The Viscous Burgers Equation is one of the simplest nonlinear advection diffusion equations.
Analytical solutions exist via the Cole-Hopf transformation. Interestingly, it is known to admit a simple traveling wave solution
under certain conditions.
Pictured right, numerical solution of the traveling wave via explicit forward-Euler method.

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The Inviscid Burgers Equation enebles the formation of "Shockwaves" due to its
lack of any diffusion terms. For certain initial conditions the waveform will develop a steep gradient which
appears to want to tumble over itself. After formation of the shock, numerical solutions breakdown entirely,
and analytic solutions (if they are avaliable) are considered invalid after the time of shock formation.

Pictured right, shockwave formation and solution breakdown, numerical solutions using 3 different orders of Adams-Bashfourth methods

time evolution of ferromagnetic materials

Consider an elementary ferromagnet which has an average magnetic moment, for example, a small sphere of iron, fixed in space. When immersed in an external magnetic field (H), the sample's magnetic moment (M) will dynamically reorient according to the Landau-Lifshitz equations. One observes both a circular precession about the magnetic field axis, and losses which dampen the motion until the spin comes into alignment with the magnetic field.