Inverse problem with final overdetermination for time-fractional differential equation in a Banach space

Abstract

We consider in a Banach space E the inverse problem(\mathbf{D}_{t}^{\alpha}u)(t)=Au(t)+\mathcal{F}(t)f,\quad t\in[0,T],u(0)=u^{0}% ,u(T)=u^{T},\,0<\alpha<1with operator A, which generates the analytic and compact α-times resolvent family {\{S_{\alpha}(t,A)\}_{t\geq 0}}, the function {\mathcal{F}(\,\cdot\,)\in C^{1}[0,T]} and {u^{0},u^{T}\in D(A)} are given and {f\in E} is an unknown element. Under natural conditions we have proved the Fredholm solvability of this problem. In the special case for a self-adjoint operator A, the existence and uniqueness theorems for the solution of the inverse problem are proved. The semidiscrete approximation theorem for this inverse problem is obtained.

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

“…At the same time, inverse problems for differential equations [14][15][16][17][18], i.e., problems for differential equations with unknown parameters and additional overdetermination conditions, arise in physics, astronomy, geophysics, chemistry, and biology, when conducting research of processes, some parameters of which are not available for direct measurements. In recent years, linear inverse problems to equations with fractional derivatives have been researched in the works [19][20][21][22][23][24][25].…”

On Local Unique Solvability for a Class of Nonlinear Identification Problems

“…Various linear inverse problems for differential equations containing Riemann-Liouville or Gerasimov-Caputo fractional derivatives were studied in papers [12][13][14][15][16]. Unique solvability issues for a nonlinear identification problem of form ( 1)- (3) with Gerasimov-Caputo derivatives and with a closed operator A, which generates an analytic resolving family of operators for a respective linear homogeneous equation, were investigated in [17].…”

Inverse problem to identify a space-dependent diffusivity coefficient in a generalized subdiffusion equation from final data

“…The analysis of such problems uses the Fourier expansion or positivity principles with the Fredholm alternative. A more general approach in a Hilbert space setting is presented in the recent paper [15].…”