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Math 6420 • PDE

Fall 2002 • Question 5

Question

Math 6420 • PDE Fall 2002 • Question 5

Consider the differential operator \(Lu=u''+k^2u\), \(k\ne 0\), subject to homogeneous boundary conditions \(u(0)=u(1)=0\). Show by direct calculation that the associated Green’s function is \[ g(x,y)= \begin {cases} \dfrac {\sin kx\,\sin k(y-1)}{k\sin k}, & 0\le x<y\le 1,\\[1.2ex] \dfrac {\sin ky\,\sin k(x-1)}{k\sin k}, & 0\le y<x\le 1, \end {cases} \] provided \(\sin k\ne 0\). Using the fact that \(L\) is self-adjoint, show that the solution to the equation \[ u''(x)+k^2u(x)=f(x) \] with inhomogeneous boundary data \(u(0)=\alpha \), \(u(1)=\beta \) is \[ u(x)=\int _0^1 g(x,y)f(y)\,dy-\frac {\alpha }{\sin k}\sin k(x-1) +\frac {\beta }{\sin k}\sin kx \] provided \(\sin k\ne 0\). What happens when \(k=m\pi \)?

Consider the differential operator \(Lu=u''+k^2u\), \(k\ne0\), subject to homogeneous
boundary conditions \(u(0)=u(1)=0\). Show by direct calculation that the associated Green's
function is
\[
g(x,y)=
\begin{cases}
\dfrac{\sin kx\,\sin k(y-1)}{k\sin k}, & 0\le x<y\le 1,\\[1.2ex]
\dfrac{\sin ky\,\sin k(x-1)}{k\sin k}, & 0\le y<x\le 1,
\end{cases}
\]
provided \(\sin k\ne0\). Using the fact that \(L\) is self-adjoint, show that the solution to
the equation
\[
  u''(x)+k^2u(x)=f(x)
\]
with inhomogeneous boundary data \(u(0)=\alpha\), \(u(1)=\beta\) is
\[
  u(x)=\int_0^1 g(x,y)f(y)\,dy-\frac{\alpha}{\sin k}\sin k(x-1)
  +\frac{\beta}{\sin k}\sin kx
\]
provided \(\sin k\ne0\). What happens when \(k=m\pi\)?

Topic Tags

Adjoint Operators and Riesz Representation Fundamental Solutions Linear Programming and Optimization Self-Adjoint and Positive Operators