Let the parabolic problem
c
(
x
,
t
,
u
)
u
t
=
a
(
x
,
t
,
u
)
u
x
x
+
b
(
x
,
t
,
u
,
u
x
)
,
0
>
x
>
1
,
0
>
t
≦
T
,
u
(
x
,
0
)
=
f
(
x
)
,
u
(
0
,
t
)
=
g
0
(
t
)
,
u
(
1
,
t
)
=
g
1
(
t
)
c(x,t,u){u_t} = a(x,t,u){u_{xx}} + b(x,t,u,{u_x}),0 > x > 1,0 > t \leqq T,u(x,0) = f(x),u(0,t) = {g_0}(t),u(1,t) = {g_1}(t)
, be solved approximately by the continuous-time collocation process based on having the differential equation satisfied at Gaussian points
ξ
i
,
1
{\xi _{i,1}}
and
ξ
i
,
2
{\xi _{i,2}}
in subintervals
(
x
i
−
1
,
x
i
)
({x_{i - 1}},{x_i})
for a function
U
:
[
0
,
T
]
→
H
3
U:[0,T] \to {\mathcal {H}_3}
, the class of Hermite piecewise-cubic polynomial functions with knots
0
=
x
0
>
x
1
>
⋯
>
x
n
=
1
0 = {x_0} > {x_1} > \cdots > {x_n} = 1
. It is shown that
u
−
U
=
O
(
h
4
)
u - U = O({h^4})
uniformly in x and t, where
h
=
max
(
x
i
−
x
i
−
1
)
h = \max ({x_i} - {x_{i - 1}})
.