Coupling RBF-based meshless method and Landweber iteration algorithm for approximating a space-dependent source term in a time fractional diffusion equation

• Published in: Journal of computational and applied mathematics Vol. 417; p. 114531
• Main Author: Salehi Shayegan, Amir Hossein
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2023
• Subjects: Inverse source problem; Landweber iteration algorithm; Lipschitz continuity; Meshless method; Radial basis functions; Time fractional diffusion equation; Radial basis functions; Inverse source problem; 35R30; Meshless method; Time fractional diffusion equation; Landweber iteration algorithm; 65M32; Lipschitz continuity
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2022.114531

The problem of determining a space-dependent source term in a time fractional diffusion equation is considered from the measured data at the final time. To find an approximation of the source term, a methodology involving minimization of the cost functional is applied. Also, in order to construct the Landweber iteration algorithm, an explicit formula for the gradient of the cost functional J is given via the solution of an adjoint problem. The resulting adjoint problem is treated by a radial basis function method for spatial dimension and a finite difference scheme for the time fractional derivative followed by an iterative domain decomposition method to achieve a desired accuracy. In addition, Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of Landweber iteration algorithm are proved. At the end, a numerical example is given to show the validation of the iterative algorithm.