Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2219510.22363/2658-4670-2019-27-1-42-48Research ArticleSoftware for the numerical solution of first-order partial differential equationsKuzivYaroslav Yu<p>postgraduate student of Department of Applied Probability and Informatics</p>yaroslav.kuziw@yandex.ruPeoples’ Friendship University of Russia (RUDN University)15122019271424820112019Copyright © 2019, Kuziv Y.Y.2019<p>Partial differential equations of the first order, arising in applied problems of optics and optoelectronics, often contain coefficients that are not defined by a single analytical expression in the entire considered domain. For example, the eikonal equation contains the refractive index, which is described by various expressions depending on the optical properties of the media that fill the domain under consideration. This type of equations cannot be analysed by standard tools built into modern computer algebra systems, including Maple.The paper deals with the adaptation of the classical Cauchy method of integrating partial differential equations of the first order to the case when the coefficients of the equation are given by various analytical expressions in the subdomains G1, . . . , Gk , into which the considered domain is divided. In this case, it is assumed that these subdomains are specified by inequalities. This integration method is implemented as a Python program using the SymPy library. The characteristics are calculatednumerically using the Runge-Kutta method, but taking into account the change in the expressions for the coefficients of the equation when passing from one subdomain to another. The main functions of the program are described, including those that can be used to illustrate the Cauchy method. The verification was carried out by comparison with the results obtained in the Maple computer algebra system.</p>eikonalpartial differential equationSymPy[M. Born, E. Wolf, Principles of optics. Electromagnetic theory of propagation, interference and diffraction of light, 6th Edition, Elsevier, 1980.][B. Enquist, O. Runborg, Computational high frequency wave propagation, 6th Edition, Cambridge University Press, 2003.][A. A. Egorov, L. A. Sevastianov, Structure of modes of a smoothly irregular integrated-optical four-layer three-dimensional waveguide, Quantum Electronics 39 (6) (2009) 566. doi:10.1070/QE2009v039n06ABEH013966.][A. A. Egorov, K. P. Lovetsky, L. A. Sevastianov, A. L. Sevastianov, Simulation of guided modes (eigenmodes) and synthesis of a thin-film generalised waveguide Luneburg lens in the zero-order vector approximation, Quantum Electronics 40 (9) (2010) 830-836. doi:10.1070/QE2010v040n09ABEH014332.][A. A. Egorov, L. A. Sevastianov, A. L. Sevastianov, Method of adiabatic modes in research of smoothly irregular integrated optical waveguides: zero approximation, Quantum Electronics 44 (2) (2014) 167-173. doi:10.1070/QE2014v044n02ABEH015303.][A. D. Polyanin, V. E. Nazaikinskii, Handbook of linear partial differential equations for engineers and scientists, 2nd Edition, CRC Press, Boca Raton, London, 2016.][E. Goursat, Cours d’analyse mathématique, 3rd Edition, Vol. 2, Gauthier-Villars, Paris, 1918.][PDEplot for Maple (2019). URL http://www.maplesoft.com][Python library for symbolic mathematics SymPy (2019). URL http://www.sympy.org][M. D. Malykh, On integration of the first order differential equations in finite terms, IOP Conf. Series: Journal of Physics: Conf. Series 788 (2017) 012026. doi:10.1088/1742-6596/788/1/012026.][IarKuz at Github (2019). URL https://github.com/IarKuz/PostGradeCode/blob/MF_Solver_PDE/MF_Solver_PDE.ipynb][E. Hairer, G. Wanner, S. P. Nørsett, Solving ordinary differential equations, 3rd Edition, Vol. 1, Springer, New York, 2008.][W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical recipes in C: the art of scientific computing, 2nd Edition, Cambridge University Press, 1992.][J. Lock, Scattering of an electromagnetic plane wave by a Luneburg lens. II. Wave theory, Journal of the Optical Society of America A: Optics Image Science and Vision 25 (2008) 2980-2990. doi:10.1364/JOSAA.25.002980.][S. Cornbleet, Geometrical optics reviewed: A new light on an old subject, Proceeding of the IEEE 71 (1983). doi:10.1109/PROC.1983.12620.]