An inverse problem for a family of time fractional diffusion equations

We consider a diffusion equation involving fractional derivative in time of order β (0<β<1) with a nonlocal boundary condition involving a parameter α>0. A bi-orthogonal system of functions constructed from two Riesz basis of L2(0,1) is used to prove the existence and uniqueness of classical solution of the direct problem. The inverse problem of determination of the temperature distribution and the unknown source term is considered. The inverse problem is proved to be well-posed in the sense of Hadamard whenever an overdetermination condition of the final temperature is given.

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