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In this paper, we propose an approximate analytical solution of the problem of nonlinear diffusion of the current density in a high-temperature superconducting plate with current transport. It is obtained by the technique of self-similar solution. The construction of this solution highlights a characteristic time of penetration Tp whose limit for large n is the model of Bean. We compare our solution to the ones obtained using COMSOL multiphysics. We study the influence of variation of the magnetic induction on time penetration and the influence of the n factor on time penetration.