
622 SECOND-ORDER HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE
where
¯
J
ν
(z) = 2
ν
Γ(1 + ν)z
−ν
J
ν
(z), Γ(ν) =
Z
∞
0
e
−s
s
ν−1
ds.
⊙ Literature: M. M. Smirnov (1975).
3.
∂
2
w
∂t
2
+
2a
t
∂w
∂t
= t
m
∂
2
w
∂x
2
.
Domain: −∞ < x < ∞. Cauchy problem.
Initial conditions are prescribed:
w = f (x) at t = 0,
t
2a
∂
t
w = g(x) at t = 0.
Solution for 0 ≤ 2a < 1 and m > 0:
w(x, t) =
Γ(2β)
Γ
2
(β)
Z
1
0
f
x +
2
2 + m
t
2+m
2
(2ξ − 1)
ξ
β−1
(1 − ξ)
β−1
dξ
+
Γ(2 − 2β)t
1−2a
(1 − 2a)Γ
2
(1 − β)
Z
1
0
g
x +
2
2 + m
t
2+m
2
(2ξ − 1)
ξ
−β
(1 − ξ)
−β
dξ,
where
β =
m + 4a
2(m + 2)
, Γ(z) =
Z
∞
0
e
−s
s
z−1
ds.
⊙ Literature: M. M. Smirnov (1975).
4. t
2
∂
2
w
∂t
2
+ kt
∂w
∂t
= a
2
∂
2
w
∂x
2
+ b
∂w
∂x
+ cw.
The substitution t = Ae
τ
(A 6= 0) leads to a constant coefficient equation ...