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Applied Calculus of Variations for Engineers, 2nd Edition
book

Applied Calculus of Variations for Engineers, 2nd Edition

by Louis Komzsik
June 2014
Intermediate to advanced content levelIntermediate to advanced
233 pages
5h 42m
English
CRC Press
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
104 Applied calculus of variations for engineers
L(G(P, Q)) = δ(P Q),
where δ is the Dirac function. Let us work with the two-dimensional Poisson’s
equation of the form
Δu(x, y)=f(x, y).
Here L(x, y)=Δ=
2
and its Green’s function is
G(P, Q)=
1
2π
ln(r),
where
r =
(x
p
x
q
)
2
+(y
p
y
q
)
2
.
Green’s theorem’s generic form (a consequence of Gauss’ divergence theorem)
may be written as
Ω
(u
2
v v
2
u)dΩ=
Γ
(u
∂v
∂n
v
∂u
∂n
)dΓ.
Using Green’s function in place of v we obtain
Ω
(u
2
G G
2
u)dΩ=
Γ
(u
∂G
∂n
G
∂u
∂n
)dΓ.
By definition
L(G(P, Q)) =
2
G(P, Q)=δ(P Q),
and due to the characteristics of the Dirac function the first term on the
left-hand side reduces to u(x, y). Substituting ...
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Publisher Resources

ISBN: 9781482253597