Inverse problems for a time fractional diffusion equation in the Schwartz‐type distributions

Abstract

We find the conditions for the unique solvability of the inverse problem of the solution's initial data determination for a time fractional diffusion equation with a Schwartz‐type distribution in the right‐hand side and the inverse problem of identifying the equation's source under given Schwartz‐type distribution in the initial condition. We use a time‐integral overdetermination condition.

Inverse problem for a time‐fractional differential equation with a time‐ and space‐integral conditions

“…for any 𝜑 ∈ S 𝛾,(a) ( R n ) , t ∈ [0, T], and (F 1 , 𝜑 0 ) = 0 (F 1 satisfies the compatibility condition). Substituting (19) and (20) in (18), we get the linear homogeneous equation . □…”

“…Note that source inverse problems for time-fractional diffusion equations under overdetermination condition of kind (4) were studied, for example, in [10][11][12]14] and the inverse problems with an unknown reaction coefficient, for example, in [16,20]. Inverse problems for such equations with a time-integral overdetermination conditions of kind (3) were investigated, for example, in [13,17,18]. For second-order semilinear ultraparabolic equation, the inverse problem with a time-dependent unknown function in its reaction coefficient and two unknown space-time-dependent functions in its right-hand side was studied in [22].…”

Identification of a Time-Dependent Source Term in a Nonlocal Problem for Time Fractional Diffusion Equation

“…Inverse source problems for TFDe have been intensively investigated by many researchers under various initial, boundary and overdetermination conditions. The inverse problem of finding of a space-dependent source term from final temperature distribution was considered in [4,6,14,15,21,22,30,37] and recovering a space-dependent source term from total energy measurement has been discussed in [8,24,25]. For a TFDe inverse problem of identification a time-dependent source term from temperature measurement at the selected point in the spatial domain was considered in [5,19] and determining a time-dependent source term from integral type overdetermination condition was studied in [2,3,16,18].…”