Green's Functions and Propagators

    Any complete set of functions, not just eigenfunctions, can be used to construct a solution to a linear PDE if we know how these functions evolve in time. Dirac delta functions are a perfectly acceptable alternative to the trigonometric eigenfunctions that were studied previously. The time evolution of these delta functions is called a Green (or Green's) function.

  
Figure: Decomposition of sawtooth into Green's functions.

We can approximate a Dirac delta function using a narrow Gaussian if we are careful not to make the width, a, smaller than the grid spacing in the simulation.

Figure shows how an initial saw-tooth disturbance is decomposed into delta functions.

• Green's function Visualization
  Determine how a narrow delta function located in the center of the medium evolves in time under the action of the classical wave, diffusion and Schrödinger equations. Select the equation type and then select contour plot. Use [Pref]|[Speed] to slow down the the display and to obtain good time resolution. Stop the simulation after the first rendering of the contours.

  a.

  Can you guess the analytic form of the function seen on the contour plot?

  b.

  Continue to run the program until the disturbance reaches the walls. Can you still guess the analytical form?

  c.

  Place the initial Gaussian off center. Do any of the contour plots just shift when the initial position changes or does the character of the contour plot change? Can you guess the analytical form even with arbitrary initial position for any of the PDEs?

  You have found the Green's function for a particular PDE if you were able to answer the last question.

Clearly Green's functions exist for all of the linear PDEs in Table ; you just displayed them as contour plots in the previous exercise! In practice, it is often difficult to find an analytic expression for these functions even if they are known to exist. Green's functions depend on the position of the delta function, , at initial time, , and must predict the disturbance at any other point in the medium, x, at any subsequent time, t. The disturbance will, of course, only depend on the elapsed time, .

The Green's function can now be used to calculate the wavefunction at any subsequent time if the initial disturbance, , is known.

Since the Green's function for time-dependent PDEs allows one to calculate the wavefunction at any time, , using the initial conditions, , these functions are sometimes referred to as propagators.

• Maxwell Reciprocity
    Maxwell reciprocity states that the response at due to a delta function at is equal to the response at due to a delta function at --- provided that the elapsed times are the same:

  

  Use detector graphs to demonstrate this symmetry. Set up the system with a narrow initial Gaussian and a detector at another point. Run the simulation and then freeze the graph using the blue attribute button on the detector graph. Create another detector graph and reverse the positions of the delta function and the detectors. Notice that the two detector graphs are identical.

A formal representation of Green's functions is not hard to derive . Substituting the definition of the Fourier coefficients, , into an eigenfunction expansion of a solution, we obtain

If we interchange the order of the summation and integration and compare the result to eq:GreenSol, we obtain the Green's function as a sum over eigenfunctions.

• Finite Space Functions
  Use a function plotting package such as MathCAD, Maple, or Mathematica to plot the eigenfunction expansion of the Green's function for the diffusion equation for different values of .

  

  Does the expansion work best for short or long times? Close to or far from the boundary? Compare your result to contour plots produced by WAVE .

    
  Table: Green's functions for various PDEs with boundaries at infinity.
  

• Infinite Space Functions
  The Green's function expansion simplifies considerably if the medium is very large so that boundaries can be ignored. The summation in eq:GreenExpan becomes an integral which can be evaluated for the wave equation, Schrödinger, and diffusion equations. Compare the short time evolution of a narrow Gaussian located near the center of the medium to the functions listed in Table .