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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
1158 SEPARATION OF VARIABLES AND INTEGRAL TRANSFORM METHODS
with the genera l initial condition (15.1.1.3) and the homogeneous boundary conditions
w = 0 at x = 0, w = 0 at x = l. (15. 1.2.6)
The func tion ψ(t) in the particular solution (15.1.1.5) is found from (15.1.2. 2), where γ(t) = 0:
ψ
n
(t) = exp(λ
n
t). (15.1.2.7)
The functions ϕ
n
(x) are deter mined by solving the eigenvalue problem (15.1.1.6), (15.1.1.8) with
a(x) = 1, b(x) = c(x) = 0, s
1
= s
2
= 0, k
1
= k
2
= 1, x
1
= 0, and x
2
= l:
ϕ
′′
xx
+ λϕ = 0; ϕ = 0 at x = 0, ϕ = 0 at x = l.
So w e obtain the eigenfunctions and eigenvalues:
ϕ
n
(x) = sin
nπx
l
, λ
n
=
l
2
, n = 1, 2, . . . (15.1.2.8)
The solution of p r ...
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Publisher Resources

ISBN: 9781466581494