Features of mathematical modeling of physical processes with elements of differential equations

Abstract

Mathematical modeling plays a key role in modern science and technology, as it allows describing complex physical, technical and natural processes in a formalized form. It allows predicting the behavior of systems under different conditions without conducting expensive or dangerous experiments. With the help of mathematical models, it is possible to analyze the influence of individual parameters, optimize the characteristics of objects and increase the efficiency of technological processes. Mathematical modeling is the basis of computer simulations and digital twins of real systems. It is of particular importance for the study of micro- and nanoscale phenomena, where direct experimental measurements are difficult or impossible. The paper considers the application of differential equations as a key tool for mathematical modeling of physical processes. Their role in classical and quantum mechanics, electrodynamics, optics, thermodynamics, and statistical physics is analyzed. Particular attention is paid to modern trends in nanophysics and nanothermophysics, where differential equations allow describing quantum effects, thermal processes, electron transport, and wave phenomena in nanostructures. Both analytical and numerical methods for solving equations are considered, in particular the use of hyperbolic equations, the Boltzmann equation for phonons, and modern approaches with computational methods and neural networks (PINNs). The work emphasizes the importance of studying such mathematical models for the development of analytical and critical thinking, the ability to assess the adequacy of models, and well-founded conclusions about the behavior of physical systems. The relevance of the work lies in the combination of fundamental physics and modern nanotechnologies, which opens up opportunities for optimizing nanodevices, predicting their properties, and improving the efficiency of energy and heat transfer at the nanoscale. The work may be useful for students, graduate students, and researchers involved in mathematical modeling, nanophysics, heat transfer in nanomaterials, and computational methods in physics.