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 : R-0(+) --> R, E(t) = integral (1)(0) u(x, t) dx, and F : [0, 1] --> R, F(x) = integral (infinity)(0) u(x, t) dt. We perform the numerical 0 solution applying the so-called (sigma, theta)-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. (C) 2001 Elsevier Science Ltd. All rights reserved.
heat equation numerical solution qualitative properties