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

doi:10.1007/s42967-023-00319-0

Abstract

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 ^cD^α_tu-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

CLC Number:

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.

TrendMD

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)