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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2016 • Question 12

Question

Math 6410 • ODE Fall 2016 • Question 12

The fundamental solution for the Laplacian in two dimensional space is \(G(\mathbf {x},\mathbf {y})=\frac {1}{2\pi }\ln (\abs {\mathbf {x}-\mathbf {y}})\), where \(\mathbf {x}=(x_1,x_2)\). What is the Green’s function for the Laplacian operator on the first quadrant \(0<x_1,x_2<\infty \), subject to boundary conditions \(G(x_1,0)=0\), \(G_{x_1}(0,x_2)=0\)? Be sure to verify the validity of your solution.

Raw LaTeX

The fundamental solution for the Laplacian in two dimensional space is
\(G(\mathbf{x},\mathbf{y})=\frac{1}{2\pi}\ln(\abs{\mathbf{x}-\mathbf{y}})\), where
\(\mathbf{x}=(x_1,x_2)\). What is the Green's function for the Laplacian operator on
the first quadrant \(0<x_1,x_2<\infty\), subject to boundary conditions
\(G(x_1,0)=0\), \(G_{x_1}(0,x_2)=0\)? Be sure to verify the validity of your solution.

Topic Tags

Boundary Value Problems and Sturm-Liouville Theory Fundamental Solutions Laplace Equation Poincare Maps and Periodic Solutions