Browsing by Author "Kalimeris, Konstantinos"
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Item Open Access Dispersion estimates for the boundary integral operator associated with the fourth order Schrödinger equation posed on the half line(Element d.o.o., 2021-09) Özsarı, Türker; Alkan, Kıvılcım; Kalimeris, KonstantinosIn this paper, we prove dispersion estimates for the boundary integral operator associated with the fourth order Schr¨odinger equation posed on the half line. Proofs of such estimates for domains with boundaries are rare and generally require highly technical approaches, as opposed to our simple treatment which is based on constructing a boundary integral operator of oscillatory nature via the Fokas method. Our method is uniform and can be extended to other higher order partial differential equations where the main equation possibly involves more than one spatial derivatives.Item Open Access Existence of unattainable states for Schrödinger type flows on the half-line(Oxford University Press, 2023-12-01) Özsarı, Türker; Kalimeris, KonstantinosWe prove that the solutions of the Schrödinger and biharmonic Schrödinger equations do not have the exact boundary controllability property on the half-line by showing that the associated adjoint models lack observability. We consider the framework of L2 boundary controls with data spaces H−1(R+) and H−2(R+) for the classical and biharmonic Schrödinger equations, respectively. The lack of controllability on the half-line contrasts with the corresponding dynamics on a finite interval for a similar regularity setting. Our proof is based on an argument that uses the sharp fractional time trace estimates for solutions of the adjoint models. We also make several remarks on the connection of controllability and temporal regularity of spatial traces.Item Open Access Numerical computation of Neumann controls for the heat equation on a finite interval(IEEE, 2024-01) Kalimeris, Konstantinos; Özsarı, Türker; Dikaios, NikolaosThis article presents a new numerical method, which approximates Neumann type null controls for the heat equation and is based on the Fokas method. This is a direct method for solving problems originating from the control theory, which allows the realization of an efficient numerical algorithm that requires small computational effort for determining the null control with exponentially small error. Furthermore, the unified character of the Fokas method makes the extension of the numerical algorithm to a wide range of other linear partial differential equations and different type of boundary conditions straightforward.