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In the magnetic diffusion problem, a magnetic diffusion equation is coupledby an Ohmic heating energy equation. The Ohmic heating can make the magneticdiffusion coefficient (i. e., the resistivity) vary violently, and make thediffusion a highly nonlinear process. For this reason, the problem is normallyvery hard to be solved analytically. In this article, under the condition of astep-function resistivity and a constant boundary magnetic field, wesuccessfully derived an exact solution for this nonlinear problem, which shouldbe an interesting thing in the area of partial differential equations. What'smore, the solution could serve as a valuable benchmark example for testingsimulation methods of the magnetic diffusion problem.

An exact solution for the magnetic diffusion problem with a step-function resistivity model