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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2004 • Question 11

Question

Math 6410 • ODE Fall 2004 • Question 11

Distributions. Suppose \(f(\xi ),g(\xi )\in C^2_0(\mathbb {R})\) are compactly supported twice continuously differentiable functions. Then \[ u(x,t)=f(x+t)+g(x-t) \tag {2} \] is a classical solution of the wave equation \[ u_{tt}-u_{xx}=0\qquad \text {for }-\infty <x,t<\infty . \tag {3} \] Show that if \(f(\xi ),g(\xi )\in C_0(\mathbb {R})\) are merely compactly supported continuous functions, then (2) still defines a solution to (3) in the sense of distributions.

Raw LaTeX

\textbf{Distributions.}
Suppose \(f(\xi),g(\xi)\in C^2_0(\R)\) are compactly supported twice continuously
differentiable functions. Then
\[
  u(x,t)=f(x+t)+g(x-t) \tag{2}
\]
is a classical solution of the wave equation
\[
  u_{tt}-u_{xx}=0\qquad \text{for }-\infty<x,t<\infty. \tag{3}
\]
Show that if \(f(\xi),g(\xi)\in C_0(\R)\) are merely compactly supported continuous
functions, then (2) still defines a solution to (3) in the sense of distributions.

Topic Tags

Distributions and Weak Solutions Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions Wave Equation and Energy Methods