  In the magnetic diffusion problem, a magnetic diffusion equation is coupled
by an Ohmic heating energy equation. The Ohmic heating can make the magnetic
diffusion coefficient (i. e., the resistivity) vary violently, and make the
diffusion a highly nonlinear process. For this reason, the problem is normally
very hard to be solved analytically. In this article, under the condition of a
step-function resistivity and a constant boundary magnetic field, we
successfully derived an exact solution for this nonlinear problem, which should
be an interesting thing in the area of partial differential equations. What's
more, the solution could serve as a valuable benchmark example for testing
simulation methods of the magnetic diffusion problem.
