7. Diffusive Partial Differential Equations

• Scattering problems in quantum mechanics: reflection and transmission

• The time-dependent Schrodinger Equation: wave packets

• Start with something simpler: the Diffusion Equation

• Discretization and differencing of the diffusion equation

• Exercise 7.1:   Solution by explicit differencing. The goal is to reproduce a portion of the fundamental solution of the diffusion equation.

  Take the diffusion coefficient to be D = 1, and work on a uniform grid of 201 points running from x = -10 to x = 10. Start with the fundamental solution at t = 1 and integrate the system forward in time to t = 10 in steps of size dt. Plot your solution at t = 10 for dt = 0.005, 0.01, 0.05, 0.1.

  Here are some starting points in C++ and Python.