RUDN Mathematicians Developed a Software Package for Solving Nonlinear Differential Equations

RUDN mathematicians developed a universal software package to study and solve complex nonlinear differential equations and their systems. The development may come useful in hydrodynamics, gas dynamics, mechanics, elementary particle physics, plasma physics, and elastic theory. The article was published in the  Computer Physics Communications journal.

Many physical processes, such as string vibration or heat transfer from one part of a solid body to another, are mathematically described by partial differential equation. RUDN mathematicians have developed a set of tools to analyze such equations and their systems. Previously, such differential equations were solved using triangular decomposition methods that helped break an equation down to simpler parts. The authors of the work used another approach called Thomas decomposition. In the course of the tests it proved to be superior to other methods.

The study consisted of 4 stages. During the first stage RUDN mathematicians found examples of differential systems to analyze the required calculations volume and efficiency. On the second stage the authors used Thomas decomposition and adapted it to a computer algebra system. Then the team adjusted the package to the existing methods of differential equations decomposition. On the final stage the authors described their development for the users. RUDN mathematicians also provided examples of practical use of the package, namely the solution of Navier-Stokes and Burgers equations for ideal liquid that is the basis of thermodynamics.

"Our work gave the users a powerful universal set of tools to study and solve complex nonlinear equations and systems based on Thomas decomposition. The work provides a universal algorithmic method in the form of a software product with convenient UI", says Vladimir Gerdt, a co-author of the work, PhD in Physics and Mathematics, and a research associate at RUDN.

The development of RUDN mathematicians will expand the database of existing algorithms and simplify many processes in physics, physical chemistry, and IT. The package will help find solutions of differential equations and their systems faster and with higher precision than earlier methods.

More information:
Vladimir P. Gerdt et al. The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs, Computer Physics Communications (2018). DOI: 10.1016/j.cpc.2018.07.025

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