Regularization by projection for a backward problem of the time-fractional diffusion equation

Time-fractional diffusion equations, which frequently appear in the diffusion process, describe the continuous time random walk phenomena. In this paper we investigate a backward time-fractional diffusion equation where one wants to extract the initial temperature distribution from the observation data provided along the final time <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>t</m:mi> <m:mo>=</m:mo> <m:mi>T</m:mi> </m:mrow> </m:math> t=T . Based upon the eigenfunction expansion, an analytical solution is deduced. However, such an approach does not depend continuously on the observation data. Hence, we propose a regularization by projection method where the truncated level plays the role of the regularization parameter. Under appropriate regularity assumptions of the exact solution, a uniform error estimate with an optimal convergence rate between the reconstructed solution and the exact one is obtained both for a priori and a posteriori parameter choice strategies. Finally, numerical examples are presented to illustrate the validity and effectiveness of the proposed method.