Variational Regularization For The Inverse Problem of a Nonlinear Time-Fractional Diffusion Equations

Abstract

The nonlinear time-fractional diffusion equations were considered in the 2D domain, and the physical information in initial state <inlineformula><texmath id="M2">\begin{document}$ u(0,y,t) $\end{document}</texmath><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-42-0168_M2.jpg"></graphic><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-42-0168_M2.png"></graphic></alternatives></inlineformula> of the material was recovered from the measured data in the final state. This problem is seriously ill-posed, that is, the solution to this problem does not continuously depend on the measured data. Therefore, a variational regularization method was proposed to construct the approximate solution to the problem, and the convergence error estimates of the exact and approximate solutions were obtained under the assumption of the priori bounds on the exact solutions. Finally, a numerical example was given to verify the effectiveness of the proposed method.