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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
856 SECOND-ORDER ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES
and different boundary conditions in y. We assume that the coefficients of equation (1) and
the boundary conditions (2) meet the requirement
a(x), b(x), c(x) are continuous functions; a(x) > 0, |s
1
| + |k
1
| > 0, |s
2
| + |k
2
| > 0,
where x
1
x x
2
.
In the general case, the Green’s function can be represented as
G(x, y, ξ, η) = ρ(ξ)
X
n=1
u
n
(x)u
n
(ξ)
ku
n
k
2
Ψ
n
(y, η; λ
n
). (3)
Here,
ρ(x) =
1
a(x)
exp
Z
b(x)
a(x)
dx
, ku
n
k
2
=
Z
x
2
x
1
ρ(x)u
2
n
(x) dx, (4)
and the λ
n
and u
n
(x) are the eigenvalues and eigenfunctions of the homogeneous boundary
value problem for the ordinary differential equation
a(x)u
′′
xx
+ b(x)u
x
+ [λ +
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Publisher Resources

ISBN: 9781466581494