ML rgr ML ML ML (,, ) σ τ τ u + + τ σ τ v + + τ τ σ + + (.) uv,,,, σ, σ, σ, τ, τ, τ t (Hook) σ λθ + ε, τ γ σ λθ + ε, τ γ σ λθ + ε, τ γ λ, E ν ν λ E, E ( + ν)( ν) ( + ν) Θ Θ ε + ε + ε (.) ε, ε, ε, γ, γ, γ u u v ε, γ + v v ε, γ + (.) u ε, γ + (.)(.)(.) (.) (.)
u Θ u u u ( λ + ) + + + v Θ v v v ( λ + ) + + + (.) Θ ( λ + ) + + + (.) u + + v + + + + (.),, S ( λ + ) + + + + + + uv,, S uv, SH (.8) (.8) S S (.9), S S λ +, S (.) Laplac + + (.) (.) ML (.9) (.)(.)(.)
ML ML (.8) ( λ + ) + + (.),, + +,, + +,, (.) (.) p u v K + + u p v p p,, (.) uv,,,, p K K c c : (.)(.)(.),,, (.)(.) ( λ + ) + + ( λ + ) (.),, + + S S, S, S + + SH SH, SH, SH (.) (.)
, S, SH,, S SH,, S SH ML (.)(.) ( λ + ) ( λ + ) S ( λ + ) ( λ + ) S ( λ + ) ( λ + ) S ( + + ) (.8) ( + + ) (.9) S ( + + ) ( + + ) S ( + + ) ( + + ) S SH SH SH ( ) + + ( ) + + ( ) + + SH SH SH (.) ML S SH ( ) λ + S SH (.) ML
(.)(.) (.)(.) ( λ ) ( λ ) + + + + + t + + + + (.) + +,,, (.),, ( λ ) ( λ ) + + + + + t + + + + + + (.) + + + ( + + ) Γ Γ Γ,,,, (.)(.)(.) S + + + t + + S + + S + + S + + + ( + + ) Γ Γ SH + + + t + + SH + + SH + + SH + + + ( + + ) Γ Γ (.)(.). 8 (,, ).,,,,, (.) (.)
8 o l l l. 8,, Χ, W, W, W (.) a, b, c, W, W, W (.) [ 8 ], (.) Χ W W W { 8} { } 8 { } 8 { 8} { 8} { 8} W W W,,, W W W W W W W W W (.8) (.9)
{ } 8 { 8} { 8} { } W W W 8 { 8} { } 8,, 8 (.) l l l 8 + l l l 8 + l + l l 8 l + l l 8 ( )( )( ) ( )( )( ) ( )( )( ) ( )( )( ) l 8 l + l + l 8 l + l + l 8 + l + l 8 l 8 + l + l ( )( )( ) ( )( )( ) ( )( )( ) ( )( )( ) (.),,,, (.) (.),(.)(.) W t W + W + W + ( λ ) ( λ ) ( λ ) ( λ ) + W + + W + W + W + W + W + W + W + W + W W Γ+ W Γ+ Γ Γ W Γ Γ (.) (.),(.)(.),(.) (.) W, W, W, W M + K F (.)
8, m m M m m m (.) ( ) ( ) ( ) ( ) λ λ λ λ + + + + m k k k k m K k m k m k k k (.) F Γ Γ Γ Γ Γ Γ (.) (.),(.) + M K F (.8) + M K F (.9) S S S S m k k k k m K k m k m (.) F Γ Γ Γ Γ Γ Γ (.) SH SH SH SH m k k k k m K k m k m (.)
F Γ Γ Γ Γ Γ Γ Χ Χ (.) (.)(.8)(.9) M + K F (.) M + K F (.) M + K F (.) ML ML (.8)(.)(.8) (.8) ( λ ) ( λ ) + + + t + ( λ + ) + ( λ + ) + ( λ ) ( λ ) + + + (.) ( + + ) ( + + ) + + + + ( + + ) + +,,,,, (.),, ( λ ) ( λ ) + + + + ( λ + ) + ( λ + ) + ( λ ) ( λ ) + + + + + + + + + + + + ( + + ) ( + + ) Γ Γ (.) (.)(.)(.) 9
S S + + + + S + + + S + + S + + S + + + ( + + ) ( + + ) Γ Γ SH SH + + + + SH + + + SH + + SH + + SH + + + ( + + ) ( + + ) Γ Γ (.) (.) (.)(.).,,,,,, (.),, Χ W, W, W, [ 8] (.) Χ W W W,, Χ Χ, W W, W W, W W Χ W W W (.) (.) { } 8 { 8} { } 8 (.8) (.)(.)(.)
W W W W W W W W W + W + W ( ) ( λ + + λ + ) ( ) ( λ + + λ + ) ( ) ( ) λ + + λ + + W + W + W + W + W + W + W W Γ+ Γ W Γ Γ+ W Γ Γ + + (.9) W, W, W, W, W, W ML ML ML ML ML (.9) M + K F (.) M ML m m m m m m (.) K ML ( λ ) m ( λ ) k ( λ + ) m ( λ + ) k ( λ + ) ( λ + ) + + m k k k k m k k k m k k k m (.)
ML F ML ML (.) (.),(.) ML ML ML ML ML + M K F (.) ML ML ML ML ML + M K F (.) S S S S ML S S m k m k m k k k k m K k k k m k k k m (.) ML F ML ML (.) SH SH SH SH ML SH SH m k m k m k k k k m K k k k m k k k m (.8)
F ML ML Χ Χ Χ ML Χ Χ Χ (.9) (.) I I I M+ + K I I I M,, K,, (.) (.) 8. u U, v, W (.) U W { u u u u u u u u8} { v v v v v v v v } 8 { } 8 (.)(.) (.)(.)(.) Χ U W U m k k k m k k k W m k Χ m, k, k, k k, k (.) (.) (.) (.) (.)(.)
ϕ (.) m k k k U m k k k,, ϕ m k W Χ ML ML M,, K ϕϕ ϕϕ ϕϕ (.8) M M,, K K (.9) ϕϕ ϕϕ ϕϕ M,, K ML M M,, K K ϕ ϕ ϕ M M,, K K ϕ ϕ ϕ (.) (.),(.8),(.9)(.),(.),(.)ML M + K F (.) M + K F (.) M + K F (.) O O O O O M + K F (.) O { Χ } O ML O { Χ (.) Χ Χ Χ (.) } (.) O O O O + (.) Q O O O O O O M K, Q M F (.) (.)(.)
a () t () t I I I I + (.) Q I I I I, () t, I I I Q I M K M a () t (.)(.8)(.9) + Q() t (.9) uvuv,,,,, ML,,,,,,,,,,,,,,,,,,,, ML 8,,,,,,,,,,,,,,,,, ML (.9) Q, (.8) I H R, Q a () t, (.) H R I I I I M K, H M ML (.9)Rug Kutta (.9) Rug Kutta + + ( ) + + + (.) ( ) t + Q t + + Q + t + + Q + t ( ( + ) + Q+ ) ( t), + ( t + t) t Q Q Q Q + Q Q + ( t ), t, ( t t) + + (.) t t (.)(.)
t + + t + ( L + L + L) + + ( L + L + L + L) (.) + + ( S + S + S + S) L t + H + R + a t t L t H + L + + + R + L + a R + t t t t L t H + L + + + L + R + L + L + a R R + t L t H( + L) + + t + L t t t + R + t LR + + L+ RLR + LR+ HL+ a+ (.) L + H + R R ( ) L t + RL + HL R R (.) S tl R S tl + + HL + RS ( t ) R S tlr + t + L+ HL + RS S tlr + t + L+ HL + RS (.)