PDE with Fourier Transform

Description

This applet is intended to give some physical insight into using Fourier Transforms to solve partial differential equations. The equation shown here is which is a convection diffusion equation with proportional forcing. To give more insight only a single frequency component is shown. The subscripts indicate partial derivatives. The equation coefficients can be adjusted with the labeled sliders. The frequency and time can also be adjusted with the labeled sliders. And the time can be animated with the play button. The right side graph shows the frequency domain and the left side graph shows the physical domain. The initial condition is a adjusted by moving the point in the frequency domain graph. The initial condition is where is on the -axis and is on the -axis. The other points on the frequency domain graph are the current time Fourier Transform, Real and Imaginary components of the function and its two derivatives. The Fourier Transform of the partial differential equation is where is the Fourier Transform at frequency . This is now an ordinary partial differential equation with constant coefficients and the solution including the initial condition is . The physical space solution is the inverse Fourier Transform of . An outline of the solution is
  1. Take the Fourier Transform of the partial differential equation replacing each -derivative with a multiplication by
  2. Take the Fourier Transform of the initial condition.
  3. Solve the constant coefficient ordinary differential equation initial value problem in the frequency domain.
  4. Take the inverse Fourier Transform to get the physical solution at the given time.
This procedure works for linear partial differential equations with constant coefficients and results can be added together if the initial condition contains more frequencies.

Advection Diffusion equation with proportional forcing

Activities

  1. Move the point around the plane to see how the physical space curve changes.
  2. Play the solution to see how it progresses in time. Note the movements in the complex plane.
  3. Adjust the coefficients and try to decide which one of the coefficients controls the following: Movement of the Sine wave left or right Diffusion or flattening of the Sine wave Amplification of the Sine wave
  4. Vary the frequency and observe how it impacts the speed and diffusion rates.