Numerical approximation of a Cauchy problem for a parabolic partial differential equation

Abstract:

A procedure for the numerical approximation of the Cauchy problem for the following linear parabolic partial differential equation is defined: \[ \begin {array}{*{20}{c}} {{u_t} - {{(p(x){u_x})}_x} + q(x)u = 0,\quad 0 < x < 1,0 < t \leqslant T;\quad u(0,t) = {f_1}(t),} \hfill \\ {0 < t \leqslant T;\quad u(1,t) = {f_2}(t),\quad 0 < t \leqslant T;\quad p(0){u_x}(0,t) = g(t),} \hfill \\ {0 < {t_0} \leqslant t \leqslant T;\quad |u(x,t)| \leqslant M,\quad 0 \leqslant x \leqslant 1,0 \leqslant t \leqslant T.} \hfill \\ \end {array} \] The procedure involves Galerkin-type numerical methods for related parabolic initial boundary-value problems and a linear programming problem. Explicit a priori error estimates are presented for the entire discrete procedure when the data ${f_1}$, ${f_2}$, and g are known only approximately.