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We present analytic formulations for studying the energetic behavior of hysteretic magnetic materials. One formulation reduces the full nonlinear diffusion problem to a linear problem through an optimization procedure. A second formulation attempts to approximate the magnetic permeability tensor by a complete set of functions. By means of scalar product defined in the function space, we obtain a series of linear nonhomogeneous diffusion equations. We analyze for the vector case qualitatively and give solutions for a one-dimensional field configuration. For the scalar case, we investigate two different magnetic materials and, for simplicity, we approximate the relevant hysteresis cycles by a closed polygonal. A scalar Preisach model, numerically treated, is used as a benchmark.