A procedure for the numerical approximation of the Cauchy problem for the following linear parabolic partial differential equation is defined:
\[
u
t
−
(
p
(
x
)
u
x
)
x
+
q
(
x
)
u
=
0
,
0
>
x
>
1
,
0
>
t
⩽
T
;
u
(
0
,
t
)
=
f
1
(
t
)
,
0
>
t
⩽
T
;
u
(
1
,
t
)
=
f
2
(
t
)
,
0
>
t
⩽
T
;
p
(
0
)
u
x
(
0
,
t
)
=
g
(
t
)
,
0
>
t
0
⩽
t
⩽
T
;
|
u
(
x
,
t
)
|
⩽
M
,
0
⩽
x
⩽
1
,
0
⩽
t
⩽
T
.
\begin {array}{*{20}{c}} {{u_t} - {{(p(x){u_x})}_x} + q(x)u = 0,\quad 0 > x > 1,0 > t \leqslant T;\quad u(0,t) = {f_1}(t),} \hfill \\ {0 > t \leqslant T;\quad u(1,t) = {f_2}(t),\quad 0 > t \leqslant T;\quad p(0){u_x}(0,t) = g(t),} \hfill \\ {0 > {t_0} \leqslant t \leqslant T;\quad |u(x,t)| \leqslant M,\quad 0 \leqslant x \leqslant 1,0 \leqslant t \leqslant T.} \hfill \\ \end {array}
\]
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_1}
,
f
2
{f_2}
, and g are known only approximately.