Non-Local Problem in Time for the Barenblatt–Zheltov–Kochina Type Fractional Equations
Аннотация
This work is devoted to the study of a non-local problem for the abstract Barenblatt–Zheltov–Kochina type equation $$D_{t}^{\rho}u(t)+A(1+\gamma D_{t}^{\rho})u(t)=f$$ , $$\gamma\geq 0$$ , $$\rho\in(0,1]$$ , namely, the time condition is specified in the form of an integral of the solution to the equation. Here $$D_{t}$$ stands for the Caputo fractional derivative, $$A$$ is an arbitrary positive self-adjoint operator acting in a Hilbert space. The main goal of this work is to study the influence of parameter $$\gamma$$ on the correctness of the problem. The inverse problem of determining the right-hand side of the equation is also studied. The additional condition we use guarantees both the existence and uniqueness of a solution to the inverse problem. Let us pay attention to the fact that the results obtained in this work are new for classical equations too, i.e., for case $$\rho=1$$ .