In this paper, we discuss a free boundary problem occurring in axisymmetric filtration with nonhomogeneous anisotropic medium, which can be transformed into a mixed boundary value problem for nonlinear elliptic complex equations. By using Newton imbedding method, approximate solutions of the mixed boundary Value problem are obtained, and if proper conditions hold, we have also given error estimates of the approximate solutions. In this paper, we discuss a free boundary problem occurring in axisymmetric filtration with nonhomogeneous anisotropic medium, which can be transformed into nonlinear boundary complex problem. obtained, and if proper conditions hold, we have also given error estimates of the approximate solutions.