<p>The electromagnetic fields in a cylindrical metamaterial waveguide with Kerr-type nonlinearity are governed by the nonlinear Helmholtz equation. For such a cylindrical core, governing equation is derived for the axial electric field, and assuming cylindrical symmetry, the resulting nonlinear Bessel Differential Equation is obtained for the radial component of the electric field. Four numerical methods are used, among them the Freezing Nonlinearity method used for solving ordinary differential equation as a novel application as far as we know. Results of the numerical approaches have been compared among themselves and with the results of analytic methods both for nonlinear and linear Bessel equations.</p>