北京大学 数学物理方法 课件01
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Superpositionability of Solutions General Solution of PDE’s
1
2
d’Lambert
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Laplace
∂ 2u ∂ 2u + =0 ∂x2 ∂y 2
Superpositionability of Solutions General Solution of PDE’s
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
∂ 2 u(x, y) =0 ∂x2 u(x, y) = c1 (y) + xc2 (y) c1 (y) c2 (y) ∂ 2 u(x, y) =0 ∂x∂y u(x, y) = c1 (x) + c2 (y) c1 (x) c2 (y)
Laplace
∂ 2u ∂ 2u + =0 ∂x2 ∂y 2
Superpositionability of Solutions General Solution of PDE’s
Superpositionability of Solutions General Solution of PDE’s
u L[u] = f
L[u] = f
u
u1
u2 L[u1 ] = 0
L[u] = 0 L[u2 ] = 0 c 1 u1 + c 2 u2 L[c1 u1 + c2 u2 ] = 0
Superpositionability of Solutions General Solution of PDE’s
L[u] = f u L f
f ≡0
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
u1
u2 L[u1 ] = f u1 − u2
L[u] = f L[u2 ] = f
L[u1 − u2 ] = 0
= +
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
C. S. Wu
Outline
1
2
d’Lambert
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
u1
u2 L[u1 ] = f u1 − u2
L[u1 − u2 ] = 0
= +
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
L[u] = f L[u2 ] = f
L[u1 − u2 ] = 0
= +
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
u L[u] = f
L[u] = f
1
2
d’Lambert
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
L
L ∂ 2u 2 2 ∂2 −a u = f L ≡ 2 −a2 2 ∂t ∂t ∂u ∂ −κ 2 u = f L ≡ −κ 2 ∂t ∂t Poisson
∂ 2 u(x, y) =0 ∂x2 u(x, y) = c1 (y) + xc2 (y) c1 (y) c2 (y) ∂ 2 u(x, y) =0 ∂x∂y u(x, y) = c1 (x) + c2 (y) c1 (x) c2 (y)
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
Superpositionability of Solutions General Solution of PDE’s
∂ 2 u(x, y) =0 ∂x2 u(x, y) = c1 (y) + xc2 (y) c1 (y) c2 (y) ∂ 2 u(x, y) =0 ∂x∂y u(x, y) = c1 (x) + c2 (y) c1 (x) c2 (y)
u
u1
u2 L[u1 ] = 0
L[u] = 0 L[u2 ] = 0 c 1 u1 + c 2 u2 L[c1 u1 + c2 u2 ] = 0
c1
c2
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Laplace
∂ 2u ∂ 2u + =0 ∂x2 ∂y 2
Superpositionability of Solutions General Solution of PDE’s
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
∂ 2 u(x, y) =0 ∂x2 u(x, y) = c1 (y) + xc2 (y) c1 (y) c2 (y) ∂ 2 u(x, y) =0 ∂x∂y u(x, y) = c1 (x) + c2 (y) c1 (x) c2 (y)
ξ = x + iy, η = x − iy
∂u ∂u ∂ξ ∂u ∂η ∂u ∂u = + = + ∂x ∂ξ ∂x ∂η ∂x ∂ξ ∂η ∂u ∂u ∂ξ ∂u ∂η ∂u ∂u = + =i − ∂y ∂ξ ∂y ∂η ∂y ∂ξ ∂η
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
L[u] = f u L f
f ≡0
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Laplace
∂ 2u ∂ 2u + =0 ∂x2 ∂y 2
Superpositionability of Solutions General Solution of PDE’s
ξ = x + iy, η = x − iy
∂u ∂u ∂ξ ∂u ∂η ∂u ∂u = + = + ∂x ∂ξ ∂x ∂η ∂x ∂ξ ∂η ∂u ∂u ∂ξ ∂u ∂η ∂u ∂u = + =i − ∂y ∂ξ ∂y ∂η ∂y ∂ξ ∂η
c1
c2
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
u1
u2 L[u1 ] = f u1 − u2
Superpositionability of Solutions General Solution of PDE’s
L[u] = f u L f
f ≡0
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
wavepropagationinin?nitestringdlambertsolution22?u?u?a20?t2?x2xatx?at?u?u??u??u?u?x??x??x???u?u??u??u?ua??t??t??t??c
Outline
2007
C. S. Wu
Outline
1
2
d’Lambert
2 2
u=f
L≡
2
L[α1 u1 +α2 u2 ] = α1 L[u1 ] + α2 L[u2 ]
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
L
L ∂ 2u 2 2 ∂2 −a u = f L ≡ 2 −a2 2 ∂t ∂t ∂u ∂ −κ 2 u = f L ≡ −κ 2 ∂t ∂t Poisson
2 2
u=f
L≡
2
L[α1 u1 +α2 u2 ] = α1 L[u1 ] + α2 L[u2 ]
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
ξ = x + iy, η = x − iy
∂u ∂u ∂ξ ∂u ∂η ∂u ∂u = + = + ∂x ∂ξ ∂x ∂η ∂x ∂ξ ∂η ∂u ∂u ∂ξ ∂u ∂η ∂u ∂u = + =i − ∂y ∂ξ ∂y ∂η ∂y ∂ξ ∂η
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
u1
u2 L[u1 ] = f1 L[u2 ] = f2 c 1 u1 + c 2 u2 L[c1 u1 + c2 u2 ] = c1 f1 + c2 f2
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String
Superpositionability of Solutions General Solution of PDE’s
L[u] = f u L f
f ≡0
C. S. Wu
Main Properties of Linear PDE’s Wave Propagation in Infinite String