Dirichlet problem for Helmholtz equations in gyrotropic elliptical region with longitudinal magnetization

We have formulated and solved the Dirichlet problem for Helmholtz equations of electromagnetic waves, propagating in an elliptical cylinder filled with longitudinally magnetized ferrite which is described by a second-rank tensor. It is assumed that the cylinder has an infinitely conductive wall. To solve the boundary value problem of Helmholtz equations for longitudinal components of electromagnetic waves, we have used the method of shortening the initial differential equation and the method of variables separation. The solution of the above boundary value problem in elliptic coordinates is associated with the use of even and odd ordinary and modified Mathieu functions of the first kind. Using the results obtained, we have determined all the components of electromagnetic waves for even and odd solutions. Applying the Dirichlet condition to the components of electromagnetic waves and solving the system of linear homogeneous algebraic equations, we have obtained dispersion equations. The found dispersion equations of electromagnetic waves are of great practical importance and allow studying the propagation of hybrid waves in this region.