10.7 Variational Problems of Functionals with Series-Type …
733
− u[B m×
(n − m)· (E∇) n−m (E∇) T
n−m+1,n ] m
· δ A
− u(E∇) n−m (E∇) T
n−m+1,n
m
× A T n
· δ B
(5)
F n
· ∇ × δu A m
× B = ∇ ·
δu A m
× B ×
(n − 1)· F
+ F × ∇ n
· ( A m
× Bδu + uδ A m
× B + u A m
× δ B)
= ∇ ·
δu A m
× B ×
(n − 1)· F
+ F × ∇ n
· A m
× Bδu
+ u[B m×
(n − m)· (F × ∇) n−m (F × ∇) T
n−m+1,n ] m
· δ A
+ u(F × ∇) n−m (F × ∇) T
n−m+1,n
m
× A T n
· δ B
(6)
Substituting Eqs. (3)–(6) into Eq. (2), we obtain
δ J =
V
∂ F
∂u
+ (C − D · ∇ − E∇ + F × ∇) n
· A m
× B
δu
+
∂ F
∂ A
+ u B n m×
(n − m)·
C n−m C
T
n−m+1,n − ( D · ∇) n−m ( D · ∇)
T
n−m+1,n − (E∇) n−m (E∇)
T
n−m+1,n
+ (F × ∇) n−m (F × ∇)
T
n−m+1,n
m
· δ A
+
∂ F
∂ B
+ u
C n−m C
T
n−m+1,n − ( D · ∇) n−m ( D · ∇)
T
n−m+1 − (E∇) n−m (E∇)
T
n−m+1
+ (F × ∇) n−m (F × ∇)
T
n−m+1
m
× A
T
n
· δ B
dV
+
V
∇ · (δu A m
× B n
· D) + ∇ · (δu A m
× B n − 1
· E) + ∇ ·
δu A m
× B ×
(n − 1)· F
dV
(7)
The Euler equations of the functional are
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ F
∂u +
⎛
⎜
⎝
∂ F
∂u A m
×
B
−
∂ F
∂∇u A m
×
B
· ∇ −
∂ F
∂∇·u A m
×
B
∇ +
∂ F
∂∇×u A m
×
B
× ∇
⎞
⎟
⎠ n
· A m
× B = 0
∂ F
∂ A + u B m×
(n − m)·
⎡
⎢
⎢
⎣
⎛
⎜
⎝
∂ F
∂u A m
×
B
⎞
⎟
⎠
n−m
⎛
⎜
⎝
∂ F
∂u A m
×
B
⎞
⎟
⎠
T
n−m+1,n
−
⎛
⎜
⎝
∂ F
∂∇u A m
×
B
· ∇
⎞
⎟
⎠
n−m
⎛
⎜
⎝
∂ F
∂∇u A m
×
B
· ∇
⎞
⎟
⎠
T
n−m+1,n
−
⎛
⎜
⎝
∂ F
∂∇·u A m
×
B
∇
⎞
⎟
⎠
n−m
⎛
⎜
⎝
∂ F
∂∇·u A m
×
B
∇
⎞
⎟
⎠
T
n−m+1,n
+
⎛
⎜
⎝
∂ F
∂∇×u A m
×
B
× ∇
⎞
⎟
⎠
n−m
⎛
⎜
⎝
∂ F
∂∇×u A m
×
B
× ∇
⎞
⎟
⎠
T
n−m+1,n
⎤
⎥
⎥
⎦ = 0
∂ F
∂ B + u
⎡
⎢
⎢
⎣
⎛
⎜
⎝
∂ F
∂u A m
×
B
⎞
⎟
⎠
n−m
⎛
⎜
⎝
∂ F
∂u A m
×
B
⎞
⎟
⎠
T
n−m+1,n
−
⎛
⎜
⎝
∂ F
∂∇u A m
×
B
· ∇
⎞
⎟
⎠
n−m
⎛
⎜
⎝
∂ F
∂∇u A m
×
B
· ∇
⎞
⎟
⎠
T
n−m+1,n
−
⎛
⎜
⎝
∂ F
∂∇·u A m
×
B
∇
⎞
⎟
⎠
n−m
⎛
⎜
⎝
∂ F
∂∇·u A m
×
B
∇
⎞
⎟
⎠
T
n−m+1,n
+
⎛
⎜
⎝
∂ F
∂∇×u A m
×
B
× ∇
⎞
⎟
⎠
n−m
⎛
⎜
⎝
∂ F
∂∇×u A m
×
B
× ∇
⎞
⎟
⎠
T
n−m+1,n
⎤
⎥
⎥
⎦
m
× A
T = 0
(8)
The natural boundary conditions are