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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
22.3. Analytical Solutions of Mathematical Problems 1307
Here X = (x, y), X
0
= (x
0
,y
0
), the function v(X, X
0
) is the free space Green’s function (does not
depend o n the bo undary conditions), and the function w(X ,X
0
) satisfies the Laplace equation and the
boundary con ditions (and is regular at X = X
0
); i.e.,
2
w(X,X
0
) = 0 in V and w(X, X
0
) = v (X ,X
0
)
(i.e., G(X ,X
0
) = 0) on S for the Dirichlet boundary cond itions.
It is well known that the 2D free space function v(X, X
0
) is
v(X,X
0
) =
1
4π
ln
(x x
0
)
2
+ (y y
0
)
2
.
If to v(X,X
0
) we add the function
w(X,X
0
) =
1
4π
ln
(x x
0
)
2
+ (y + y
0
)
2
,
which satisfies the Laplace equa tion
2
w(X,X
0
) = 0 in V and is regular ...
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Publisher Resources

ISBN: 9781466581494