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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2017 • Question 8

Question

Math 6410 • ODE Spring 2017 • Question 8

Suppose the wave equation \[ \frac {\partial ^2u}{\partial t^2}-c^2\nabla ^2u=0,\qquad x\in \mathbb {R}^n,\qquad t>0, \] subject to initial conditions \(u(x,0)=0\), \(\frac {\partial u}{\partial t}(x,0)=h(x)\), for \(x\in \mathbb {R}^n\) has a solution denoted by \(u_h\). Find (and verify) the solution of the same equation subject to the initial conditions \(u(x,0)=g(x)\), \(\frac {\partial u(x,0)}{\partial t}=h(x)\), for \(x\in \mathbb {R}^n\) in terms of \(u_h\) and \(u_g\).

Raw LaTeX

Suppose the wave equation
\[
  \frac{\partial^2u}{\partial t^2}-c^2\nabla^2u=0,\qquad
  x\in\R^n,\qquad t>0,
\]
subject to initial conditions \(u(x,0)=0\), \(\frac{\partial u}{\partial t}(x,0)=h(x)\),
for \(x\in\R^n\) has a solution denoted by \(u_h\). Find (and verify) the solution of
the same equation subject to the initial conditions
\(u(x,0)=g(x)\), \(\frac{\partial u(x,0)}{\partial t}=h(x)\), for \(x\in\R^n\) in terms
of \(u_h\) and \(u_g\).

Topic Tags

Distributions and Weak Solutions Elliptic Equations Poincare Maps and Periodic Solutions Wave Equation and Energy Methods