A symbolic method for solving the initial boundary value problem for an inhomogeneous continuum transfer equation on a graph

The relevance of the study is due to the need to obtain analytical expressions of approximate solutions to complex technical problems, the mathematical description of which leads to boundary value problems for systems of differential equations in network-like domains and, in particular, on graphs. The article presents the formulation of an initial boundary value problem for an inhomogeneous continuum transfer equation in an n-dimensional network-like region. In the case of n=1, a symbolic method for solving the initial boundary value problem under consideration on a tree graph is proposed. 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 used) and the subsequent application of the Laplace transform to the resulting differential-difference system. A block diagram of the algorithm is presented, and a description of the structure of the software package based on the developed algorithm is given. The software package is developed in the Java programming language. To enter 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 with a step-by-step demonstration of the calculation results is considered. The materials of the article are of practical value for specialists in the field of analysis of applied problems of network hydrodynamics, thermal engineering, as well as analysis of diffusion processes in biophysics.

1. Pokorny Yu.V., Penkin O.M., Pryadiev V.L., et al. Differentsial'nye uravneniya na geometricheskikh grafakh. Moscow: FIZMATLIT; 2004. 272 p. (In Russ.).

2. Podval’ny S.L., Provotorov V.V., Hoang V.N., Tran Z. Point optimization of the laminar flow of a viscous fluid in a network carrier. Vestnik Voronezhskogo gosudarstvennogo tekhnicheskogo universiteta = Bulletin of Voronezh State Technical University. 2022;18(5):7–16. (In Russ.).

3. Provotorov V.V. K voprosu postroeniya granichnykh upravlenii v zadache o gashenii kolebanii sistemy «machta-rastyazhki». Sistemy upravleniya i informatsionnye tekhnologii. 2008;(2-2):293–297. (In Russ.).

4. Provotorov V.V. Optimum control of parabolic system with the distributed parameters on the graph. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaya matematika. Informatika. Protsessy upravleniya = Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2014;(3):154–163. (In Russ.).

5. Provotorov V.V., Provotorova E.N. Optimal control of the linearized Navier-Stokes system in a netlike domain. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaya matematika. Informatika. Protsessy upravleniya = Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2017;13(4):431–443.

6. Zhabko A.P., Shindyapin A.I., Provotorov V.V. Stability of weak solutions of parabolic systems with distributed parameters on the graph. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaya matematika. Informatika. Protsessy upravleniya = Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2019;15(4):457–471. DOI: 10.21638/11702/spbu10.2019.404.

7. Rybakov M.A. Solving the problem of transferring a continuous medium over a network medium in symbolic form. Modelirovanie, optimizatsiya i informatsionnye tekhnologii = Modeling, Optimization and Information Technology. 2023;11(4). (In Russ.). URL: https://moitvivt.ru/ru/journal/pdf?id=1451. DOI: 10.26102/2310-6018/2023.43.4.010 [Accessed 15th February 2024].

8. Malaschonok N.A., Rybakov M.A. Symbolic-numerical solution of systems of linear ordinary differential equations with required accuracy. Programmirovanie = Programming and Computer Software. 2013;39(3):38–46. (In Russ.).

9. Malaschonok G.I., Rybakov M.A. Solving systems of linear differential equations and calculation of dynamic characteristics of control systems in a web service MathPartner. Vestnik Tambovskogo universiteta. Seriya: Estestvennye i tekhnicheskie nauki = Tambov University Reviews. Series Natural and Technical Sciences. 2014;19(2):517–529. (In Russ.).

10. Rybakov M.A. Computation general and particular solutions of the inhomogeneous system of ordinary differential equations with constant coefficients. Vestnik Tambovskogo universiteta. Seriya: Estestvennye i tekhnicheskie nauki = Tambov University Reviews. Series Natural and Technical Sciences. 2012;17(2):552–565. (In Russ.).