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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
1104 SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS
12.15.2 Decompositions of the Nondissipative Thermoelasticity
Equations with f = 0
Decomposition based on two stream functions.
Each solution of the nondissipative thermoelasticity equations (2)–(3) with f = 0 can be
represented in the form
u = ϕ + v, T =
ρ
α
c
2
1
ϕ ϕ
tt
,
v = (v
1
, v
2
, v
3
), v
1
= ψ
(1)
y
, v
2
= ψ
(1)
x
+ ψ
(2)
z
, v
3
= ψ
(2)
y
,
(4)
where ψ
(1)
= ψ
(1)
(x, t) and ψ
(2)
= ψ
(2)
(x, t) are some solutions of the wave equation
ψ
tt
c
2
2
ψ = 0, (5)
and the function ϕ = ϕ(x, t) is a solution of the fourth-order equation
ϕ
tttt
[a + c
2
1
+ (αβ)]∆ϕ
tt
+ ac
2
1
∆∆ϕ = 0. (6)
Equation (6) can be represented in the form
(
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Publisher Resources

ISBN: 9781466581494