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In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related toD  t  αC   ⁢  u   -   ∇  ⋅   (   k  ⁢   (  x  )   ⁢   ∇  ⁡  u    )=  f{{}^{C}D_{t}^{\alpha}u-\nabla\cdot(k(x)\nabla u)=f}   , where the functionk  =   k  ⁢   (  x  ){k=k(x)}    is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function  k . At the first step of the methodology, we give a stability result corresponding to connectivity of  k  and  u  which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given.

Identification of the diffusion coefficient in a time fractional diffusion equation