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

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

{}^{\text{c}}{\text{D}}_t^{\alpha }u-\text{div}(g(x,\mathrm{\nabla }u))=f(x,u)+\sigma (x,u)\dot{w}(t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${^\text c}\text D_t^{\alpha }u-\text{div}(g(x,\nabla u))=f(x,u)+\sigma (x,u){\dot{w}}(t)$$\end{document}

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.

Keywords

Inverse problem / Fractional differential equation / Stochastic equation / Quasi solution / Multiplicative noise / Caputo fractional derivative / Mathematical Sciences / Statistics

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Shanghai University