In this research, we consider neural network-algorithms for option pricing. We use the Black-Scholes model and the lifted Heston model. We derive the option pricing partial differential equation (PDE), which we solve with a neural network, and the conditional characteristic function of the stock price which leads to the option price with the COS method. We consider two neural network-algorithms: the Deep Galerkin Method (DGM) and the Time Deep Nitsche Method (TDNM). We extend the TDNM to be able to solve the option pricing PDE by splitting the PDE operator in a symmetric part and an asymmetric part. The splitting method is more stable than transforming the Black-Scholes option pricing PDE to the symmetric heat equation. The DGM can predict options prices perfectly in the Black-Scholes model even when $r$ and $\sigma$ are added as variables to the neural network. In the lifted Heston model, the DGM can predict option prices perfectly for small dimensions, but has larger errors for larger dimensions. The TDNM is as good as DGM for small times to maturity and volatilities, but has larger errors for large times to maturity and volatilities.

In this research a study of the response of a simply supported microbeam subject to an electric actuation is presented. A perturbation method called the method of multiple scales is explained and used to solve our problem. A model concerning the mid-plane stretching and an electric force with a direct and alternating current component is formulated. The method of multiple scales is used to construct a solution that is valid for a long time after the initial conditions. The effect of the frequency of the alternating current was studied by performing a stability analysis. The results show that for frequencies close to the eigenfrequency of the homogeneous problem, there is no stable equilibrium and resonance occurs. Furthermore, a start was made to study the effect of the damping coefficient. The results show that a smaller damping will always lead to resonance on a very small time scale. The results also indicate larger oscillations and an small increase of the importance of the non-linear terms for smaller damping. All results are validated by comparing them with a numeric solution and show excellent agreement.