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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
276 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
2
. Fundamental solution:
E (x, t) =
1
2
πat
exp
x
2
4at
+ bt
.
Reduction to the heat equation. Remarks on the Green’s functions.
The substitution w(x, t) = e
bt
u(x, t) leads to the nonhomogeneous heat equation
∂u
∂t
= a
2
u
∂x
2
+ e
bt
Φ(x, t),
which is discussed in S ection 3.1.2 in detail. The initial condition for the new variable u
remains the same, and the nonhomogeneous part in the boundary conditions is m ultiplied
by e
bt
. Taking this into account, one can easily solve the original equation subject to the
initial and boundary conditions considered in Section 3.1.2.
In all the boundary value
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Publisher Resources

ISBN: 9781466581494