Inverse source problem in a space fractional diffusion equation from the final overdetermination

• Published in: Applications of mathematics (Prague) Vol. 64; no. 4; pp. 469 - 484
• Main Authors: Salehi Shayegan, Amir Hossein, Tajvar, Reza Bayat, Ghanbari, Alireza, Safaie, Ali
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2019; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Classical and Continuum Physics; Convexity; Diffusion; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Optimization; Theoretical; space fractional diffusion equation; adjoint problem; inverse source problem; weak solution theory; 65N21; Lipschitz continuity; 65N20
• DOI: 10.21136/AM.2019.0251-18
• ISSN: 0862-7940

We consider the problem of determining the unknown source term f = f ( x,t ) in a space fractional diffusion equation from the measured data at the final time u ( x,T ) = ψ ( x ). In this way, a methodology involving minimization of the cost functional J ( f ) = ∫ l 0 ( u ( x, t ; f ) t=T − ψ ( x )) 2 d x is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and convergence of the sequence { J′ ( f ( n ) )}, where f ( n ) is the n th iteration of a gradient like method. At the end, the convexity of the Fréchet derivative is given.