Solving the problem of transferring a continuous medium over a network medium in symbolic form

When studying the evolutionary processes of transferring a continuous medium over network media, special emphasis is placed on the issues of the existence and approximate finding of solutions to initial boundary value problems for differential systems of equations, the formalisms of which describe mathematical models of these processes. In engineering practice, such models are usually considered linear or allow linearization (a classic example is linearized Navier-Stokes systems). The core idea is based on the use of symbolic mathematics theory tools which determined the entire direction of the research; it predetermines the understanding of the transfer phenomena patterns in the branching places (nodal places) of the process carrier and the subsequent mathematical description of such phenomena in terms of differential or other relationships. The paper presents a mathematical model of an evolutionary network-like process of continuum transfer (linear differential system) and its corresponding differential-difference system obtained by semi-sampling the differential system with respect to a time variable. To prove the solvability of the latter and empirically determine the approximations of the solution to the original differential system, methods of symbolic mathematics are used. At the same time, an algorithm for finding a symbolic-numerical solution to a differential-difference system and approximations of solutions to the initial boundary value problem for the continuum transport equation are proposed and validated. The algorithm is based on the approximation of the partial derivative with respect to a time variable by a difference ratio (a two-layer approximation scheme is utilized) and the subsequent application of the Laplace transform to the resulting differential-difference system. A block diagram of the algorithm is presented; a description of the software complex structure based on the developed algorithm is given. The software package is developed using the Java programming language. To upload the initial data of the initial-boundary value problem and output the solution, the web interface of the software package based on the Spring framework is used. To illustrate the operation of the software package, an example of solving an initial-boundary value problem is considered with a step-by-step demonstration of the calculation results. The presented method can be used in the analysis of applied problems of network hydrodynamics, heat engineering, as well as the analysis of diffusion processes in biophysics.

1. Samarskiy A.A., Gulin A.V. Numerical methods of mathematical physics. Moscow, Nauchnyi mir; 2000. 315 p. (In Russ.).

2. Kulyabov D.S., Kokotchikova M.G. Analytical review of symbolic computing systems. Vestnik RUDN. Seriya Matematika. Informatika. Fizika = Bulletin of the RUDN, Mathematics Series. Computer science. Physics. 2007;(1-2):38–45.

3. Pokorny Yu.V., Penkin O.M., Pryadiev V.L., etc. Differential equations on geometric graphs. Moscow, FIZMATLIT; 2004. 272 p. (In Russ.).

4. Sviridyuk G.A., Shemetova V.V. Hoff equations on graphs. Differents. uravneniya = Differents. equations. 2006;42;(1):126–131. (In Russ.).

5. Volpert A.I., Khudyaev S.I. Analysis in classes of discontinuous functions and equations of mathematical physics. Moscow, Nauka; 1975. 395 p. (In Russ.).

6. Karpov D.V. Graph Theory. St. Petersburg: V.A. Steklov Mathematical Institute of the Russian Academy of Sciences, St. Petersburg Branch; 2021. 559 p. (In Russ.).

7. Provotorov V.V. On the issue of constructing boundary controls in the problem of damping vibrations of the "mast-stretching" system. Sistemy upravleniya i informatsionnye tekhnologii = Management systems and information technologies. 2008;32 (2.2):293–297. (In Russ.).

8. Zhabko A.P., Shindyapin A.I., Provotorov V.V. Stability of weak solutions of parabolic systems with distributed parameters on the graph. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2019;15(4):457–471. DOI: 10.21638/11702/spbu10.2019.404.

9. Malashonok N.A., Rybakov M.A. Symbolic-numerical solution of systems of linear ordinary differential equations with the required accuracy. Zhurnal “Programmirovanie” RAN = Journal “Programming" RAS. 2013;3:38–46. (In Russ.).

10. Malashonok G.I., Rybakov M.A. Solution of systems of linear differential equations and calculation of dynamic characteristics of control systems in the MathPartner web service. Vestnik Tambovskogo universiteta. Ser. Estestvennye i tekhnicheskie nauki = Bulletin of the Tambov University. Ser. Natural and technical sciences. 2014;19(2):517–529. (In Russ.).

11. Rybakov M.A. On finding general and particular solutions of an inhomogeneous system of ordinary linear differential equations with constant coefficients. Bulletin of the Tambov University. Ser. Natural and technical sciences. 2012;17(2):552–565. (In Russ.).