Inverse problems for a perturbed time fractional diffusion equation with final overdetermination

Abstract

Inverse problems to recover a space‐dependent factor of a source term and an initial condition in a perturbed time fractional diffusion equation containing an additional convolution term from final data are considered. Existence, uniqueness, and stability of solutions to these problems are proved.

An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations

“…∀δ ∈ (t 0 , T). Applying d n dt n to (8), using the second condition in (1) and rearranging the terms, we obtain the following integral equation of the first kind for u| (0,t 0 ) :…”

“…Quite often in the inverse source problem, the goal is to determine a source that is either a spaceor time-dependent function. The space-dependent source term is usually reconstructed based on the final time overdetermination condition [6][7][8][9][10][11]. The time-dependent source term can be recovered from additional boundary measurements [7] or from integral conditions [12,13].…”

An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

“…Identification of a space-dependent source factor h(x) in a source function of the form F(x, t) = h(x)q(x, t) from final overdetermination are studied in [17,19,[23][24][25], where different assumptions on the known source factor q(x, t) are discussed. Concerning the generalized subdiffusion equation, various types of inverse problems for such equations are studied in [26][27][28].…”

Inverse Problems for a Generalized Subdiffusion Equation With Final Overdetermination

“…Often parameters of models are unknown. Then additional observations are performed and inverse problems solved to reconstruct unknown quantities [12,13,16,17,20,21]. In the present paper we consider two inverse problems (IPs) that use final observation data: IP1 is to identify a space-dependent factor f of a source term g(t, x)f (x); IP2 is to reconstruct a coefficient r(x) of a linear reaction term.…”

“…IP1 for fractional and perturbed fractional diffusion equations is studied in several papers. Theoretical and numerical results are obtained in the particular case g = g(t) [7,17,18,26] and in the case g = g(t, x) [30,32]. In latter papers the existence and uniqueness of solutions are proved for almost all scalar diffusion coefficients.…”