In this paper on the example of the nonlinear heat conduction equation with the help of asymptotic homogenization method has been proposed a new approach to solving of the nonlinear partial differential equations of the elliptic type with rapidly oscillating coefficients. It has been formulated a cell problem with nonlinearity parameter. The solution of this problem is defined the effective characteristics of the composite medium. Analytical dependences for determination of the effective thermo-physical properties have been received for the composite materials with cubic cell periodicity and with inclusions of spherical or cylindrical form with interphase layer. Also that has been received for the layered composites with three layers in the cell, when thermal conductivity coefficients can be expressed as the multiplication of two factors, one of which depends on the spatial variables, and the second depends on the parameter of nonlinearity. The comparison of the calculating results in the frameworks of the proposed method with «exact» solution for the layered medium is given.