# Loewner Equation Simulation

The Loewner equation is the initial value problem $$\frac{\partial}{\partial t}g_t(z)=\frac{2}{g_t(z)-\lambda(t)},\quad g_0(z)=z$$ for $z\in\mathbb{H}$ with $\lambda:[0,T]\to\mathbb{R}$. The function $\lambda(t)$ is called the driving function. The only time that a solution fails to exist is when the denominator is zero. Let $K_t=\{z\in\mathbb{H}:\lambda(s)=g_s(z)\text{ for some }s\leq t\}$. This set is called a hull. The following SageMath code simulates the hull corresponding to a given driving function. In order to simulate a hull, input a driving function, final time, and number of samples. This will automatically update when the cursor leaves an input box.

The following Mathematica code can be downloaded and used to simulate the hulls from the Loewner equation: