Searcharxiv⌕ Search

arXiv subjects

Konstantinos Kalimeris

Publications and source records attributed to Konstantinos Kalimeris.

15 recordsLinked to original sources
arXiv2026

The unified transform for Burgers' equation: Application to unsaturated flow in finite interval

In this paper, we focus on one-dimensional vertical infiltration, assuming constant diffusivity and a quadratic relationship between hydraulic conductivity and water content. Under these assumptions, Richards' equation reduces to Burgers' equation, which we then linearize via the Hopf-Cole transformation. This turns the initial boundary value problem into a diffusion equation on a finite interval with mixed boundary conditions. To solve it, we use the Unified Transform Method (also known as the Fokas method). This approach gives an explicit integral representation of the solution, and when evaluated numerically, the results match classical Fourier series solutions exactly, but with better convergence and stability. Two examples from hydrological applications are examined.

math.AP↗
arXiv2025

Estimating parameters of the diffusion model via asymptotic expansions

A broad class of inverse problems deals with determining certain parameters, from measurement data, in models which are associated to certain partial differential equations. In this work we focus on the heat equation on a finite interval and we determine the dimensionless diffusion parameter from a single measurement. Our results extend to estimating additional parameters of the initial-boundary value problem, such as the length of the interval and/or the time required for the solution to achieve a specific state. Our approach relies on the asymptotic solution of an integral equation: The formulation of this integral equation is based on the solution of the direct problem via the Fokas method; the solution of this equation is achieved through the asymptotic evaluation of the associated integrals which yield an effective approximate solution, supported by numerical verifications. We apply these approximations to well-established problems in soil science and we compare our results with existing ones, displaying clear improvement.

math.AP↗
arXiv2025

A new approach for the analysis of evolution partial differential equations on a finite interval

We show that, for certain evolution partial differential equations, the solution on a finite interval $(0,\ell)$ can be reconstructed as a superposition of restrictions to $(0,\ell)$ of solutions to two associated partial differential equations posed on the half-lines $(0,\infty)$ and $(-\infty,\ell)$. Determining the appropriate data for these half-line problems amounts to solving an inverse problem, which we formulate via the unified transform of Fokas (also known as the Fokas method) and address via a fixed point argument in $L^2$-based Sobolev spaces, including fractional ones through interpolation techniques. We illustrate our approach through two canonical examples, the heat equation and the Korteweg-de Vries (KdV) equation, and provide numerical simulations for the former example. We further demonstrate that the new approach extends to more general evolution partial differential equations, including those with time-dependent coefficients. A key outcome of this work is that spatial and temporal regularity estimates for problems on a finite interval can be directly derived from the corresponding estimates on the half-line. These results can, in turn, be used to establish local well-posedness for related nonlinear problems, as the essential ingredients are the linear estimates within nonlinear frameworks.

math.AP↗
arXiv2025

Fokas method for linear convection-diffusion equation with time-dependent coefficients and its extension to other evolution equations

In this paper, we study a linear convection-diffusion equation with time-dependent coefficients on a bounded interval, motivated by associated physical problems. We apply and adapt the Unified Transform Method (UTM), a.k.a. Fokas Method, which handles both time-varying coefficients and nonzero boundary data, to obtain an explicit integral formula for the solution. Next, we study well-posedness of the model in fractional Sobolev spaces and prove spatial and temporal regularity estimates, showing that the smoothing effect of the heat operator is still prevalent even when coefficients depend on time. Finally, we extend this approach to obtain the solution for several evolution equations with time-dependent coefficients, in one space variable.

math.AP↗
arXiv2025

Estimating properties of a homogeneous bounded soil using machine learning models

This work focuses on estimating soil properties from water moisture measurements. We consider simulated data generated by solving the initial-boundary value problem governing vertical infiltration in a homogeneous, bounded soil profile, with the usage of the Fokas method. To address the parameter identification problem, which is formulated as a two-output regression task, we explore various machine learning models. The performance of each model is assessed under different data conditions: full, noisy, and limited. Overall, the prediction of diffusivity $D$ tends to be more accurate than that of hydraulic conductivity $K.$ Among the models considered, Support Vector Machines (SVMs) and Neural Networks (NNs) demonstrate the highest robustness, achieving near-perfect accuracy and minimal errors.

physics.geo-ph↗
arXiv2024

Rainfall infiltration: Direct and Inverse problems on a linear evolution equation

Originating from the mathematical modelling of rainfall infiltration, we derive the solution of an initial-boundary value problem of a linear evolution partial differential equation, by using the Fokas method. We present numerical examples which correspond to specific physical rainfall problems. Based on this formalism we present an effective algorithm for the associated null-controllability problem, namely we numerically derive a family of boundary controls that steer the solution to the desired flat final state. Finally, a regularisation scheme allows the derivation of relatively small controls, in cases where this is necessary.

math.AP↗
arXiv2024

An analytical solution for vertical infiltration in bounded profiles

In this study, we derive an analytical solution to address the problem of one-dimensional vertical infiltration within bounded profiles. We consider the Richards equation together with various boundary conditions, simulating different scenarios of water application onto the surface of a homogeneous and bounded medium. To solve the corresponding initial boundary value problem over a finite interval, we apply the unified transform, commonly known as the Fokas method. Through this methodology, we obtain an integral representation that can be efficiently and directly computed numerically, yielding a convergent scheme.

math.AP↗
arXiv2023

Wave scattering in 1D: D'Alembert-type representations and a reconstruction method

We derive the extension of the classical d'Alembert formula for the wave equation, which provides the analytical solution for the direct scattering problem for a medium with constant refractive index; this is achieved by employing results obtained via the Fokas method. This methodology is further extended to a medium with piecewise constant refractive index, providing the apparatus for the solution of the associated inverse scattering problem. Hence, we provide an exact reconstruction method which is valid for both full and phaseless data.

math.AP↗
arXiv2023

Numerical computation of Neumann controls for the heat equation on a finite interval

This paper 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 realisation 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 PDEs and different type of boundary conditions straightforward.

math.NA↗
arXiv2021

Dispersion estimates for the boundary integral operator associated with the fourth order Schrödinger equation posed on the half line

In this paper, we prove dispersion estimates for the boundary integral operator associated with the fourth order Schrödinger 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.

math.AP↗
arXiv2020

An elementary proof of the lack of null controllability for the heat equation on the half line

In this note, we give an elementary proof of the lack of null controllability for the heat equation on the half line by employing the machinery inherited by the unified transform, known also as the Fokas method. This approach also extends in a uniform way to higher dimensions and different initial-boundary value problems governed by the heat equation, suggesting a novel methodology for studying problems related to controllability.

math.OC↗
arXiv2018

Explicit asymptotics for certain single and double exponential sums

By combining classical techniques together with two novel asymptotic identities contained in [FL], we analyse certain single sums of Riemann-zeta type. In addition, we analyse Euler-Zagier double exponential sums for particular values of $Re\{u\}$ and $Re\{v\}$ and for a variety of sets of summation, as well as particular cases of Mordell-Tornheim double sums. Some of these results are used in [F] where a novel approach to the Lindelöf hypothesis is presented.

math.CA↗
arXiv2012

Photoacoustic imaging in attenuating acoustic media based on strongly causal models

In this paper we derive time reversal imaging functionals for two strongly causal acoustic attenuation models, which have been proposed recently. The time reversal techniques are based on recently proposed ideas of Ammari et al for the thermo-viscous wave equation. Here and there an asymptotic analysis provides reconstruction functionals from first order corrections for the attenuating effect. In addition, we present a novel approach for higher order corrections.

math.AP↗