Inverse problem for 2b-order differential equation with a time-fractional derivative - Lopushansky, A.O.; Lopushanska, H.P. (Carpathian Mathematical Publications, 2019)
We study the inverse problem for a differential equation of order 2b with the Riemann-Liouville fractional derivative of order β(0,1) in time and given Schwartz type distributions in the right-hand sides of the equation and the initial condition. The problem is to find the pair of functions (u, g): a generalized solution u to the Cauchy problem for such equation and the time dependent multiplier g in the right-hand side of the equation. As an additional condition, we use an analog of the integral condition (u(·,t),φ_0(·))=F(t), t[0,T], where the symbol (u(·,t),φ_0(·)) stands for the value of an unknown distribution u on the given test function φ_0 for every t[0,T], F is a given continuous function. We prove a theorem for the existence and uniqueness of a generalized solution of the Cauchy problem, obtain its representation using the Green's vector-function. The proof of the theorem is based on the properties of conjugate Green's operators of the Cauchy problem on spaces of the Schwartz type test functions and on the structure of the Schwartz type distributions. We establish sufficient conditions for a unique solvability of the inverse problem and find a representation of anunknown function g by means of a solution of a certain Volterra integral equation of the second kind with an integrable kernel.