! > 0 U− > 0 u! (x, µ) 1 µ∂x u! + ! (u! − #u! $) = 0 , u! (−1, +µ) = U− , u! (+1, −µ) = U+ , (T ) #u! $(x) = u! ∈ L∞ ([−1, 1]2 ) % 1 −1 U+ > 0 u! ≡ |x| < 1 , |µ| < 1 , 0 < µ < 1, 0 < µ < 1, u! (x, µ) dµ . 2 !>0 x &→ u! (x, µ) C1 [−1, 1] ! u! (x, µ) = & !k uk (x, µ) . k≥0 u0 u1 u2 u0 U (x) = U+ + U− U+ − U− + x , |x| < 1 . 2 2 v! (x, µ) = u! (x, µ) − U (x) + !µU $ U (x) x (v! − #v! $(2L2 ([−1,1]2 ) ≤ C0 !3 U+ U− [−1, 1]2 C0 v! S! (x, µ) = 1! (v! (x, µ) − #v! $(x)) √ '% 1 (1/2 2 2 |v! (x, µ)| ≤ !|µ||U | + S! (x, µ) dx |µ| −1 $ U$ v! (x, µ) α>0 % C1 1 −1 % 1 −1 1[α,1] (|µ|)v! (x, µ)2 dxdµ ≤ C1 !2 |U $ |2 + C2 C0 ! α2 C2 α>0 % 1 −1 % α −α v! (x, µ)2 dxdµ ≤ 2C0 !3 + 8α(v! (2L2 ([−1,1]2 ) ! ∈]0, 1] √ (u! − U (x)(L2 ([−1,1]2 ) ≤ C3 |U $ | ! C3 ν>0 (0, 1) ∂u ∂2u −ν 2 =0 (x, t) ∈ (0, 1) × R+ ∗ ∂t ∂x + u(t, 0) = u(t, 1) = 0 t ∈ R∗ u(0, x) = u0 (x) x ∈ (0, 1), u0 (x) ∆x = 1/(N + 1) > 0 [0, 1] (tn , xj ) = (n∆t, j∆x) unj θ ∈ [0, 1] u(t, x) un+1 − unj j ∆t n≥1 un0 = − n ≥ 0, j ∈ {0, 1, ..., N + 1}. (tn , xj ) θ ν ) n+1 θ(uj+1 − 2un+1 + un+1 j j−1 ) (∆x)2 * +(1 − θ)(unj+1 − 2unj + unj−1 ) = 0 unN +1 =0 ∆t > 0 N u0j = u0 (xj ) j ∈ {1, ..., N }, unj 1 ≤ j ≤ N c = ν∆t/(∆x)2 Un ∆t AU n+1 = BU n A N ×N B M U n+1 A Un L∞ L2 U n+1 − U n + KU ∗ = 0, ∆t U ∗ = θU n+1 + (1 − θ)U n K 2U ∗ = (U n+1 + U n ) + (2θ − 1)(U n+1 − U n ). 2U ∗ N & j=1 ∆x|un+1 |2 ≤ j L2 N & j=1 1/2 ≤ θ ≤ 1 ∆x|u0j |2 .