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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
406 SECOND-ORDER PARABOLIC EQUATIONS WITH TWO SPACE VARIABLES
where
G(x, y, ξ, η, t) =
1
4πat
exp
(x ξ)
2
+ (y η)
2
4at
exp
(x + ξ)
2
+ (y η)
2
4at

.
Literature: H. S. Carslaw and J. C. Jaeger (1984).
Domain: 0 x < , −∞ < y < . Second boundary value problem.
A half-plane is considered. The following conditions are prescribed:
w = f (x, y) at t = 0 (initial condition),
x
w = g(y, t) at x = 0 (boundary condition).
Solution:
w(x, y, t) =
Z
0
Z
−∞
f(ξ, η) G(x, y, ξ, η, t)
a
Z
t
0
Z
−∞
g(η, τ ) G(x, y, 0, η, t τ) dτ,
where
G(x, y, ξ, η, t) =
1
4πat
exp
(x ξ)
2
+ (y η)
2
4at
+ exp
(x + ξ)
2
+ (y η)
2
4at

.
Domain: 0 x < , −∞ < y < . Third boun ...
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Publisher Resources

ISBN: 9781466581494