1. Laplace equation's solution for (x,y) plane on a circle x>0, y>0 in polar coordinates with the conditions :
u(r=0,μ) = sinμ/sin∏/8 + sin μ/2 + cos μ/2 if 0≤μ≤π/2
u(r,μ=0) = √r if 0≤r≤1
u(r,μ=∏/2) = r^1/4 + √(2r) if 0≤r≤1
The equation is linear in u.
Use the variable separation method.
Sorry for my bad english, I hope it's understandable.
Are you sure about the first condition? I suspect you want
instead. Also, the last condition currently reads
, the ^1 is redundant.
Recall the Laplacian in polar:
. Separate variables you see
, or a sum of these, are what you should try.