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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2002 • Question 9

Question

Math 6420 • PDE Fall 2002 • Question 9

Use Fourier transforms to show that the solution of the diffusion equation \[ \frac {\partial \phi }{\partial t}=\frac {\partial ^2\phi }{\partial x^2},\qquad -\infty <x<\infty \] with initial data \(\phi (x,0)=h(x)\) takes the form \[ \phi (x,t)=\int _{-\infty }^{\infty } g(x-x',t)h(x')\,dx' \] where \[ g(x,t)=\frac {e^{-x^2/4Dt}}{2\sqrt {\pi Dt}}. \] Determine \(\lim _{t\to 0}g(x,t)\).

Raw LaTeX

Use Fourier transforms to show that the solution of the diffusion equation
\[
  \frac{\partial\phi}{\partial t}=\frac{\partial^2\phi}{\partial x^2},\qquad
  -\infty<x<\infty
\]
with initial data \(\phi(x,0)=h(x)\) takes the form
\[
  \phi(x,t)=\int_{-\infty}^{\infty} g(x-x',t)h(x')\,dx'
\]
where
\[
  g(x,t)=\frac{e^{-x^2/4Dt}}{2\sqrt{\pi Dt}}.
\]
Determine \(\lim_{t\to0}g(x,t)\).

Topic Tags

Fourier Transform and Sampling Theory Heat Equation and Parabolic Equations Linear Programming and Optimization Real Integral Evaluation by Complex Methods