Skip to Main Content
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
266 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
Domain: 0 x l. First boun dary value problem.
The following conditions are prescribed:
w = f (x) at t = 0 (initial condition),
w = g
1
(t) at x = 0 (boundary condition),
w = g
2
(t) at x = l (boundary condition).
Solution:
w(x, t) =
2
l
X
n=1
sin
nπx
l
exp
an
2
π
2
t
l
2
M
n
(t),
where
M
n
(t) =
Z
l
0
f(ξ) sin
nπξ
l
+
anπ
l
Z
t
0
exp
an
2
π
2
τ
l
2
g
1
(τ) (1)
n
g
2
(τ)
dτ.
Remark 3.2. Using the relations [see Prudnikov, Brychkov, and Marichev (1986)]
X
n=1
sin
n
=
π ξ
2
(0 < ξ < 2π);
X
n=1
(1)
n1
sin
n
=
ξ
2
(π < ξ < π),
one can transform the solution to
w(x, t) = g
1
(t) +
x
l
g
2
(t) g
1
(t)
+
2
l
X
n=1
sin(λ
n
x) exp(
2
n
t)R
n
(t), λ
n
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Differential Equations, 2nd Edition

Differential Equations, 2nd Edition

Steven G. Krantz

Publisher Resources

ISBN: 9781466581494