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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
894 SECOND-ORDER ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
is a necessary condition for a solution of the nonhomogeneous problem to exist. The solu-
tion is then given by
w =
m1
X
n=1
A
n
λ
n
λ
m
w
n
+
X
n=m+1
A
n
λ
n
λ
m
w
n
+ Cw
m
, A
n
=
1
kw
n
k
2
Z
V
Φw
n
dV,
where C is an arbitrary constant and kw
n
k
2
=
R
V
w
2
n
dV .
3
. If λ = λ
m
and
R
V
Φw
m
dV 6= 0, then the boundary value problem for the nonhomoge-
neous equation has no solution.
Remark 10.4. If to each eigenvalue λ
n
there are cor responding p
n
mutually orthogonal eigen-
functions w
(s)
n
(s = 1, . . . , p
n
), then the solution is written as
w =
X
n=1
p
n
X
s=1
A
(s)
n
λ
n
λ
w
(s)
n
, where A
(s)
n
=
1
kw
(s)
n
k
2
Z
V
Φw
(s)
n
dV, kw
(s)
n
k
2
=
Z
V
w
(s)
n
2
dV
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Publisher Resources

ISBN: 9781466581494