IDENTIFICATION OF UNKNOWN COEFFICIENT IN TIME FRACTIONAL PARABOLIC EQUATION WITH MIXED BOUNDARY CONDITIONS VIA SEMIGROUP APPROACH

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problem of identifying the unknown coefficient k(x) in the linear time fractional parabolic equation D^α_t u(x, t) = (k(x)u_x)_x, 0 < α ≤ 1, with mixed boundary conditions u(0, t) = ψ₀(t), u_x(1, t) = ψ₁(t). Our aim is the investigation of the distinguishability of the input-output mapping Φ[·] : K → C[0, T ], via semigroup theory. This work shows that if the null space of the semigroup Tα,α(t) consists of only zero function, then the input-output mapping Φ[·] has distinguishability property. Also, the value k(0) of the unknown function k(x) is determined explicitly. In addition to these the boundary observation f(t) can be shown as an integral representation. This also implies that the mapping Φ[·] : K → C [0, T ] can be described in terms of the semigroup.