Laplace equation in polar coordinate

[NapoleonZ]

Urr+(1/r)*Ur+(1/r^2)*Uθθ=0

a<r<b, 0<θ<w

with the conditions
U(r,0)=U1
U(r,w)=U2
U(a,θ)=0
U(b,θ)=f(θ)

[tiny-tim]

Hi NapoleonZ! Welcome to PF! :smile:
Urr+(1/r)*Ur+(1/r2)*Uθθ=0

a<r<b, 0<θ<w

with the conditions
U(r,0)=U1
U(r,w)=U2
U(a,θ)=0
U(b,θ)=f(θ)

Show us what you've tried, and where you're stuck, and then we'll know how to help. :smile:

[NapoleonZ]

I have got two sets of solutions

U(r,θ)=A*ln(r)+B*
U(r,θ)=(C*r^λ+D/r^λ)*(E*sinλθ+F*cosλθ)

My problem is the boundary conditions are nonhomogeneous, with which I cannot work out the coefficients.

[tiny-tim]

I have got two sets of solutions

U(r,θ)=A*ln(r)+B*
U(r,θ)=(C*r^λ+D/r^λ)*(E*sinλθ+F*cosλθ)

My problem is the boundary conditions are nonhomogeneous, with which I cannot work out the coefficients.

Hi NapoleonZ! :smile:

i] I'm a little confused about the conditions … U(a,θ)=0 seems incompatible with U(r,0)=U1
and U(r,w)=U2, if the conditions are continuous

ii] Doesn't the condition U(a,θ)=0 make it fairly clear what E and F are (unless λ = 0)?

[NapoleonZ]

Hi NapoleonZ! :smile:

i] I'm a little confused about the conditions … U(a,θ)=0 seems incompatible with U(r,0)=U1
and U(r,w)=U2, if the conditions are continuous

ii] Doesn't the condition U(a,θ)=0 make it fairly clear what E and F are (unless λ = 0)?


i] I'm confused too.
ii] No always, actually D=-C*a^(2n)