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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2005 • Question 2

Question

Math 6420 • PDE Fall 2005 • Question 2

Consider the distribution \(g\) defined by \(\langle g,\phi \rangle =\int _{-\infty }^{\infty } e^{-|x|}\phi (x)\,dx\), for all test functions \(\phi \).

  1. Calculate the second derivative \(g''\) explicitly. Prove \(g''\) is a distribution of order zero.
  2. Find a differential equation for which \(g\) is a Green’s function.

Raw LaTeX

Consider the distribution \(g\) defined by
\(\langle g,\phi\rangle=\int_{-\infty}^{\infty} e^{-|x|}\phi(x)\,dx\), for all test
functions \(\phi\).
\begin{enumerate}
\item[(a)] Calculate the second derivative \(g''\) explicitly. Prove \(g''\) is a distribution of
order zero.
\item[(b)] Find a differential equation for which \(g\) is a Green's function.
\end{enumerate}

Topic Tags

Distributions and Weak Solutions Fundamental Solutions Linear Programming and Optimization Real Integral Evaluation by Complex Methods