The inverse problem of identifying the time-independent source term and initial value simultaneously for a time-fractional diffusion equation is investigated. This inverse problem is reformulated into an operator equation based on the Fourier method. Under a certain smoothness assumption, conditional stability is established. A standard Tikhonov regularisation method is proposed to solve the inverse problem. Furthermore, the convergence rate is given for an a priori and a posteriori regularisation parameter choice rule, respectively. Several numerical examples, including one-dimensional and two-dimensional cases, show the efficiency of our proposed method.

This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C 1,1 boundary. It is an extension of anearlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C ∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces a nearly optimal convergence rate for the method of quasi-reversibility introduced in [Lattès and Lions, Dunod (1967)] to solve the ill-posed Cauchy problems.