Ідентифікація параметрів однієї дробово-диференціальної моделі міграції розчинних речовин

• Дата: 2019
• Автори: Богаєнко, Всеволод Олександрович, Булавацький, Володимир Михайлович, Гладкий, Анатолій Васильович
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Kamianets-Podilskyi National Ivan Ohiienko University 2019
• Онлайн доступ: http://mcm-tech.kpnu.edu.ua/article/view/173658
• Назва журналу: Mathematical and computer modelling. Series: Technical sciences

The paper deals with the problem of identification of model parameters in the case of mathematical modeling of fractional-differential dynamics of anomalous process of convective diffusion of soluble substances under steady-state profile groundwater filtration with a free surface. We describe the process of mass transfer using a model containing a generalized fractional derivative of Caputo-Gerasimov with respect to the time variable while the filtration process is considered in the potential velocity field. Since the filtration domain is a domain with a partially unknown boundary, the solution of the problem is performed using an anticipatory transition to a completely determined complex potential domain with a known characteristic flow function. We pose the problem of identification of the values of the parameters of a generalized fractional derivative based on the measurements of substance concentration. Such an approach allows us to more adequately describe the processes of mass transfer in environments with a complex spatial and temporal structure, including soils, in the situation of significant costs needed for their exact geophysical analysis. Taking into account the complexity of the solution of inverse problems for differential equations with fractional derivatives, the fixed quantity and continuity of optimized parameters, it is proposed to use a meta-heuristic particle swarm optimization algorithm for their identification. The paper briefly describes the finite-difference method of the approximate solution of the direct problem, poses the problem of parameters identification, and describes the modification of the used particle swarm optimization algorithm. We present the results of computer experiments that show the efficiency of the particle swarm optimization algorithm for determining the parameters of the fractional derivative, as well as the fact that, depending on the type of functional parameter of the generalized fractional derivative, the model allows describing both «ultra-slow» and «ultra-fast» diffusion modes.