Consider the heat equation that describes a disc, of radius
R=3. ∂t∂u=Δu,u=u(r,θ,t),0≤r≤3,0≤θ≤π/2,t≥0,u∣r=3=10+2cos2θ,u∣t=0=100.
Find the stationary solution.
u[infinity](r,θ).
Hint In polar coordinates, bounded solutions to Δu=0 have the form
u(r,θ)=a0+∑m=1[infinity]rm(amcos(mθ)+bmsin(mθ)).

(General solutions to Δu=0 in R2, in polar coordinates, have the form
u(r,θ)=a0+A0lnr+∑m=1[infinity]rm(ancos(mθ)+bmsin(mθ))+∑m=1[infinity]r−m(Amcos(mθ)+βmsin(m ,

but for bounded solutions terms with r−m,m∈N, and lnr, are discarted.)