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Question

Math 6420 • PDE

Fall 2015 • Question 8

Question

Math 6420 • PDE Fall 2015 • Question 8

Solve Laplace’s equation \[ \phi _{xx}+\phi _{yy}=0 \quad \text {in the domain}\quad D=\{(x,y)\in \mathbb {R}^2:\ x^2+y^2>1,\ x^2+(y+1)^2<4\} \] subject to the boundary condition \[ \phi (x,y)=a\text { when }x^2+y^2=1 \qquad \text {and}\qquad \phi (x,y)=b\text { when }x^2+(y+1)^2=4 \] (\(a\) and \(b\) are real parameters; the two circles touch each other; express your answer in terms of the original real variables \(x\) ans \(y\)).

Raw LaTeX

Solve Laplace's equation
\[
  \phi_{xx}+\phi_{yy}=0
  \quad\text{in the domain}\quad
  D=\{(x,y)\in\mathbb{R}^2:\ x^2+y^2>1,\ x^2+(y+1)^2<4\}
\]
subject to the boundary condition
\[
  \phi(x,y)=a\text{ when }x^2+y^2=1
  \qquad\text{and}\qquad
  \phi(x,y)=b\text{ when }x^2+(y+1)^2=4
\]
(\(a\) and \(b\) are real parameters; the two circles touch each other; express your answer
in terms of the original real variables \(x\) ans \(y\)).

Topic Tags

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