Extension of the Duhamel principle for the heat equation with Dezin's initial condition

The classical Duhamel principle for the heat equation is extended to the case when the initial condition $u(x,0) = f(x)$ is replaced by the nonlocal A. Dezin's condition $\mu u(0) - u(T) = f(x), \mu \neq 1$. To this end three types of operational calculi are developed: 1) operational calculus for $\frac{d}{dt}$ with the Dezin's functional, 2) operational calculus for $\frac{d^2}{dx^2}$ in a segment $[0, a]$ with boundary conditions $u(0) = 0$ and $u(a) = 0$, and 3) a combined operational calculus for functions $u(x,t)\textrm{ in } C(\Delta), \Delta=[0,a]\times[0,T]$.