Kateryna Marynets

We study a nonlinear fractional differential equation, defined on a finite and infinite interval. In the finite interval setting, we attach initial conditions and parameter-dependent boundary conditions to the problem. We apply a dichotomy approach, coupled with the numerical-analytic method, to analyze the problem and to construct a sequence of approximations. Additionally, we study the existence of bounded solutions in the case when the fractional differential equation is defined on the half-axis and is subject to asymptotic conditions. Our theoretical results are applied to the Arctic gyre equation in the fractional setting on a finite interval.

This work allows proving that the action of fractional derivatives and fractional integrals on periodic functions does not preserve the periodicity of any period. This result is proved not only for one type of fractional operator but also for the wide class of generalized fractional operators based on the Sonine condition, a class that encompasses the majority of the fractional operators commonly used. Moreover, for several specific fractional operators, we provide explicit representations of the derivatives and integrals of the sine function, showing that they are composed of a local periodic term and a non-local term, which is the cause of the loss of periodicity.

We present recent results in study of a mathematical model of the sea-breeze flow, arising from a general model of the ’morning glory’ phenomena. Based on analysis of the Dirichlet spectrum of the corresponding Sturm–Liouville problem and application of the Fredholm alternative, we establish conditions of existence/uniqueness of solutions to the given problem.

We introduce a successive approximations method to study one fractional periodic boundary value problem of the Hilfer-Prabhakar type. The problem is associated to the corresponding Cauchy problem, whose solution depends on an unknown initial value. To find this value we numerically solve the so-called ’determining system’ of algebraic or transcendental equations. As a result, we determine an approximate solution of the studied problem, written in a closed form. Finally, we evaluate efficiency of our method on a nonlinear numerical example.