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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1996 • Question 4

Question

Math 6420 • PDE Fall 1996 • Question 4
Let \(Lu(x)=\int_0^x(t-x)u(t)\,dt\). **(a)** Find the resolvent operator associated with the integral equation \(u=f(x)+Lu\). **(b)** Solve the integral equation \(u=1+Lu\). **(c)** Show that the integral equation \(u=1+Lu\) is equivalent to a differential equation and give the equation.

Raw LaTeX

Let \(Lu(x)=\int_0^x(t-x)u(t)\,dt\).
\begin{enumerate}
\item[(a)] Find the resolvent operator associated with the integral equation
\(u=f(x)+Lu\).
\item[(b)] Solve the integral equation \(u=1+Lu\).
\item[(c)] Show that the integral equation \(u=1+Lu\) is equivalent to a differential
equation and give the equation.
\end{enumerate}
\end{enumerate}

Topic Tags

Cauchy Integral Formula and Integral Theorems Linear Programming and Optimization Resolvent Sets and Spectral Bounds Spectral Theory of Operators