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.

We study a system of non-linear fractional differential equations, subject to integral boundary conditions. We use a parametrization technique and a dichotomy-type approach to reduce the original problem to two “model-type” fractional boundary value problems with linear two-point boundary conditions. A numerical-analytic technique is applied to analytically construct approximate solutions to the “model-type” problems. The behaviour of these approximate solutions is governed by a set of parameters, whose values are obtained by numerically solving a system of algebraic equations. The obtained results are confirmed by an example of the fractional order problem that in the case of the second order differential equation models the Antarctic Circumpolar Current.

We study a parameter-dependent non-linear fractional differential equation, subject to Dirichlet boundary conditions. Using the fixed point theory, we restrict the parameter values to secure the existence and uniqueness of solutions, and analyse the monotonicity behaviour of the solutions. Additionally, we apply a numerical-analytic technique, coupled with the lower and upper solutions method, to construct a sequence of approximations to the boundary value problem and give conditions for its monotonicity. The theoretical results are confirmed by an example of the Antarctic Circumpolar Current equation in the fractional setting.

We consider a nonlinear Neumann boundary value problem which is derived for the Antarctic Circumpolar Current. By the theory of topological degree, we prove the existence results for the problem with semilinear oceanic vorticity term. We also construct the approximate solutions for such a nonlinear model.

We present here exact solutions to the equations of geophysical fluid dynamics that depict inviscid flows moving in the azimuthal direction on a circular path, around the globe, and which admit a velocity profile below the surface and along it. These features render this model suitable for the description of the Antarctic circumpolar current (ACC). The governing equations we work with–taken to be the Euler equations written in spherical coordinates–also incorporate forcing terms which are generally regarded as means that ensure the general balance of the ACC. Our approach allows for a variable density (depending on the depth and latitude) of discontinuous type which divides the water domain into two layers. Thus, the discontinuity gives rise to an interface. The velocity in both layers and the pressure in the lower layer are determined explicitly, while the pressure in the upper layer depends on the free surface and the interface. Functional analytical techniques render (uniquely) the surface and interface-defining functions in an implicit way. We conclude our discussion by deriving relations between the monotonicity of the surface pressure and the monotonicity of the surface distortion that concur with the physical expectations. A regularity result concerning the interface is also derived.

We study a nonlinear fractional boundary value problem (BVP) subject to non-local multipoint boundary conditions. By introducing an appropriate parametrization technique we reduce the original problem to an equivalent one with already two-point restrictions. Using a notion of Chebyshev nodes and Lagrange polynomials we construct a successive iteration scheme, that converges to the exact solution of the non-local problem for particular values of the unknown parameters, which are calculated numerically.

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 a corresponding Sturm-Liouville problem and application of the Fredholm alternative, we establish conditions of existence/uniqueness of solutions to the given problem.

We study a boundary value problem for a Caputo-type fractional differential equation subjected to periodic boundary conditions. For an auxiliary problem with the simplified right-hand side, we explicitly construct its unique solution. In addition, based on the theory of the topological index, we prove existence of at least one solution to the original problem.

This paper proposes a modelling of the Antarctic Circumpolar Current (ACC) by means of a two-point boundary value problem. As the major means of exchange of water between the great ocean basins (Atlantic, Pacific and Indian), the ACC plays a highly important role in the global climate. Despite its importance, it remains one of the most poorly understood components of global ocean circulation. We present some recent results on the existence and uniqueness of solutions of a two-point nonlinear boundary value problem that arises in the modeling of the flow of the (ACC) (see discussions in [4-9]).

We study a boundary value problem for a system of the third order semi-linear partial differential equations with nonlocal boundary conditions. We establish sufficient conditions of existence, uniqueness, regularity and sign-preserving property of solutions of the studied problem and construct an iterative method for its approximation.

We use a numerical-analytic technique to construct a sequence of successive approximations to the solution of a system of fractional differential equations, subject to Dirichlet boundary conditions. We prove the uniform convergence of the sequence of approximations to a limit function, which is the unique solution to the boundary value problem under consideration, and give necessary and sufficient conditions for the existence of solutions. The obtained theoretical results are confirmed by a model example.

We present an approach that facilitates the generation of explicit solutions to atmospheric Ekman flows with a height-dependent eddy viscosity. The approach relies on applying to the governing equations, of Sturm–Liouville type, a suitable Liouville substitution and then reducing the outcome to a nonlinear first-order differential equation of Riccati type.