Appl. 310.2 (2005): 557-578.
• [21] A.N. Carvalho, J.W. Cholewa, J.D.M. Nascimento, On the continuation of solutions of non-autonomous semilinear parabolic problems, Proc Edinb Math Soc. 59.1 (2016): 17-55.
• [22] Chavez A, Fractional diffusion equation to describe Lévy ?ights, Phys Lett A. 239.1-2 (1998): 13-16.
• [23] W.S. Chung, Fractional newton mechanics with conformable fractional derivative, J Comput Appl Math. 290 (2015): 150-158. December.
• [24] W.S. Chung, H. Hassanabadi, Dynamics of a Particle in a Viscoelastic Medium with Conformable Derivative, International Journal of Theoretical Physics. 56.3 (2017): 851-862.
• [25] R. F. Curtain, P. L. Falb, Stochastic differential equations in Hilbert space, Journal of Di?erential Equations, 10.3 (1971): 412-430.
• [26] L. Debbi, Well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on the torus and on bounded domains, J. Math. Fluid Mech. 18.1 (2016): 25-69.
• [27] J.L.A. Dubbeldam, A. Milchev, V.G. Rostiashvili, T.A. Vilgis, Polymer translocation through a nanopore: A showcase of anomalous diffusion, Physical Review E, 76.1 (2007): 010801.
• [28] S. Fedotov, V. Méndez, Non-Markovian model for transport and reactions of particles in spiny dendrites, Physical review letters, 101.21 (2008): 218102.
• [29] A. Jaiswal, D. Bahuguna, Semilinear conformable fractional di?erential equations in banach spaces, Differ Equ Dyn Syst, 27.1 (2019): 313-325.
• [30] Y. Jiang, T. Wei, X. Zhou, Stochastic generalized Burgers equations driven by fractional noises, J. Differ. Equ. 252.2 (2012): 1934-1961.
• [31] I. Golding, E. C. Cox, Physical nature of bacterial cytoplasm, Physical review letters, 96.9 (2006): 098102.
• [32] S. He, K. Sun, X. Mei, B. Yan, S. Xu, Numerical analysis of a fractional-order chaotic system based on conformable fractional-order derivative, Eur Phys J Plus. 132.1 (2017): 1-11.
• [33] S. Hanot, S. Lyonnard, S. Mossa (2016), Sub-diffusion and population dynamics of water con?ned in soft environments, Nanoscale, 8.6 (2016): 3314-3325.
• [34] G. Hu, Y. Lou and P.D. Christofides, Dynamic output feedback covariance control of stochastic dissipative partial di?er- ential equations, Chem. Eng. Sci. 63.18 (2008): 4531-4542.
• [35] Y. Kantor, M. Kardar, Anomalous dynamics of forced translocation, Physical Review E, 69.2 (2004): 021806.
• [36] AR. Khalil, A. Yousef, M. Sababheh, A new definition of fractional derivative, J Comput Appl Math, 264 (2014): 65-70. [37] P.D. Lax, Functional Analysis, Wiley Interscience, New York, (2019).
• [38] C. Li, G. Chen, Chaos in the fractional order Chen system and its control, Chaos, Solitons & Fractals, 22.3 (2004): 549-554.
• [39] M. Li, JR. Wang, D. O'Regan, Existence and Ulam's stability for conformable fractional di?erential equations with constant coeficients, Bulletin Malaysian Math SciSoc, 42.4 (2019): 1791-1812.
• [40] Y. Li, Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, Journal of Differential Equations, 266.6 (2019): 3514-3558.
• [41] J. Liang, X. Qian, T. Shen, S. Song, Analysis of time fractional and space nonlocal stochastic nonlinear Schrodinger equation driven by multiplicative white noise, Journal of Mathematical Analysis and Applications, 466.2 (2018): 1525- 1544.
• [42] V.F. Morales-Delgado, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, M.A. Taneco-Hernández, Fractional conformable deriva- tives of Liouville-Caputo type with low-fractionality, Physica A. 503 (2018): 424-438.
• [43] M. Maheri, N.M. Arifin, Synchronization of two different fractional-order chaotic systems with unknown parameters using a robust adaptive nonlinear controller, Nonlinear Dynamics, 85.2 (2016): 825-838.
• [44] L. Martínez, J.J. Rosales, C.A. Carreño, J.M. Lozano, Electrical circuits described by fractional conformable derivative, Int J Circuit TheoryAppl. 46.5 (2018): 1091-1100.
• [45] R. Metzler, J. Klafter, When translocation dynamics becomes anomalous, Biophysical journal, 85.4 (2003): 2776-2779.
• [46] J.C. Pedjeu, G.S. Ladde, Stochastic fractional differential equations: Modeling, method and analysis, Chaos Solitons Fractals, 45.3 (2012): 279-293.
• [47] I. Podlubny, Fractional differential equations, Academic Press, San Diego, (1998).
• [48] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer (2017).
• [49] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Naukai Tekhnika, Minsk, (1987).
• [50] F. Santamaria, S. Wils, E. De Schutter, G.J. Augustine, The diffusional properties of dendrites depend on the density of dendritic spines, European Journal of Neuroscience, 34.4 (2011): 561-568.
• [51] N.H. Tuan, T. Caraballo, T.N. Thach, On terminal value problems for bi-parabolic equations driven by Wiener process and fractional Brownian motions, Asymptotic Analysis, (2020): 1-32.
• [52] N.H. Tuan, N.H. Tuan, D. Baleanu, T.N. Thach, On a backward problem for fractional diffusion equation with Riemann- Liouville derivative, Mathematical Methods in the Applied Sciences, 43.3 (2020): 1292-1312.
• [53] B. Xin, W. Peng, Y. Kwon, Y.Liu, Modeling, discretization, and hyperchaos detection of conformable derivative approach to a financial system with market con?dence and ethics risk, Adv Difference Equ, 1 (2019): 1-14.
• [54] S. Yang, L. Wang, S. Zhang, Conformable derivative: Application to non-Darcian flow in low-permeability porous media, Appl. Math. Lett., 79 (2018): 105-110.
• [55] M. Yavuz, B. Yaskiran, Conformable Derivative Operator in Modelling Neuronal Dynamics, Applications & Applied Mathematics, 13.2 (2018). [56] W. Zhong, L. Wang, Basic theory of initial value problems of conformable fractional differential equations, Adv. Difference Equ., 2018.1 (2018): 1-14.
• [57] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Academic Press (2016).
• [58] Y. Zhou, Attractivity for fractional differential equations, Appl. Math. Lett., 75 (2018): 1-6.
• [59] HW. Zhou, S. Yang, SQ. Zhang, Conformable derivative approach to anomalous diffusion, Physica A: Stat Mech Its Appl, 491 (2018): 1001-1013.
• [60] G. Zou, B. Wang, Stochastic Burgers' equation with fractional derivative driven by multiplicative noise, Computers & Mathematics with Applications, 74.12 (2017): 3195-3208.
• [61] G. Zou, G. Lv, J.-L. Wu (2018). Stochastic Navier-Stokes equations with Caputo derivative driven by fractional noises. Journal of Mathematical Analysis and Applications, 461.1 (2018): 595-609.
• [62] G.A. Zou, B. Wang, Y. Zhou, Existence and regularity of mild solutions to fractional stochastic evolution equations, Mathematical Modelling of Natural Phenomena, 13.1 (2018): 15.
• [63] F.S. Bachir, S. Abbas, M. Benbachir, M. Benchohra, Hilfer-Hadamard Fractional Differential Equations, Existence and Attractivity, Advances in the Theory of Nonlinear Analysis and its Application, 5.1 (2021): 49-57.
• [64] A. Salim, M. Benchohra, J. Lazreg, J. Henderson, Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Non-Instantaneous Impulses in Banach Spaces , Advances in the Theory of Nonlinear Analysis and its Application, 4.4 (2020): 332-348.
• [65] Z. Baitichea, C. Derbazia, M. Benchohrab, ψ-Caputo Fractional Differential Equations with Multi-point Boundary Con- ditions by Topological Degree Theory, Results in Nonlinear Analysis, 3.4 (2020): 166-178.