Regularization of a backward problem for composite fractional relaxation equations

A backward problem for composite fractional relaxation equations is considered with Caputo's fractional derivative, which covers as particular case of Basset problem that concerns the unsteady motion of a particle accelerating in a viscous fluid in fluid dynamics. Based on a spectral problem, the representation of solutions is established. Next, we show the maximal regularity for the corresponding initial value problem. Due to the mildly ill-posedness of current backward problem, the fractional Landweber regularization method will be applied to discuss convergence analysis and error estimates.