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
3.1. Constant Coefficient Equations 281
2
. On passing from t, x to the new variables t, z = x + bt, we obtain the nonhomogeneous
heat equation
∂w
∂t
= a
2
w
∂z
2
+ Φ(z bt, t),
which is treated in Section 3.1.2.
3
. For all first boundary value problems, the Green’s function can be represented as
G
b
(x, ξ, t) = exp
b
2a
(ξ x)
b
2
4a
t
G
0
(x, ξ, t),
where G
0
(x, ξ, t) is the Green’s function for the heat equation that corresponds to b = 0.
Domain: −∞ < x < . Cauchy problem.
An initial condition is prescribed:
w = f (x) at t = 0.
Solution:
w(x, t) =
Z
−∞
f(ξ)G(x, ξ, t) +
Z
t
0
Z
−∞
Φ(ξ, τ )G(x, ξ, t τ) dτ,
where
G(x, ξ, t) =
1
2
πat
exp
b(ξ x)
2a
b
2
t
4a
(x ξ)
2
4at
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