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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2009 • Question 5

Question

Math 6420 • PDE Fall 2009 • Question 5

  1. Let \(f(x)=|x|\). Find its first and second derivatives using operational calculus, that is, the theory of distributions.
  2. Using Green’s functions, find the solution to \[ \frac {d^2u}{dx^2}=f(x),\qquad u(0)=u(1)=0. \] Use the reproducing property of the delta distribution to formally verify that your solution satisfies the equation.

Raw LaTeX

\begin{enumerate}
\item[(a)] Let \(f(x)=|x|\). Find its first and second derivatives using operational calculus,
that is, the theory of distributions.

\item[(b)] Using Green's functions, find the solution to
\[
  \frac{d^2u}{dx^2}=f(x),\qquad u(0)=u(1)=0.
\]
Use the reproducing property of the delta distribution to formally verify that your solution
satisfies the equation.
\end{enumerate}

Topic Tags

Calculus of Variations Distributions and Weak Solutions Fundamental Solutions Linear Programming and Optimization