C&SA | Contents

Volume 61 >>> № 5 SEPTEMBER — OCTOBER 2025

-->

UDC 519.7::532.5+556.5-026:517.9

А.Y. Bomba
National University of Water and Environmental Engineering, Rivne, Ukraine, abomba@ukr.net, a.ya.bomba@nuwm.edu.ua

S.S. Kashtan
National University of Water and Environmental Engineering, Rivne, Ukraine, sstepanovic940@gmail.com, s.s.kashtan@nuwm.edu.ua

METHODS OF COMPLEX ANALYSIS FOR FORECASTING THE CONTROLLABLE
FLOWS OF SURFACE AND FILTRATION WATERS

Abstract. An approach to modeling an ideal flow in a water environment limited by flow lines and equipotential surfaces, in particular, a reservoir with tributaries, is proposed. Analysis of the hydrological regime of a water body indicates a complex interaction of underground groundwater and surface water masses. In this work, the influence of both surface and underground waters supporting the main flow (for example, tributaries to the reservoir) at the initial stage is proposed to be modeled as areas (sources) of transverse disturbances to the base flow. Slow movements (close to ideal) of water in such reservoirs are considered, which can be described, as in soils, with the help of an analogue of Darcy’s law. Depending on the values of potentials (pressures) on the corresponding equipotential lines, different cases of flow formation in the reservoir (physical area) are possible, and therefore, configurations of the corresponding areas of complex potential. The modeling methodology for this kind of process is based on the development and adaptation of numerical methods of complex analysis, which enables the effective description and prediction of the dynamics of water systems at the global level and decision-making regarding the management of the process under consideration. Approximate solutions of the corresponding, so-called, inverse boundary value problems on conformal mappings (for areas with sources of perturbation of boundary flow lines that arise at certain stages of computational processes) are found using an algorithm based on the alternate parameterization of the values of conformal invariants, boundary and internal nodes of the grid area using the ideas of the block iteration method. The proposed approach, in addition to calculating the water mass interface curves, provides the ability to simultaneously determine the characteristic flow function, complex potential, total discharge, and values of various types of flows. It also enables the construction of a hydrodynamic grid in a given area, calculation of the velocity field, and control of the process.

Keywords: nonlinear boundary-value problems, conformal mappings, modeling, flow control, baseflow, numerical methods, intelligent data analysis.

full text

REFERENCES

1. Xie J., Liu X., Jasechko S. et al. Majority of global river flow sustained by groundwater. Nature Geoscience. 2024. Vol. 17, Iss. 8. P. 770–777. https://doi.org/10.1038/s41561-024-01483-5.
2. Yang C., Condon L.E., Maxwell R.M. Unravelling groundwater-stream connections over the continental United States. Nature Water. 2025. Vol. 3, Iss. 1. P. 70–79. https://doi.org/10.1038/s44221-024-00366-8.
3. Golden H.E., Christensen J.R., McMillan H.K. et al. Advancing the science of headwater streamflow for global water protection. Nature Water. 2025. Vol. 3, Iss. 1. P. 16–26. https://doi.org/10.1038/s44221-024-00351-1.
4. Бомба А.Я., Каштан С.С. Нелінійні обернення крайових задач на конформні відображення з потенціалом керування. Математичні методи та фізико-механічні поля. 2002. Т. 45, № 3. С. 69–76. URL: http://journals.iapmm.lviv.ua/ojs/index.php/MMPMF/article/view/2925.
5. Bomba A.Y., Kashtan S.S., Skopetskii V.V. Nonlinear inverse boundary-value problems of conformal mapping with a controlling potential. Cybernetics and Systems Analysis. 2004. Vol. 40, N 1. P. 58–65. https://doi.org/10.1023/B:CASA.0000028100.70341.57.
6. Bomba A.Y., Yaroshchak S.V. Complex approach to modeling of two-phase filtration processes under control conditions. Journal of Mathematical Sciences. 2012. Vol. 184, Iss. 1. P. 56–68. https://doi.org/10.1007/s10958-012-0852-x.
7. Bomba A.Y., Terebus A.V. A spatial generalization of the method of conformal mappings for the solution of model boundary-value filtration problems. Journal of Mathematical Sciences. 2012. Vol. 187, Iss. 5. P. 596–605. https://doi.org/10.1007/s10958-012-1086-7.
8. Malachivskyy P.S., Pizyur Y.V. Chebyshev approximation of the steel magnetization characteristic by the sum of a linear expression and an arctangent function. Mathematical Modeling and Computing. 2019. Vol. 6, N 1. P. 77–84. https://doi.org/10.23939/mmc2019.01.077.
9. Gurevich M.I. Theory of jets in ideal fluids. New York; London: Academic Press, 1965. 585 p. https://doi.org/10.1016/C2013-0-12561-5.
10. Polubarinova-Kochina P.Y. Theory of ground water movement. Princeton, New Jersey: Princeton University Press, 1962. 613 p. https://doi.org/10.1016/0022-1694(63)90025-8.
11. McMillan H., Araki R., Bolotin L., Kim D.-H., Coxon G., Clark M., Seibert J. Global patterns in observed hydrologic processes. Nature Water. 2025. Vol. 3, Iss. 4. P. 497–506. https://doi.org/10.1038/s44221-025-00407-w.
12. Bloomfield J.P., Gong M., Marchant B.P., Coxon G., Addor N. How is Baseflow Index (BFI) impacted by water resource management practices. Hydrology and Earth System Sciences. 2021. Vol. 25, Iss. 10. Р. 5355–5379. https://doi.org/10.5194/hess-25-5355-2021.
13. Chernukha O., Bilushchak Y., Shakhovska N., Kulhїnek R. A numerical method for computing double integrals with variable upper limits. Mathematics. 2022. Vol. 10, Iss. 1. Article number 108. https://doi.org/10.3390/math10010108.
14. Lawrentjew M.A., Schabat B.W. Methoden der komplexen Funktionentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften, 1967. 846 s. https://doi.org/10.1002/zamm.19680480222.
15. Samarskii A.A. The theory of difference schemes. Boca Raton: CRC Press, 2001. 761 p. https://doi.org/10.1201/9780203908518.
16. Ortega J.M., Rheinboldt W.C. Iterative solution of nonlinear equations in several variables. New York; London: Academic Press, 1970. 555 p. https://doi.org/10.1016/C2013-0-11263-9.
17. Polozhii G.N. The method of summary representation for numerical solution of problems of mathematical physics. Oxford; London: Pergamon Press, 1965. 283 p. https://doi.org/10.1016/C2013-0-05391-1.
18. Lyashko I.I., Lyashko S.I., Semenov V.V. Control of pseudo hyperbolic systems by the concentrated impacts. Journal of Automation and Information Sciences. 2000. Vol. 32, Iss. 12. Р. 23–36. https://doi.org/10.1615/JAutomatInfScien.v32.i12.40.
19. Lyashko S.I., Klyushin D.A., Timoshenko A.A., Lyashko N.I., Bondar E.S. Optimal control of intensity of water point sources in unsaturated porous medium. Journal of Automation and Information Sciences. 2019. Vol. 51, Iss. 7. Р. 24–33. https://doi.org/10.1615/JAutomatInfScien.v51.i7.20.
20. Bomba A.Ya., Boychura M.V. Methods of complex analysis in identification problems. Rivne: NUVGP, 2020. 188 p. URL: http://ep3.nuwm.edu.ua/18899.
21. Baranovsky S.V., Bomba A.Y. Decision making in predicting the dynamics of viral infection considering diffusion-convective migration of active factors via several ways under immunotherapy. Cybernetics and Systems Analysis. 2024. Vol. 60, N 4. P. 561–570. https://doi.org/10.1007/s10559-024-00696-1.
22. Petryk M.R., Khimich O.M., Petryk M.M., Fraissard J.P. Experimental and computer simulation studies of dehydration on microporous adsorbent of natural gas used as motor fuel. Fuel. 2019. Vol. 239. P. 1324–1330. https://doi.org/10.1016/j.fuel.2018.10.134
23. Bohaienko V.O., Bulavatsky V.M. Fractional-fractal modeling of filtration-consolidation processes in saline saturated soils. Fractal and Fractional. 2020. Vol. 4, Iss. 4. P. 2–12. https://doi.org/10.3390/fractalfract4040059.
24. Bulavatsky V.M. On some generalizations of the bi-ordinal Hilfer’s fractional derivative. Cybernetics and Systems Analysis. 2024. Vol. 60, N 4. P. 541–552. https://doi.org/10.1007/s10559-024-00694-3.

UDC 519.7::532.5+556.5-026:517.9

METHODS OF COMPLEX ANALYSIS FOR FORECASTING THE CONTROLLABLE FLOWS OF SURFACE AND FILTRATION WATERS

REFERENCES

METHODS OF COMPLEX ANALYSIS FOR FORECASTING THE CONTROLLABLE
FLOWS OF SURFACE AND FILTRATION WATERS