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. The solution takes four parameters as input: the fixed magnetic boundary \(B_0\), \(\eta _{\text {S}}\) and \(\eta _{\text {L}}\) that are resistivities below and above the critical energy density of a material, and the critical energy density \(e_{\text {c}}\) of the material. The solution curve B(x, t) possesses the characteristic of a sharp front, and its evolution obeys the usual self-similar rule with the similarity variable \(x/\sqrt{t}\).

在磁扩散问题中，磁扩散方程与欧姆热能方程耦合。欧姆加热会使磁扩散系数（即电阻率）剧烈变化，使扩散成为高度非线性的过程。因此，这个问题通常很难通过分析来解决。在本文中，在阶跃函数电阻率和恒定边界磁场的条件下，我们成功地导出了该非线性问题的精确解。该解采用四个参数作为输入：固定磁边界\(B_0\)、\(\eta _{\text {S}}\)和\(\eta _{\text {L}}\) ，即电阻率低于和高于材料的临界能量密度，以及材料的临界能量密度\(e_{\text {c}}\) 。解曲线B ( x ,  t )具有尖锐前沿的特点，其演化遵循通常的自相似规则，相似变量为\(x/\sqrt{t}\)。