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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1995 • Question 5

Question

Math 6420 • PDE Fall 1995 • Question 5

Let \(f(x)\) be in the Hilbert space \(L^2(-\infty ,\infty )\). Construct and use a Green’s function to find the solution in \(L^2(-\infty ,\infty )\) of the equation \[ u''-\alpha ^2u=f(x),\qquad (\alpha >0). \]

Raw LaTeX

Let \(f(x)\) be in the Hilbert space \(L^2(-\infty,\infty)\).
Construct and use a Green's function to find the solution in \(L^2(-\infty,\infty)\) of the
equation
\[
  u''-\alpha^2u=f(x),\qquad (\alpha>0).
\]

Topic Tags

Distributions and Weak Solutions Fundamental Solutions Hilbert Spaces and Orthogonality Linear Programming and Optimization