SymDR: Symbol Computer Algebra Library for Generation of Classical and Approximate Dispersion Relations for Systems of Partial Differential Equations

• Published in: Lobachevskii journal of mathematics Vol. 46; no. 1; pp. 1 - 12
• Main Authors: Arendarenko, M. S., Dzhanbekova, A. R., Kotov, S. V., Malyutin, M. S., Savvateeva, T. A., Samoylov, M. V., Utyupina, V. Y., Stoyanovskaya, O. P.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.01.2025; Springer Nature B.V
• Subjects: Algebra; Analysis; Computer algebra; Continuum mechanics; Geometry; Libraries; Mathematical analysis; Mathematical Logic and Foundations; Mathematical models; Mathematics; Mathematics and Statistics; Numerical models; Parameters; Partial differential equations; Plasma physics; Probability Theory and Stochastic Processes; Wave dispersion; 2010 Mathematics Subject Classification: 34A45, 34K07; SymPy; partial differential equation; dispersion relation; approximate dispersion relation; symbolic calculations
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080224608579

Mathematical models of numerous processes in continuum mechanics (CM), plasma physics (PP) and astrophysics (AP) are partial differential equations (PDEs). When developing computer models, these equations are replaced by discrete equations that are solved numerically. In order to investigate mathematical and numerical models of CM, PP and AP, the technique of constructing dispersion relations has been developed. Using dispersion relations allows one to derive particular solutions to systems of PDEs, to investigate the stability of solutions for continuous and discrete models, to estimate the order of approximation and the rate of convergence for discrete models, and to establish of the optimal numerical parameters of a discrete model. Dispersion relations describe wave processes (i.e., processes of perturbation transfer with a velocity different from the velocity of matter) in media. The classical dispersion relation is a nonlinear algebraic equation (relating the wave parameters, namely the wave number and the wave frequency, ), which corresponds to a continuous system of PDEs. There is a technique that allows one to derive a dispersion relation (classical or approximate, respectively) for a continuous or discrete CM, PP, and AP model. This paper presents a symbolic computer algebra library, developed by the authors, which automates this technique. The current version supports the use of nonstationary models with a single spatial variable both for continuum and finite-difference notation. The library is written in Python using the SymPy symbolic computing package and is available at https://pypi.org/project/symdr/ .