Abstract: The transmission characteristics of electromagnetic wave in the waveguide system,filled with nearly- zero- index- metamaterials and loaded with dielectric defects,belongs to the problem of Helmholtz equation with non- homogeneous boundary value in the source of the method of mathematics. It can be solved by a variety of methods for solving partial differential equations. In this paper,the separation variables method is used for solving two typical examples. Furthermore,the software named COMSOL multi- physics,which is based on the finite element method,is used for simulation,and the correctness of theoretical analysis is certified. For more general problems,numerical calculation and simulation by COMSOL are more commonly applied.

Key words: nearly-zero-index-metamaterials, problem of Helmholtz equation with boundary values, separation variables method, numerical method