The Role of Domain Size and Boundary Conditions in Mathematical Modeling of Railway Tracks

In developing a mathematical model of a railway track, the question of determining the dimensions of the modeling domain inevitably arises. If the modeling area is too small, boundary effects may significantly influence the results, reducing their accuracy. Conversely, excessively large areas can increase computational complexity without substantial improvements in accuracy. An optimal choice of dimensions enables the balancing of computational costs and accuracy. Solving this problem is non-trivial, as it depends on numerous factors, primarily the type of mathematical model and the problem being addressed. In most cases, preference is given to minimal domain sizes that ensure the approach’s adequacy. The aim of this study is to justify the dimensions of the modeling domain by addressing such tasks as load scaling, introducing additional boundary conditions, and making relevant assumptions. The main object of the study is the minimum adequate longitudinal length of the track for the spatial model. The research is based on the analytical application of modern approaches in the theory of elasticity. The results are analyzed using mathematical methods, such as modeling the railway track through the propagation of elastic waves and finite element modeling. These findings can be applied to a wide range of problems related to the mathematical modeling of the stress–strain state of railway tracks.