Identification of source term for the ill-posed Rayleigh–Stokes problem by Tikhonov regularization method

• Published in: Advances in difference equations Vol. 2019; no. 1; pp. 1 - 20
• Main Authors: Binh, Tran Thanh, Nashine, Hemant Kumar, Long, Le Dinh, Luc, Nguyen Hoang, Nguyen, Can
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 08.08.2019; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Computational fluid dynamics; Difference and Functional Equations; Exact solutions; Fractional derivative; Functional Analysis; Ill posed problems; Ill-posed problem; Mathematics; Mathematics and Statistics; Numerical methods; Ordinary Differential Equations; Partial Differential Equations; Rayleigh–Stokes problem; Regularization; Regularization methods; Tikhonov regularization method; 47H10; 35K05; Ill-posed problem; 47J06; Tikhonov regularization method; 35K99; Rayleigh–Stokes problem; Fractional derivative
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-019-2261-7

In this paper, we study an inverse source problem for the Rayleigh–Stokes problem for a generalized second-grade fluid with a fractional derivative model. The problem is severely ill-posed in the sense of Hadamard. To regularize the unstable solution, we apply the Tikhonov method regularization solution and obtain an a priori error estimate between the exact solution and regularized solutions. We also propose methods for both a priori and a posteriori parameter choice rules. In addition, we verify the proposed regularized methods by numerical experiments to estimate the errors between the regularized and exact solutions.