Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 4: 1350 - 1363

DOI: https://doi.org/10.1007/s42967-023-00319-0

ORIGINAL PAPERS

Quasi Solution of an Inverse Fractional Stochastic Nonlinear Partial Differential Equation of Parabolic Type

  • T. Nasiri ,
  • A. Zakeri ,
  • A. Aminataei
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  • Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

Received date: 2023-01-16

  Revised date: 2023-06-20

  Accepted date: 2023-09-14

  Online published: 2023-12-26

Supported by

The authors declare that no funds, grants, or other support was received during the preparation of this manuscript.

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Abstract

In this paper, the existence theorem for a quasi solution of an inverse fractional stochastic parabolic equation driven by multiplicative noise in the form cDαtu-div(g(x,▽u))=f(x,u)+σ(x,u)ẇ(t) is given. In this equation, the fractional derivative is considered in the Caputo sense. Also, the random function g is unknown and should be determined. To identify the unknown coefficient, the minimization and stochastic variational formulation methods in a fractional stochastic Sobolev space are used. Indeed, we obtain a stability estimation and then prove the continuity of the minimization functional using obtained stability estimation. These results show the existence of the quasi solution for the mentioned problem.

Key words： Inverse problem; Fractional differential equation; Stochastic equation; Quasi solution; Multiplicative noise; Caputo fractional derivative

Cite this article

T. Nasiri , A. Zakeri , A. Aminataei . Quasi Solution of an Inverse Fractional Stochastic Nonlinear Partial Differential Equation of Parabolic Type[J]. Communications on Applied Mathematics and Computation, 2025 , 7(4) : 1350 -1363 . DOI: 10.1007/s42967-023-00319-0

References

[1] Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
[2] Benchaabane, A., Sakthivel, R.: Sobolev-type fractional stochastic differential equations with non-Lipschitz coefficients. Comput. Appl. Math. 312, 65-73 (2017)
[3] Chen, Z.Q., Kim, K.H., Kim, P.: Fractional time stochastic partial differential equations. Stochastic Process. Appl. 125, 1470-1499 (2015)
[4] Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differential Equations 22(3), 558-576 (2005)
[5] Ford, N., Xiao, J., Yan, Y.: A finite element method for time fractional partial differential equations. Fract. Calc. Appl. Anal. 14(3), 454-474 (2011)
[6] Guo, B., Pu, X., Huang F.: Fractional Partial Differential Equations and Their Numerical Solutions. World Scientific (2015)
[7] Hafiz, F.M.: The fractional calculus for some stochastic process. Stoch. Anal. Appl. 22(2), 507-523 (2004)
[8] Hasanov, A., Mueller, J.L.: A numerical method for backward parabolic problems with non-selfadjoint elliptic operators. Appl. Numer. Math. 37(1/2), 55-78 (2001)
[9] Hasanov, A.: Inverse coefficient problems for monotone potential. Inverse Prob. 13, 1265-1278 (2013)
[10] Hassanov, A., Li, Z.: An inverse coefficiemt problem for a nonlinear parabolic variational inequality. Appl. Math. Lett. 21(6), 563-570 (2008)
[11] Kilbas, A.A., Sirvastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)
[12] Kim, I., Kim, K., Lim, S.: A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives. Ann. Prob. 47, 2087-2139 (2019)
[13] Le Mehaute, A., Machado, T., Sabatier, J. C.: Fractional differentiation and its applications. In: FDA’04, Proceedings of the First IFAC Workshop, International Federation of Automatic Control, ENSEIRB, Bordeaux, France, July 19-21 (2004)
[14] Mijena, J.B., Nane, E.: Space-time fractional stochastic partial differential equations. Stochastic Process. Appl. 125, 3301-3326 (2015)
[15] Niu, P., Helin, T., Zhang, Z.: An inverse random source problem in a stochastic fractional diffusion equation. Inverse Probl. (2020). https://doi.org/10.1088/1361-6420/ab532c
[16] Noonan, J.P., Polchlopek, H.M.: An Arzela-Ascoli type thorem for random functions. Int. J. Math. Math. 14, 789-796 (1991)
[17] Nouy, A.: A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 196, 4521-4537 (2007)
[18] Oksendal, B.K.: Stochastic Differential Equations: an Introduction with Applications. Springer-Verlag, Heidelberg, New York (2003)
[19] Ou, Y., Hasanov, H., Liu, Z.H.: Inverse coefficient problems for nonlinear parabolic differential equations. Acta Math. Sin. (Engl. Ser.) 24, 1617-1624 (2008)
[20] Rozanov, Y.A.: Random Fields and Stochastic Partial Differential Equations. Springer, Kluwer Academic, New York (1998)
[21] Pedjeu, J.-C., Ladde, G.S.: Stochastic fractional differential equations: modeling, method and analysis. Chaos, Solitons & Fractals 45, 279-293 (2012)
[22] Sakthivel, R., Revathi, P., Ren, Y.: Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. 81, 70-86 (2013)
[23] Salehi Shayegan, A.H., Zakeri, A.: A numerical method for determining a quasi solution of a backward time-fractional equation. Inverse Probl. Sci. Eng. 26(8), 1130-1154 (2017)
[24] Shen, T., Huang, J.: Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials. Nonlinear Anal. Theory Methods Appl. 110, 33-46 (2014)
[25] Shen, T., Huang, J.: Well-posedness of the stochastic fractional Boussinesq equation with Lévy noise. Stoch. Anal. Appl. 33, 1092-1114 (2015)
[26] Sohail, A., Chighoub, F., Li, Z.: Spectral analysis of the stochastic time-fractional-KdV equation. Alex. Eng. J. 57, 2509-2514 (2018)
[27] Zou, G.: A Galerkin finite element method for time-fractional stochastic heat equation. Comput. Math. Appl. 75, 4135-4150 (2018)
[28] Zou, G., Lv, G., Wu, J.L.: Stochastic Navier-Stokes equations with Caputo derivative driven by fractional noises. J. Math. Anal. Appl. 461, 595-609 (2018)
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