TY - JOUR

T1 - SymDR: Symbol Computer Algebra Library for Generation of Classical and Approximate Dispersion Relations for Systems of Partial Differential Equations

AU - Arendarenko, M. S.

AU - Dzhanbekova, A. R.

AU - Kotov, S. V.

AU - Malyutin, M. S.

AU - Savvateeva, T. A.

AU - Samoylov, M. V.

AU - Utyupina, V. Y.

AU - Stoyanovskaya, O. P.

N1 - Algorithm development and software implementation were performed by A.R. Dzhanbekova, S.V. Kotov, M.S. Malyutin, M.V. Samoylov, and V.Y. Utyupina within the framework of the event Great Mathematical Workshop-2024, library verification was performed by M.S. Arendarenko and T.A. Savvateeva with the support of the Mathematical Center in Akademgorodok, agreement with the Ministry of Science and Higher Education of the Russian Federation no. 075-15-2022-282, verification set was developed by O.P. Stoyanovskaya founded by the Russian Science Foundation project no. 23-11-00142.

PY - 2025/5/30

Y1 - 2025/5/30

N2 - Abstract: 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/.

AB - Abstract: 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/.

KW - SymPy

KW - approximate dispersion relation

KW - dispersion relation

KW - partial differential equation

KW - symbolic calculations

UR - https://www.mendeley.com/catalogue/658fbbc4-b33c-3708-8bb7-06df61613c13/

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-105007086392&origin=inward&txGid=77751cdd02f59eed47ffd7ceb8da3178

U2 - 10.1134/S1995080224608579

DO - 10.1134/S1995080224608579

M3 - Article

VL - 46

SP - 1

EP - 12

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 1

ER -