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 a general filter method for constructing regularized solution, and the convergence rate of this method also has been investigated.

We consider a non-standard inverse heat conduction problem in a bounded domain which appears in some applied subjects. We want to know the surface temperature in a body from a measured temperature history at a fixed location inside the body. This is an exponentially ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we introduce the two new classes of quasi-type methods and iteration methods to solve the problem and prove that our methods are stable under both a priori and a posteriori parameter choice rules. An appropriate selection of a parameter in the scheme will get a satisfactory approximate solution. Furthermore, if we use the discrepancy principle we can avoid the selection of the a priori bound. MSC: 35K05; 35K99; 47J06; 47H10