
266 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
◮ Domain: 0 ≤ x ≤ l. First boun dary value problem.
The following conditions are prescribed:
w = f (x) at t = 0 (initial condition),
w = g
1
(t) at x = 0 (boundary condition),
w = g
2
(t) at x = l (boundary condition).
Solution:
w(x, t) =
2
l
∞
X
n=1
sin
nπx
l
exp
−
an
2
π
2
t
l
2
M
n
(t),
where
M
n
(t) =
Z
l
0
f(ξ) sin
nπξ
l
dξ +
anπ
l
Z
t
0
exp
an
2
π
2
τ
l
2
g
1
(τ) −(−1)
n
g
2
(τ)
dτ.
Remark 3.2. Using the relations [see Prudnikov, Brychkov, and Marichev (1986)]
∞
X
n=1
sin nξ
n
=
π −ξ
2
(0 < ξ < 2π);
∞
X
n=1
(−1)
n−1
sin nξ
n
=
ξ
2
(−π < ξ < π),
one can transform the solution to
w(x, t) = g
1
(t) +
x
l
g
2
(t) − g
1
(t)
+
2
l
∞
X
n=1
sin(λ
n
x) exp(−aλ
2
n
t)R
n
(t), λ
n