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2 1L ((−L;L)) V =H ((−L;L)) a (u,v)∈V×V

Z L

2a(u,v) = c ∂ u(x)·∂ v(x)+a(x)·u(x)·v(x) dxx x

−L

S ⊂S ⊂S ···S ⊂V0 1 2 m

u ∈S (P )m m m

2 d (u (t),v) +a(u (t),v) = 0, ∀ v∈Sm H m H m t

u (0) =f (f ∈S )m m m m

u (0) =g (g ∈S )m,t m m m

f g f gm m

x u um

t Δt = T/n t = nΔtmax n

n = 0;1;2···n n mmax max

n nU u (t ) V =u (t )m n m,t n

1 n+1 n n 2 n+1 n (U −U ;v) −(Δt)·(V ;v) +(Δt) a βU + −β U ;v = 0 H H 2

n+1 n n+1 n(V −V ;v) +(Δt)·a(γU +(1−γ)U ;v) = 0 H 0 U =f m 0V =gm

β γ

m m{Ψ } Smj j=0

G = (Ψ ;Ψ ′) a A = (a(Ψ ;Ψ ′))m j j ′ m j j ′j,j j,j

n n 0 0 n nC D C D U Vm m

m mf g {Ψ }m m j j=0

1 n+1 n n 2 n+1 n G (C −C −(Δt)D )+(Δt) A βC + −β C = 0 m m 2

n+1 n n+1 nG (D −D )+(Δt)A (γC +(1−γ)C ) = 0 m m 0 0 C =C m 0 0D =Dm

P

m n nu (t ,x) = C ψ (x) Cm n jj=0 j j

u x = x t = nT/n ψj n max j

2ψ (x ) =δ (j,k)∈{0;1;2;···;m} δ = 1 j =kj k jk jk

then

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3−j j/2u∈C ([0;T);D(A )) j = 0;1;2;3

Πm

Sm

0 0 1 1|u (t )−u(t )|≤C{|u −Π u |+|u −Π u |m n n m mm m Rt (3)n+|(I−Π )u(t )|+ |(I−Π )u (s)|+Δt|u (s)| ds}m n m tt t0

A D(A)

u

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k+1−j j/2 1/2u C ([0;T);D(A ) j = 0;1;2;···k+1 V =D(A )

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c

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f

−3 1

f f

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L 60

x = 0 f Ω2

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u (K ),(K ),...,(K )0 1 k

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(

j/2A u (T ), j0

(j−1)/2A u (T ), j0

Q

2 2u∈D(A)⇐⇒u∈ H (Ω ) u (T ) (T )i 0 1i=1

x = 0

∞g C

Q3/2 2 3u ∈ D(A )⇐⇒ u ∈ H (Ω ) u (T ) (T ) Aui 0 1i=1

(T )0

a2′′ + ′′ −u (0 )−u (0 ) = u(0)

2c

∞f f C

f(0) = 0 f

u(t,·)

t (P) M(t)

σ(t) u(t,·) t

Z

1 2 RM(t) = x|u(t,x)| dx 2 |u(t,x)| dx

Z 1 2 2 σ(t) = R (x−M(t)) |u(t,x)| dx

2|u(t,x)| dx

M σ t

a = 0 a > 0 g =g = 01 2 1 2

f : Ω → k∈{1;2}k k

quan

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9(f ,f )∈D(A) f (−∞;0)1 2 1

f ≡ 02

Ω (u ,u )1 1 2√Q2 + 2 2 1 + 0 +C ( , L (Ω ))∩C ( ,D( A))∩C ( ,D(A))kk=1

+∗(t,x) ×Ω1

h √ i1 −1 2 2u (t,x) = F cos( c ω t)Ff (ω)1 1ω → xπ

√

2 2c ω −a2

q√

a22 2 c ω −a ω≥ 2 2 c q q√

a a2 2 2 2i a −c ω − ≤ω≤2 2 2c c q√ a2 2 2 − c ω −a ω≤−2 2c

+∗(t,x) ×Ω u (t,x) =2 2

√ !Z ∞ √ 1 2 21 i c ω −a x 2ωc2c2 2− cos( c ω t)e √ Ff (ω)dω12 22π −∞ ωc+ c ω −a2

2a =a =m1 2

q1 −1 2 2 2u (t,x) = F cos( m +c ξ t)Ff(ξ)1 ξ → x

π

f

Z Z

1 1i(ξx−ω(ξ)t) i(ξx+ω(ξ)t)u (t,x) = e Ff(ξ)dξ + e Ff(ξ)dξ1

2π 2π

√

2 2 2ω(ξ) = m +c ξ (ξx−ω(ξ)t)

′x/t =ω (ξ)

2c ξ′ω (ξ) = √

2 2 2m +c ξ

ω

[ξ ;ξ ] Ff [ξ ;ξ ]1 2 1 2

′ ′t u(.,t) [ω (ξ )t;ω (ξ )t]1 2

t

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10

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