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Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
book

Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition

by Andrei D. Polyanin, Vladimir E. Nazaikinskii
December 2015
Intermediate to advanced content levelIntermediate to advanced
1643 pages
59h 33m
English
Chapman and Hall/CRC
Content preview from Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
5.4. Equations with n Space Variables 553
Solution:
w(x, t) =
Z
t
0
Z
V
Φ(y, τ) G(x, y, t, τ) dV
y
+
Z
V
f(y) G(x, y, t, 0) dV
y
+
n
X
k=1
Z
t
0
Z
S
(k)
a
k
(α
k
, τ)
g
k
(y, τ )
∂y
k
G(x, y, t, τ )
y
k
=α
k
dS
(k)
y
n
X
k=1
Z
t
0
Z
S
(k)
a
k
(β
k
, τ)
h
k
(y, τ)
∂y
k
G(x, y, t, τ )
y
k
=β
k
dS
(k)
y
dτ,
where
dV
y
= dy
1
dy
2
. . . dy
n
, dS
(k)
y
= dy
1
. . . dy
k1
dy
k+1
. . . dy
n
,
S
(k)
= {α
m
y
m
β
m
for m = 1, . . . , k1, k+1, . . . , n}.
The Green’s function can be represented in the product form
G(x, y, t, τ ) =
n
Y
k=1
G
k
(x
k
, y
k
, t, τ). (1)
Here, the G
k
= G
k
(x
k
, y
k
, t, τ) are auxiliary Green’s functions that, for t > τ 0, satisfy
the one-dimensional linear homogeneous equations
∂G
k
∂t
a
k
(x
k
, t)
2
G
k
∂x
2
k
b
k
(x
k
, t)
∂G
k
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Publisher Resources

ISBN: 9781466581494