Some integral properties of the heat equation

In this paper, we consider the one-dimensional heat conduction equation on the interval [0, 1]. We investigate the integrals of the solution u with respect to the space and time variables and the equivalents of the integrals in the numerical solution. We give the properties of the functions E : → , E(t) = ∫10 u(x, t) dx, and F : [0, 1] → , F(x) = ∫∞0 u(x, t)dt. We perform the numerical solution applying the so-called (σ, θ)-method [1]. We show that with the additional conditions of the nonnegativity preservation and maximum norm contractivity [2], similar statements are valid as in the continuous case.