Inverse Problem of Determining the Heat Source Densityfor the Subdiffusion Equation
Аннотация
We study the inverse problem of determining the right-hand side of a subdiffusion equation with Riemann–Liouville fractional derivative whose elliptic part has the most general form and is defined in an arbitrary multidimensional domain (with sufficiently smooth boundary). The Fourier method is used to prove theorems on the existence and uniqueness of the classical solution of the initial–boundary value problem and on the unique reconstruction of the unknown right-hand side of the equation. The concept of generalized solution is introduced and a theorem on its existence is proved. The stability of classical and generalized solutions is proved. Requirements for the initial function and for the additional condition are established under which the classical Fourier method can be applied to the inverse problem under consideration. The results obtained are also new for the classical diffusion equation.