Bohaienko V. Mathematical and computer modeling of hydrogeomigration processes with non-classical dynamics based on high-performance computational algorithms

Українська версія

Thesis for the degree of Doctor of Science (DSc)

State registration number

0524U000416

Thesis Registration Form

0524U000416.pdf

Applicant for

Vsevolod O. Bohaienko

Specialization

01.05.02 - Математичне моделювання та обчислювальні методи

Date of defense

22-11-2024

Specialized Academic Board

Д 26 194.02

V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine

Essay

Bohaienko V. O. Mathematical and computer modeling of hydrogeomigration processes with non-classical dynamics based on high-performance computational algorithms. – Qualifying scientific work under the rights of a manuscript. The dissertation for obtaining the scientific degree of Doctor of Physical and Mathematical Sciences by specialty 01.05.02 "Mathematical modeling and computational methods" (11 – Mathematics and statistics). – V.M. Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine, Kyiv, 2024. The dissertation is devoted to the development of mathematical modeling tools for anomalous hydrogeomigration processes based on the fractional-differential approach and to the increase of computational efficiency in computer modeling. In the work, fractional-differential mathematical models were built and simulations of a series of non-local processes of convective diffusion and filtration-consolidation were carried out for the first time. For the developed models a class of high-performance algorithms for solving initial-boundary problems has been built. The dissertation also considers parameter identification problems for the developed models. By comprehensively applying the obtained results in practice, the problems of modeling moisture transport under sprinkler irrigation in complex hydrogeological conditions are solved. In the first chapter, information on the current state of development of the apparatus of fractional integro-differentiation and the corresponding mathematical models of migration processes is given. In the second chapter we present the results of mathematical modeling of convective diffusion processes in a two-dimensional setting based on the models with time-fractional derivatives. The third chapter presents the results of mathematical modeling of soil filtration-consolidation processes in one-dimensional setting based on fractional-differential models. The fourth chapter is dedicated to the mathematical modeling of non-local convective diffusion processes taking into account the phenomenon of mass transfer between particles in mobile and immobile phases according to different laws of mass transfer kinetics. In the fifth chapter we consider optimized computational schemes, in particular parallel algorithms, for one- and multidimensional problems of modeling geomigration processes, containing the Caputo–Gerasimov and Caputo–Fabrizio derivatives. A series of parallel algorithms for distributed memory systems is proposed for locally one-dimensional splitting schemes in which red-black 2D block distribution of data is used. Parallel algorithms for graphic processors (GPUs) have been developed for the models with Caputo–Gerasimov derivatives for both the time and spatial variables. To increase the speed of computational schemes for modeling heat and mass transfer processes based on the models with the Caputo–Gerasimov derivative for the time variable, a procedure for its approximation with a given accuracy based on expansion into series and variable separation techniques along with corresponding parallel algorithms are proposed. In the sixth chapter, a computational scheme based on the expansion into series of integral operators’ kernels is developed for fractional-differential equations containing the ψ-Caputo derivative. In order to achieve additional acceleration of computations, we propose a series of GPU-algorithms. For the case of the three-dimensional model of anomalous diffusion with the ψ-Caputo derivatives for the spatial variables, parallel algorithms for distributed memory systems are also constructed and studied. For the three-dimensional diffusion equation with the ψ-Caputo derivatives for both the time and spatial variables, the accuracy and speed of implicit finite-difference schemes and splitting schemes when applied together with the algorithms aimed at increasing the speed of computations were studied. Based on the obtained accuracy and speed estimates, an algorithm for automatic selection of the optimal computational scheme is presented. The approach of the expansion into series with subsequent separation of variables is also used to construct a scheme for calculating the values of the Atangana–Baleanu derivative. The seventh chapter is devoted to the algorithms for solving parameters identification problems for fractional-differential equations. We consider the generalized moisture transport equation stated in terms of pressure that contains the ψ-Caputo derivatives for the time and space variables. To solve the problem of finding the values of model’s numerical parameters the particle swarm optimization and genetic programming algorithm. The eighth chapter presents the results of fractional-differential modeling of moisture transport when solving problems that arise in agriculture during irrigation management.

Official opponents

Mykhaylo R. Petryk
Anatoliy Y. Doroshenko
Ivan I. Dyyak

Reviewers

Oleksandr V. Popov
Petro I. Stetsyuk
Vadim H. Tulchinsky

Research papers

Bulavatsky VM, Bohaienko VO. Mathematical Modeling of the Fractional Differential Dynamics of the Relaxation Process of Convective Diffusion Under Conditions of Planned Filtration. Cybernetics and Systems Analysis 2015; 51:886–895. doi: 10.1007/s10559-015-9781-2 [Scopus, Web of Science, zbMATH]

Bulavatsky VM, Bohaienko VO. Mathematical Modeling of the Dynamics of Nonequilibrium in Time Convection–Diffusion Processes in Domains with Free Boundaries. Cybernetics and Systems Analysis 2016; 52:427–440. doi: 10.1007/s10559-016-9843-0 [Scopus, Web of Science, zbMATH, MathSciNet]

Bohaienko VO, Bulavatsky VM, Kryvonos IuH. On Mathematical modeling of Fractional-Differential Dynamics of Flushing Process for Saline Soils with Parallel Algorithms. Journal of Automation and Information Sciences 2016; 10:1-12. doi: 10.1615/JautomatInfScien.v48.i10.10 [Scopus]

Булавацкий ВМ, Богаенко ВА. Численное моделирование дробнодифференциальной динамики процесса фильтрационно-конвективной диффузии на основе параллельных алгоритмов для кластерных систем. Доповіді НАНУ 2017; 1: 21-28. doi: 10.15407/dopovidi2017.01.021 [zbMATH, MathSciNet]

Bohaienko VO, Bulavatsky VM, Kryvonos IuH. Mathematical Modeling of Fractional-Differential Dynamics of Process of Filtration-Convective Diffusion of Soluble Substances in Nonisothermal Conditions. Journal of Automation and Information Sciences 2017; 49(4):12-25. doi: 10.1615/JautomatInfScien.v49.i4.20 [Scopus, MathSciNet]

Bulavatsky VM, Bohaienko VO. Numerical Simulation of Fractional-Differential Filtration-Consolidation Dynamics Within the Framework of Models with Non-Singular Kernel. Cybernetics and Systems Analysis 2018; 54:193–204. doi: 10.1007/s10559-018-0020-5 [Scopus, Web of Science, zbMATH, MathSciNet]

Булавацкий ВМ, Богаенко ВА. Компьютерное моделирование дробнодифференциальной динамики некоторых фильтрационно-консолидационных процессов. Доповіді НАНУ 2018; 4:16-24. doi: 10.15407/dopovidi2018.04.016 [MathSciNet]

Bohaienko VO. Parallel Algorithms for Modelling Two-Dimensional NonEquilibrium Salt Transfer Processes on the Base of Fractional Derivative Model. Fractional calculus and applied analysis 2018; 21(3):654–671. doi: 10.1515/fca-2018-0035 [Scopus, Web of Science, zbMATH, MathSciNet]

Bohaienko VO, Bulavatsky VM. Mathematical Modeling of Solutes Migration Under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative. Fractal Fract. 2018; 2(4):28. doi: 10.3390/fractalfract2040028 [Scopus, Web of Science]

Bohaienko VO. Numerical schemes for modelling time-fractional dynamics of non-isothermal diffusion in soils. Mathematics and Computers in Simulation 2019; 157: 100–114. doi: 10.1016/j.matcom.2018.09.025 [Scopus, Web of Science, DBLP, zbMATH, MathSciNet]

Булавацкий ВМ, Богаенко ВА. Компьютерное моделирование динамики процесса миграции растворимых веществ при фильтрации грунтовых вод со свободной поверхностью на основе дробно-дифференциального подхода. Доповіді НАНУ 2018; 12:21-29. doi: 10.15407/dopovidi2018.12.021 [zbMATH, MathSciNet]

Bohaienko VO. A fast finite-difference algorithm for solving space-fractional filtration equation with a generalised Caputo derivative. Computational and Applied Mathematics 2019; 38:105. doi: 10.1007/s40314-019-0878-5 [Scopus, Web of Science, DBLP, zbMATH, MathSciNet]

Богаенко ВА, Булавацкий ВМ. Компьютерное моделирование на основе нелокальной модели динамики конвективной диффузии растворимых веществ в подземном фильтрационном потоке в условиям массообмена. Международный научно-технический журнал "Проблемы управления и информатики" 2019; 3:41-53.

Bohaienko VO, Bulavatsky VM. Simplified Mathematical Model for the Description of Anomalous Migration of Soluble Substances in Vertical Filtration Flow. Fractal Fract. 2020; 4: 20. doi: 10.3390/fractalfract4020020 [Scopus, Web of Science]

Bulavatsky VM, Bohaienko VO. Some boundary-value problems of fractionaldifferential mobile-immobile migration dynamics in a profile filtration flow. Cybernetics and Systems Analysis 2020; 56(3): 410–425. doi: 10.1007/s10559-020-00257-2 [Scopus, Web of Science, zbMATH, MathSciNet]

Bohaienko VO. Parallel finite-difference algorithms for three-dimensional space-fractional diffusion equation with ψ-Caputo derivatives. Computational and Applied Mathematics 2020; 39:163. doi: 10.1007/s40314-020-01191-x [Scopus, Web of Science, DBLP, zbMATH, MathSciNet]

Bohaienko VO, Gladky AV, Romashchenko MI, Matiash TV. A Identification of fractional water transport model with ψ-Caputo derivatives using particle swarm optimization algorithm. Applied Mathematics and Computation 2021; 390:125665. doi: 10.1016/j.amc.2020.125665 [Scopus, Web of Science, DBLP, zbMATH, MathSciNet]

Bohaienko VO. Accuracy and speed of splitting methods for three-dimensional space-time fractional diffusion equation with ψ-Caputo derivatives. Mathematics and Computers in Simulation 2021; 188: 226-240. doi: 10.1016/j.matcom.2021.04.004 [Scopus, Web of Science, DBLP, zbMATH, MathSciNet]

Bohaienko VO. On the recurrent computation of fractional operator with Mittag-Leffler kernel. Applied Numerical Mathematics 2021; 162: 137-149. doi: 10.1016/j.apnum.2020.12.016 [Scopus, Web of Science, zbMATH, MathSciNet]

Bohaienko VO. Selection of ψ-Caputo derivative functional parameter in generalized water transport equation by genetic programming technique. Results in Control and Optimization 2021; 5: 100068. doi: 10.1016/j.rico.2021.100068 [Scopus, Web of Science]

Bohaienko VO, Bulavatsky VM. Fractional-fractal modeling of filtrationconsolidation processes in saline saturated soils. Fractal and Fractional 2020; 4(4): 59. doi: 10.3390/fractalfract4040059 [Scopus, Web of Science]

Bohaienko V, Gladky A. Modelling fractional-order moisture transport in irrigation using artificial neural networks. SeMA 2023. doi: 10.1007/s40324-023-00322-8 [Scopus]

Богаєнко ВО, Булавацький ВМ, Хіміч ОМ. Математичне та комп’ютерне моделювання в задачах гідрогеоміграційної динаміки. Київ: Наукова Думка, 2022.

Bohaienko VO. Numerical Integration Schemes for Finite Difference Solution of Time-Fractional Diffusion Equation with Generalized Caputo Derivative. "Інформаційні технології та комп’ютерне моделювання"; матеріали статей Міжнародної науково-практичної конференції, м. Івано-Франківськ, 14-19 травня 2018 року. Івано-Франківськ: п. Голіней О.М., 2018, с.250-253.

Bohaienko VO, Bulavatsky VM, Gladky AV. GPU algorithms for solving timefractional diffusion equation with generalised Caputo derivative with respect to a function. Fifth International Conference "High Performance Computing" HPC-UA 2018 (Ukraine, Kyiv, October 22-23, 2018), 2018, p.12-17.

Bohaienko VO. Efficient computation schemes for generalized twodimensional time-fractional diffusion equation. "Інформаційні технології та комп’ютерне моделювання"; матеріали статей Міжнародної науково-практичної конференції, м. Івано-Франківськ, 20-25 травня 2019 року. Івано-Франківськ: п. Голіней О.М., 2019, с.238-241.

Bohaienko VO. Performance of vectorized GPU-algorithm for computing ψCaputo derivative values. Hu Z., Petoukhov S., Dychka I., He M. (eds) Advances in Computer Science for Engineering and Education III. ICCSEEA 2020. Advances in Intelligent Systems and Computing, vol 1247. Cham: Springer, 2020, p. 266-275 doi: 10.1007/978-3-030-55506-1_24 [Scopus]

Bohaienko VO. Computing ψ-Caputo Fractional Derivative Values Using CUDA 10. Proceedings of the 9th International Conference "Information Control Systems & Technologies" Odessa, Ukraine, September 24–26, 2020. CEUR Workshop proceedings, vol. 2711, 2020, p. 49. [Scopus, DBLP]

Bohaienko VO, Gladky AV. On the selection of fractional-differential model of convective diffusion with mass exchange. Modeling, Control and Information Technologies: Proceedings of International Scientific and Practical Conference 2020; 4:7- 10. doi: 10.31713/MCIT.2020.02

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