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

Abstract

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 $\mu u(0)-u(T)=f(x),\mu \ne 1$. 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 $\frac{d^2}{d{x}^2}$ in a segment $[0,a]$ with boundary conditions $u(0)=0$ and $u(a)=0$, and 3) a combined operational calculus for functions $u(x,t)\text{ in }C(\mathrm{\Delta }),\mathrm{\Delta }=[0,a]\times [0,T]$.