Inverse problem with two unknown time-dependent functions for $2b$-order differential equation with fractional derivative


  • A.O. Lopushansky University of Rzeszow, 1 Prof. St. Pigonia str., 35-310, Rzeszow, Poland
  • H.P. Lopushanska Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine


distribution, fractional derivative, inverse problem, Green vector-function
Published online: 2022-06-23


We study the inverse problem for a differential equation of order $2b$ with a Riemann-Liouville fractional derivative over time and given Schwartz-type distributions in the right-hand sides of the equation and the initial condition. The generalized (time-continuous in a certain sense) solution $u$ of the Cauchy problem for such an equation, the time-dependent continuous young coefficient and a part of a source in the equation are unknown.

In addition, we give the time-continuous values $\Phi_j(t)$ of desired generalized solution $u$ of the problem on a fixed test functions $\varphi_j(x)$, $x\in \mathbb R^n$, namely $(u(\cdot,t),\varphi_j(\cdot))=\Phi_j(t)$, $t\in [0,T]$, $j=1,2$.

We find sufficient conditions for the uniqueness of the generalized solution of the inverse problem throughout the layer $Q:=\mathbb R^n\times [0,T]$ and the existence of a solution in some layer $\mathbb R^n\times [0,T_0]$, $T_0\in (0,T]$.

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How to Cite
Lopushansky, A.; Lopushanska, H. Inverse Problem With Two Unknown Time-Dependent Functions for $2b$-Order Differential Equation With Fractional Derivative. Carpathian Math. Publ. 2022, 14, 213-222.