Sea breezes are low-level atmospheric flows that move from the sea to land. They typically form during warm days when the temperature difference between the land and sea surfaces becomes large enough. This temperature difference occurs because seawater, having a higher heat capacity, heats up more slowly than land. In this thesis, we study a non dimensional mathematical model that describes the horizontal velocity of sea breezes in two regions: the Gulf of Carpentaria (Australia) and Calgary (Canada). The model was derived in [4] by writing the Navier-Stokes equations in rotating spherical coordinates. After non dimensionalisation and assuming a specific form of the solution, they obtained the non dimensional model. Furthermore, the model was complemented by two homogeneous Dirichlet boundary conditions, representing the no-slip condition at Earth’s surface and the presence of a thermal inversion layer. In [11], their work was extended by studying the corresponding Sturm-Liouville problem for various mass density functions to determine the existence and uniqueness of the solutions to the model. Building on the work in [11], we analysed the solvability of the model and determined the numerical solutions for two physically relevant mass density functions. These functions were chosen to ensure that zero is not an eigenvalue. The Fredholm alternative then implies that the solution is unique. On the other hand, if zero were an eigenvalue, then the Fredholm alternative implies that we could have an infinite number of solutions, which is physically unrealistic. For the first mass density function, the eigenvalue problem was written as the hyper geometric equation. For the Gulf of Carpentaria (where β > 0), we proved analytically that zero is not an eigenvalue. However, since the parameter β is negative for the Calgary region, we were unable to prove this analytically. Instead, we demonstrated numerically that the eigenvalues closest to zero were λ807 = −828 and λ808 = 790, confirming that zero is not an eigenvalue. Using the finite difference method, we then computed the velocity profiles for three different forcing functions. We first considered the breezes in the Gulf of Carpentaria. In the first case we assumed zero forcing, the result showed that in the absence of any forcing, sea breezes do not occur. For the second case we assumed a non-zero constant forcing, resulting in a uniform velocity profile. Lastly, with an increasing quadratic forcing function, the velocity increased quadratically with height, reaching its maximum just below the thermal inversion layer. Similarly, we computed the velocity profiles for the Calgary region. From these computations, we found that the breezes were flowing in the opposite direction and with higher velocities. These differences arise due to the Coriolis force, which forms due to Earth’s rotation. This force deflects winds to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. Furthermore, the Coriolis force is zero at the equator and grows with latitude, which explains the higher velocities. Finally, we considered the model for the second mass density function, which includes the weak effect the boundaries have on the flow. For the Gulf of Carpentaria it was again proven analytically that zero was not an eigenvalue, while for the Calgary region, the eigenvalues closest to zero were λ501 ≈ −351 and λ502 ≈ 3030. Similar to the first mass density function, we used the finite difference method to compute the velocity profiles. The results showed that the profileswere identical to those obtained with the first mass density function. This similarity occurs because the two mass density function exhibit similar parabolic behaviour. Moreover, the same forcing functions were used in both cases.

The Antarctic Circumpolar Current (ACC) is the only ocean current that flows continuously around the globe, and it plays a crucial role in global climate. It encircles Antarctica and keeps warm waters away, helping to maintain the continent’s cold environment. This report studies a simplified mathematical model to capture the ACC’s steady-state behaviour, incorporating the effects of density stratification. The model studied is originally developed by Ding, who reformulated the ocean dynamics of Arctic gyres as a second-order non-linear differential equation for a stream function u(t) that describes the flow field. This includes the effect of Earth’s rotation (Coriolis effect), represented by ω, a prescribed vorticity distribution F (u), and a stratified density function ρ(u).  

In this report, the model is adapted to describe the ACC. This is achieved by imposing Dirichlet boundary conditions representing the two latitudinal limits. This leads to a BVP defined on a finite domain, providing an appropriate mathematical framework for capturing the flow of the ACC. After simplifying the differential equation to an integral equation, our analytical analysis of the model showed that under reasonable assumptions, an unique solution exists. Before computing a numerical solution, the assumptions introduced in the analytic analysis were verified for the specific choices of F and ρ, confirming their regularity and positivity. The equation was then solved numerically using a collocation method, yielding a stream function profile, consistent with observed ACC behaviour. The numerical solution was validated against an exact solution in a special case (constant F and ρ), showing excellent agreement (error < 10−6). Stability tests confirmed that the solution changes only slightly under small perturbations in boundary conditions.  

Finally, the influence of different density stratification profiles was explored. Gentle stratification (constant or linear ρ(u)) led to a stable, symmetric jet, whereas strongly non-linear stratification (quadratic or exponential ρ(u)) resulted in oscillatory or unstable solutions. This highlights the model’s sensitivity to stratification.  

In this thesis, we study fractional differential equations with Hilfer derivative operators. Solutions are approximated using a newly developed Bernstein-splines approach and subsequently applied to the Van der Pol oscillator with fractional damping. Fractional derivatives generalize differentiation to the order α ∈ (0,∞), resulting in order α differential operators. The Hilfer-derivative of order α ∈ (0,1) and type β ∈ [0,1] is one of such operators and combines two of the most commonly used operators through parametrization: the Riemmann-Liouville-derivative (β = 0) and Caputo-derivative (β=1). The choice of β results in different kernel behavior of the operator, commonly yielding singular behavior of solutions of initial value problems (IVP's) and boundary value problems (BVP's) for β ∈ [0,1). Based on existing results for IVP's, we develop a new proof for existence and uniqueness of solutions to BVP's for Hilfer-fractional derivatives. To obtain solution approximations numerically for IVP's and BVP's, a Bernstein splines solution approach is developed and implemented, providing accurate convergence results for nonlinear IVP's and BVP's in an efficient vectorized approach. Implementation difficulties for the singular behavior of solutions for β ∈ [0,1) are successfully resolved through time-domain transformation approximation techniques. Finally, the methods are applied to numerically study the behavior of the fractionally damped Van der Pol oscillator, a nonlinear equation of interest in electrical engineering and control theory. We study the approximate limit cycle, of which behavior corresponds with existing analytical results for the Caputo-derivative (β=1). Convergence of solutions can also be obtained for Hilfer type values of β ∈ [0,1), appearing to be of little influence on the long-term limit cycle. Furthermore, when forcing is applied, we observe periodic, quasiperiodic and chaotic behavior.

Describing phylogenetic trees or networks with a polynomial is a tool to distinguish between them. In this thesis, a new polynomial for describing rooted binary internally labeled phylogenetic networks and trees is introduced based on the research of P. Liu and J. Pons et al. Two different cases are considered, one where the reticulation nodes have distinct labels λ_i and one where the reticulation nodes have the same label λ. There are a few conjectures stated about the uniqueness of the polynomial and the relation of the polynomial with the primary subtrees and their monomial. Also the folding and unfolding of a network is described. Furthermore, an algorithm is provided with which a tree can be made out of different monomials. With use of the lemma that states when a tree can be folded to a network, it can be determined if the tree can be folded to a network.