DISCRETE AND CONTINUOUS MODELS AND APPLIED COMPUTATIONAL SCIENCE

Earlier we developed a stable fast numerical algorithm for solving ordinary differential equations of the first order. The method based on the Chebyshev collocation allows solving both initial value problems and problems with a fixed condition at an arbitrary point of the interval with equal success. The algorithm for solving the boundary value problem practically implements a single-pass analogue of the shooting method traditionally used in such cases. In this paper, we extend the developed algorithm to the class of linear ODEs of the second order. Active use of the method of integrating factors and the d’Alembert method allows us to reduce the method for solving second-order equations to a sequence of solutions of a pair of first-order equations. The general solution of the initial or boundary value problem for an inhomogeneous equation of the second order is represented as a sum of basic solutions with unknown constant coefficients. This approach ensures numerical stability, clarity, and simplicity of the algorithm.

Various approaches to calculating normal modes of a closed waveguide are considered. A review of the literature was given, a comparison of the two formulations of this problem was made. It is shown that using a self-adjoint formulation of the problem of normal waveguide modes eliminates the occurrence of artifacts associated with the appearance of a small imaginary additive to the eigenvalues. The implementation of this approach for a rectangular waveguide with rectangular inserts in the Sage computer algebra system is presented and tested on hybrid modes of layered waveguides. The tests showed that our program copes well with calculating the points of the dispersion curve corresponding to the hybrid modes of the waveguide.

We describe introduced in the journal the rubric system. We describe the general structure of an IMRAD research publication. The IMRAD structure for a research article is described in detail.