Simultaneous numerical inversion of space-dependent initial condition and source term in multi-order time-fractional diffusion models

This article deals with a simultaneous reconstruction of unknown initial conditions and space-dependent source function in multi-order time-fractional diffusion problems. We discuss the existence and uniqueness of the direct problem. The problem is presented as a regularized optimization problem and converted into a variational problem. The existence of the minimizer for the optimization problem is demonstrated. For the numerical part, a modified Levenberg-Marquardt regularization approach is constructed to identify the initial condition and source function. Several numerical examples in one and two dimensions are employed to test the performance of the identification technique.