TY - JOUR
AU - Lopushansky, A.O.
AU - Lopushanska, H.P.
PY - 2022/06/23
Y2 - 2024/07/14
TI - Inverse problem with two unknown time-dependent functions for $2b$-order differential equation with fractional derivative
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 14
IS - 1
SE - Scientific articles
DO - 10.15330/cmp.14.1.213-222
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/5228
SP - 213-222
AB - <p>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.</p><p>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 $(u(\cdot,t),\varphi_j(\cdot))=\Phi_j(t)$, $t\in [0,T]$, $j=1,2$.</p><p>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 $\mathbb R^n\times [0,T_0]$, $T_0\in (0,T]$.</p>
ER -