# (Easy?) Boundary Value Problem

• Feb 28th 2011, 11:41 AM
mstrfrdmx
(Easy?) Boundary Value Problem
Hi,

I have to solve a boundary value problem that involves finding function$\displaystyle \varphi$ that is harmonic on an annulus centered at $\displaystyle 1+i$ in the plane and takes the value zero on the inner boundary (radius 1) and ten on the outer (radius 2.) Once in possession of this function, I have to evaluate it at (0,0). The solutions I was given don't give the function but just the value of phi (1/2.) I'm not getting this answer; I tried setting up a system of equations with the solution form $\displaystyle \varphi =A+\ln |z-(1+i)| +B$ but kept getting $\displaystyle A=\frac{10}{\ln 2}, B=0$ I'd appreciate at least an understanding of why their answer is right. Also, the solution should involve elementary methods (no crazy transforms please.) Thanks!
• Mar 1st 2011, 03:32 AM
Opalg
Quote:

Originally Posted by mstrfrdmx
Hi,

I have to solve a boundary value problem that involves finding function$\displaystyle \varphi$ that is harmonic on an annulus centered at $\displaystyle 1+i$ in the plane and takes the value zero on the inner boundary (radius 1) and ten on the outer (radius 2.) Once in possession of this function, I have to evaluate it at (0,0). The solutions I was given don't give the function but just the value of phi (1/2.) I'm not getting this answer; I tried setting up a system of equations with the solution form $\displaystyle \varphi =A+\ln |z-(1+i)| +B$ but kept getting $\displaystyle A=\frac{10}{\ln 2}, B=0$ I'd appreciate at least an understanding of why their answer is right. Also, the solution should involve elementary methods (no crazy transforms please.) Thanks!

Presumably you mean $\displaystyle \varphi(z) =A\ln |z-(1+i)| +B$. Your answer looks correct to me, giving $\displaystyle \varphi(0) = 5$. Are you sure that the book says 10, not 1, for the value on the outer boundary? If it really says 10, then my guess is that there's a misprint in the book.