Conjugate gradient method for simultaneous identification of the source term and initial data in a time-fractional diffusion equation

Abstract

In this paper, we consider the simultaneous inversion of the source term and the initial data for a time-fractional diffusion equation based on the additional temperatures at two fixed times $ t=T_{1} $ and $ t=T_{2} $ . The inverse problem is formulated on the basis of the Fourier method as an operator equation of the first kind. For the overdetermined system of linear equations, we apply the conjugate gradient method with an appropriate stopping criterion to solve the least-squares problems. The method is not only able to converge quickly, but also it can solve large-scale linear equations. Based on the proposed results, it can be seen that the iteration numbers play a role of a regularization parameter. Four numerical examples are provided to demonstrate the effectiveness of the proposed method.

Keywords

• ill-posedness
• stopping criterion
• conditional stability
• minimization functional
• iteration method