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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2016 • Question 10

Question

Math 6420 • PDE Spring 2016 • Question 10

Solve Laplace’s equation \[ \phi _{xx}+\phi _{yy}=0 \quad \text {in the domain}\quad D=\{z:\lvert z-1\rvert >1,\ \lvert z-2\rvert <2\} \] subject to the boundary condition \[ \phi (x,y)=a\text { when }\lvert z-1\rvert =1 \qquad \text {and}\qquad \phi (x,y)=b\text { when }\lvert z-2\rvert =2 \] (\(a\) and \(b\) are real parameters).

Raw LaTeX

Solve Laplace's equation
\[
  \phi_{xx}+\phi_{yy}=0
  \quad\text{in the domain}\quad
  D=\{z:\lvert z-1\rvert>1,\ \lvert z-2\rvert<2\}
\]
subject to the boundary condition
\[
  \phi(x,y)=a\text{ when }\lvert z-1\rvert=1
  \qquad\text{and}\qquad
  \phi(x,y)=b\text{ when }\lvert z-2\rvert=2
\]
(\(a\) and \(b\) are real parameters).

Topic Tags

Boundary Value Problems Finite Difference Boundary Value Problems Laplace Equation Linear Programming and Optimization