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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
12.5. Simplest Systems Containing Vector Functions and Operators div and curl 1069
1
. We seek a solution of problem (13) in the form (6), (8). For the function ϕ, we obtain
the Poisson equation
ϕ + g(x) = f(x), (14)
where the function g(x) is determined by formula (12).
2
. The solution of problem (13) can be represented in the form
u = ϕ + curl ψ + θ,
ϕ(x) =
1
4π
Z
V
f(x
)
|x x
|
dV
, ψ(x) =
1
4π
Z
V
A(x
)
|x x
|
dV
,
where the function θ satisfies the Laplace equation, θ = 0. The surface V may be un-
bounded if both integrals converge and decay as |x| at least at the rate of |x|
(1+ε)
,
where ε > 0.
3
. Equations (13) are often supplemented w ith the
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Publisher Resources

ISBN: 9781466581494