
3.2. Heat Equation with Axial or Central Symmetry and Related Equations 311
Solution for 0 < β < 1:
w(x, t) =
x
β
2t
Z
∞
0
f(ξ)ξ
1−β
exp
−
x
2
+ ξ
2
4t
I
−β
ξx
2t
dξ
−
2
2β−1
Γ(1 −β)
Z
t
0
g(τ) exp
−
x
2
4(t − τ)
dτ
(t − τ)
1−β
.
◮ Domain: 0 ≤ x < ∞. Third boundary value problem.
The following conditions are prescribed:
w = 0 at t = 0 (initial condition),
x
1−2β
∂
x
w + k(w
0
− w)
= 0 at x = 0 (boundary condition).
Solution for 0 < β < 1:
w(x, t) = w
0
2
2β−1
k
Γ(1 −β)
Z
t
0
ϕ(τ) exp
−
x
2
4(t − τ)
dτ
(t − τ)
1−β
,
where the function ϕ(t) is given as the series
ϕ(t) =
∞
X
n=0
(−µt
β
)
n
Γ(nβ + 1)
, µ =
2
2β−1
kΓ(β)
Γ(1 − β)
,
which is convergent for any t.
⊙ Literature: W. G. L. Sutton (1943).
3.2.6 Equation of the