Fokas Method-Based Control of the Heat Equation in the k-Plane

We propose a novel boundary value control method for the heat equation. That is, boundary values are constructed such that a given initial condition is driven into a desired final condition. It is based on the Fokas method and it has a similar foundation as the recently proposed algorithm by Kalimeris et al. (2024). However, instead of exploiting the (real) solution formula, our method uses the complex valued global relation. Using the global relation, originally an intermediary product of the Fokas method, allows us to drastically reduce the number of integrals that have to be computed numerically. Compared to the existing algorithm, similar performance but a substantially reduced runtime are found. Some extensions for the Kalimeris et al. algorithm are also proposed, which allow the algorithm to work on a wider range of initial conditions.