数学物理方法课件
一个端点自由2 2 2 ( 2 a )u ( x , t ) = 0 t x 2
(0 < x < ∞ )
设初始条件为 边界条件 应该是偶延拓
u t =0 = ( x) uxx =0
和
ux
t =0
= ψ ( x)
=0( x ≥ 0) ( x < 0)
( x) Φ( x) = ( x)
ψ ( x) Ψ ( x) = ψ ( x)x + at
( x ≥ 0) ( x < 0)
1 1 u ( x, t ) = [Φ ( x + at ) + Φ ( x at )] + ∫at Ψ (ξ )dξ 2 2a x x + at 1 1 [ ( x + at ) + ( x at )] + ∫ ψ (ξ ) d ξ 2 a x at 2 u ( x, t ) x + at 1 1 1 [ ( x + at ) + ( at x )] + ψ (ξ ) d ξ + 2 2a ∫ 2a 0
(t ≤ x / a )at x
∫0
ψ (ξ ) d ξ
(t ≥ x / a )
(t ≥ x / a )
1 2a
x + at
1 ∫at Ψ (ξ ) dξ = 2a x 1 2ax + at
x + at
∫0
1 Ψ (ξ ) dξ + 2a0
1 ∫at Ψ (ξ ) dξ = 2a xx + at
0
x + at
∫0
1 ψ (ξ ) dξ + 2a 1 2aat x
x at
∫ ψ ( ξ ) dξ
0
=
∫0
ψ (ξ ) dξ
1 1 ∫ at ψ ( ξ ) d ( ξ ) = 2a 2a ( x )
∫0
ψ (ξ ) dξ +
∫ ψ (ξ ) d (ξ )0