A numerical method for determining a quasi solution of a backward time-fractional diffusion equation

In this study, we present existence and uniqueness theorems of a quasi solution to backward time-fractional diffusion equation. To do that, we consider a methodology, involving minimization of a least squares cost functional, to identify the unknown initial data. Firstly, we prove the continuous dependence on the initial data for the corresponding forward problem and then we obtain a stability estimate. Based on this, we give the existence theorem of a quasi solution in an appropriate class of admissible initial data. Secondly, it is shown that the cost functional is Fréchet-differentiable and its derivative can be formulated via the solution of an adjoint problem. These results help us to prove the convexity of cost functional and subsequently the uniqueness theorem of the quasi solution. In addition, in order to approximate the quasi solution, WEB-spline finite element method is used. Since the obtained system of linear equations is ill-posed, we apply the Levenberg-Marquardt regularization. Finally, a numerical example is given to show the validation of the introduced method.