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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
Chapter 5
Second-Order
Parabolic Equations with
Three or More Space Variables
5.1 Heat Equation
∂w
∂t
= a
3
w
5.1.1 Problems in Cartesian Coordinates
The three-dimensional sourceless heat equation in the rectangular Cartesian system of co-
ordinates has the form
∂w
∂t
= a
2
w
∂x
2
+
2
w
∂y
2
+
2
w
∂z
2
.
It governs three-dimensional thermal phenomena in quiescent media or solids with constant
thermal diffusivity. A similar equation is used to study the corresponding three-dimensional
unsteady mass-exchange processes with constant diffusivity.
Particular solutions:
w(x, y, z, t) = Ax
2
+By
2
+Cz
2
+2a(A+ B +C)t,
w(x, y, z, t) = A(x
2
+2at)(y
2
+2at)(z
2
+2at)+ B,
w(x, y, z, t) = A
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Publisher Resources

ISBN: 9781466581494