Error Estimates for Finite Element Simulations Using Neural Networks

In this work residual error estimates are constructed using Neural Networks for Finite Element Method. These can be used to do adaptive mesh refinement. Two neural networks are developed the Multilayer Perceptron and the Transformer model. The error estimates are made for 1d poisson equations but the idea will generalise to higher dimensions as...

Reframing existence and uniqueness theorems for the non-homogeneous wave equation

This thesis is about the wave equation. The wave equation describes waves that propagate through a certain medium. The solution of the wave equation is a mathematical description of what that wave looks like. There are many fields of study where the wave equation is used to model processes that behave like travelling waves. For instance, a...

On the Influence of Constant Vertical Wind Speed on the Classical Ekman Spiral

The Ekman spiral is described by a coupled system of differential equations originally discussed by Walfrid Ekman (Ekman, 1905). This system is a simplified version of the Navier­Stokes equations. The differential equations, as discussed in Ekman’s paper, concern the currents of the ocean. However, it is also possible to interpret these...

Stochastic Genetics

In this bachelor thesis we use a stochastic model to aspire to explain biodiversity patterns in different ecosystems with selection advantage. The stochastic model we use is an extension of the mean­-field voter model where we include a selection factor. In the model individuals with two different types of alleles in two different ecosystems are...

Tanner's law in the case of partial wetting

This thesis considers the thin-film equation in partial wetting. The mobility in this equation is given by h³+λ^3-nhⁿ, where h is the film height, λ is the slip length and n is the mobility exponent. The partial wetting regime implies the boundary condition dh/dz>0 at the triple junction....

Three Way Duels: Infinite Games on the Unit Square

With the growing wealth and economy of a country, there are an increasing amount of small and big businesses. Every company has its own marketing strategy that it uses in order to lure customers away from their competition and increase their sales. Choosing the perfect time to advertise or discount several products is of essence for a company to...

On discrete and continuous state adaptive network models: with an application to self-organisation in swarming systems

We consider adaptive network models with discrete and continuous state sets obeying dynamical rules that enable application to swarming systems. The 2-state adaptive network contains a supercritical pitchfork bifurcation in the transition between ordered and disordered stationary solutions. We derive an adaptive network model that works on a...

Het toepassen van complexe methodes voor het schatten van afgeleiden

It is possible to estimate derivatives of real functions with complex methods. These complex methods can increase speed or accuracy. The functions needs to be known to determine the derivative with a complex method.

Differential Equations for Random Processes and Applications for the Balls and Bins Model

On the Rain-Wind Induced Vibrations of a Mass-Spring System

In this report, the rain-wind induced vibrations of cables are studied. This is done by modeling the cable cross-section as a mass-spring system with two time-varying masses. Thereafter, the solution of this model is approximated using a multiple timescale perturbation method. Lastly, for some choices<br/>of the time-varying masses the...