Solution of the backward problem for the space-time fractional diffusion equation related to the release history of a groundwater contaminant

• Published in: Journal of inverse and ill-posed problems Vol. 32; no. 1; pp. 107 - 126
• Main Authors: Salehi Shayegan, Amir Hossein, Zakeri, Ali, Salehi Shayegan, Adib
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 01.02.2024; Walter de Gruyter GmbH
• Subjects: 47A52; 65N21; Algorithms; Backward space-time fractional diffusion equation; computational algorithm; conjugate gradient algorithm; Contaminants; Existence theorems; Finite element method; Groundwater; Ill posed problems; Iterative algorithms; Iterative methods; Landweber iteration algorithm; Landweber's iteration algorithm; Lipschitz condition; Mathematical analysis; quasi-solution; Regularization; Uniqueness theorems
• DOI: 10.1515/jiip-2022-0054
• ISSN: 0928-0219

Finding the history of a groundwater contaminant plume from final measurements is an ill-posed problem and, consequently, its solution is extremely sensitive to errors in the input data. In this paper, we study this problem mathematically. So, firstly, existence and uniqueness theorems of a quasi-solution in an appropriate class of admissible initial data are given. Secondly, in order to overcome the ill-posedness of the problem and also approximate the quasi-solution, two approaches (computational and iterative algorithms) are provided. In the computational algorithm, the finite element method and TSVD regularization are applied. This method is tested by two numerical examples. The results reveal the efficiency and applicability of the proposed method. Also, in order to construct the iterative methods, an explicit formula for the gradient of the cost functional is given. This result helps us to construct two iterative methods, i.e., the conjugate gradient algorithm and Landweber iteration algorithm. We prove the Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of the iterative methods. At the end of the paper, a numerical example is given to show the validation of the iterative algorithms.