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A method for simulating axisymmetric waves of electric and magnetic type in dielectric or magnetic corrugated rods and systems of rings are developed. Such electrodynamic systems are attractive for application in relativistic microwave electronics and fiber optics devices and systems. The method described in this article is based on one-dimensional integral equations of the second kind with kernels having logarithmic singularities. A method is suggested for separating desired solutions from others, whose appearance is typical for problems pertaining to finding the eigenvalues for the integral equations under consideration. The results of the simulation of dispersion properties for dielectric rods with sinusoidal corrugation and a series of dielectric circular rings are presented.