952
Appendix: All Solutions to the Problems
−αu
2
(v str − c)
2
2∇v
(|∇v|
2
+ ε) 2 + γ (1 − δ)(u − u ind )
2 ∇v
|∇v|
· n
S
= 0
[∇v × (β∇u × ∇v)] · n| S = 0, [∇u × (β∇u × ∇v)] · n| S = 0
10.34 Find the Euler equations and boundary condition of the functional J =
V F(u, a, ∇ · e
u a)dV .
Solution Using Example 10.4.5, Putting f (u) = e
u , integrating u, we have f u (u) =
e
u , therefore, the Euler equations of the functional are
∂ F
∂u
− e
u a · ∇
∂ F
∂∇·e u a
= 0
∂ F
∂ a
+ e
u
∇
∂ F
∂∇·e u a
= 0
(1)
The natural boundary conditions are
∂ F
∂∇ · e u a
S
= 0
( 2 )
10.35 There is the following variational problem of the functional for diffusional
processes of heat and matter
J =
V
σ (∇T )
2
2
+ T
∇ · σ ∇T − ρC v
∂ T
∂t
dV
(1)
where, σ is the given function, find its Euler equation.
Solution Let F =
σ (∇T )
2
2
+ T
∇ · σ ∇T − ρC v
∂ T
∂t
, find the various partial
derivatives, we have
∂ F
∂ T
= ∇ · σ ∇T − ρC v
∂ T
∂t
,
∂ F
∂∇T
= σ ∇T,
∂ F
∂∇ · σ ∇T
= T
The Euler equation of the functional is
∂ F
∂ T
− ∇ ·
∂ F
∂∇T
+ ∇ · σ ∇
∂ F
∇ · σ ∇T
= 0
( 2 )
or
∇ · σ ∇T − ρC v
∂ T
∂t
= 0
( 3 )
This is the differential equation of heat conduction without heat source.
10.36 Find the Euler equation of the functional J =
V [((u)
2
− 2 f u]dV .