This paper considers a homogeneous system of functional equations that occurs when solving a spectral problem the eigenfunctions of which describe guided modes of smoothly irregular waveguides. The resulting system of functional equations is a homogeneous system of algebraic equations for any fixed argument. A symbolic method for solving systems of functional equations of this type is proposed. The method is implemented in the Maple computer algebra system. Results of numerical computations are presented to demonstrate the effectiveness of the developed symbolic-numerical method in comparison with the approach based on only numerical techniques.

This paper investigates the problem of symbolic representation for the general solution of a system of ordinary differential equations (ODEs) with symbolically defined constant coefficients in the case where some symbolic constants can vanish. In addition, the symbolic representation of eigenvectors for the system’s coefficient matrix is not unique. It is shown that standard procedures of computer algebra systems search for specific symbolic representations of eigenvectors while ignoring the other symbolic representations. In turn, the eigenvectors found by a computer algebra system can be inadequate for constructing numerical algorithms based on them, which is demonstrated by an example. We propose an algorithm for finding various symbolic representations of eigenvectors for symbolically defined matrices. This paper considers a particular system of ODEs obtained by investigating some solutions of Maxwell’s equations; however, the proposed algorithm can be applied to an arbitrary system with a normal matrix of coefficients.

EPJ Web of Conferences, open-access proceedings in physics and related sciences

In this work, the model of adiabatic waveguide modes is studied by means of computer algebra. Within the model, the solution of the system of Maxwell’s equations is reduced to a form expressed via the solution of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field. In the case of multilayer waveguides, by means of a computer algebra system, the equations are reduced to a homogeneous system of linear algebraic equations, which is studied symbolically. The condition for non-trivial solvability of the system defines a dispersion relation, which is solved by the symbolic-numerical method, while the system is solved symbolically. The paper presents solutions that describe adiabatic waveguide modes in the zeroth approximation, taking into account the small slope of the interface of the waveguide layer, which are qualitatively different from solutions that do not take into account this slope.