Applied Analysis Seminar: Kateryna Marynets

Fractional calculus is a powerful tool in mathematical modeling, that got more attention of applied mathematicians and geoscientists in the past decades and continues to develop further. Dynamical systems, involving fractional order derivatives, are able to incorporate the so-called 'memory effects', and due to the non-local nature of fractional differential operators they are usually used in modeling of the flows through porous media (e.g., the groundwater flows), sub- and super-diffusion processes etc. Additionally, a large choice of fractional derivatives and variations in their order give more flexibility in comparison to the classical integer-order models. 

Since most physical processes are nonlinear and the exact solutions of such models is, in general, impossible to find, we are interested in construction of reliable iterative methods that enable us to deal with this task. In my talk I will present one of such approaches, that is based on analysis of a system of fractional differential equations of the Caputo type, subject to different types of boundary conditions. A proper perturbation of the studied system allows us to reduce analysis of the BVP to an equivalent initial value problem, whose solutions are approximated using the numerical-analytic scheme. I will also demonstrate the applicability and effectiveness of this method on a real-world problem.