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Andreas Chatziafratis

Publications and source records attributed to Andreas Chatziafratis.

10 recordsLinked to original sources
arXiv2026

On generalised d'Alembert-type integral representations for damped wave equations on the quarter-plane

We rigorously construct and verify a posteriori new closed-form solutions for the forced Maxwell-Cattaneo-Vernotte equation (also broadly known as the damped wave equation, hyperbolic heat, and telegrapher's equation on lossy transmission lines) posed on the spatiotemporal quarter-plane with general initial and boundary data in classical function spaces. For this purpose, the modern complex-analytic unified transform method of Fokas (originally developed for elliptic PDE and evolution equations with polynomial dispersion relations) is here, for the first time, extended for analysis of hyperbolic-parabolic problems on the semi-infinite interval. Importantly, we then establish theorems which pertain to regularity, boundary and asymptotic properties of the new analytical formulae as well as to well-posedness of the addressed boundary-value problems. Notably, the nature and generality of problems considered, combined with the semi-unboundedness of the domain, induce substantial analytic challenges which demand delicate treatment, both in appropriately interpreting oscillatory integral terms of the solution formulae and in proving the proposed results. In this process, crucially, certain compatibility conditions, between initial, boundary and forcing data at the origin, are revealed, which guarantee the existence of a smooth solution across the whole domain of interest. Our explicit integral representations are of direct utility for numercal benchmarking purposes, for exploring connections with modelling in continuum mechanics, mathematical physics, biology and the natural sciences, and for the investigation of well-posedness for nonlinear counterparts too.

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arXiv2026

Integral Representations for Interface Problems for the Barenblatt-Sobolev-Galpern Pseudoparabolic Equation

We obtain novel integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous initial-boundary-value as well as interface problems for the Barenblatt-Zheltov-Kochina pseudo-parabolic equation of Sobolev-Galpern type formulated on the real line, half-line and finite interval. This fundamental partial differential equation (PDE) of mathematical physics emerges in a wide variety of natural phenomena and applied sciences including continuum mechanics, thermodynamics, chemical engineering, solid-state electronics, semi-conductor devices, battery research, and nanotechnology. A suitable implementation of a modern methodology, known as Unified Transform Method, is in force throughout this study, with particular challenges arising due to the higher-order mixed-derivative term of the PDE and the generality of the problems under consideration altogether. The boundary and interface conditions appear to be non-standard but are naturally dictated by the structure of the PDE itself. Our explicit analytical formulae directly lend themselves to future explorations of the solutions' qualitative properties such as asymptotic behavior, spatio-temporal dynamics, regularity and well-posedness. This work is expected to be of utility also in the investigation of nonlinear counterparts as well as towards the study of phase-transition phenomena and free-boundary problems, where the interface evolves dynamically according to energy balance laws.

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arXiv2026

Rigorous analysis for the Dirac system on the quarter-plane

Considered and analyzed below are fully non-homogeneous initial-boundary-value problems for the celebrated Dirac system, formulated on the spatial half-line. Analytical solution formulae are derived formally via suitable implementation of the well-known Fokas' unified transform methodology, and rigorously verified a posteriori. The latter substantial task relies on complex-analytic tools and careful interpretation of the obtained integral representations. These valid solutions are then used for investigating qualitative properties. These include boundary behavior near the axes of the domain as well as long-range asymptotics and long-time (eventual) periodicity. Notably, smoothness of the solution, both within and upto the boundary of the domain, depends heavily on certain compatibility conditions between initial, boundary and forcing data. Further results pertaining to solution's regularity and uniqueness are thence established based on the qualitative theory. The closed-form expressions reported here are also useful in the study of non-linear counterparts.

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arXiv2026

The linear Cahn-Hilliard equation with an interface

We obtain new integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous interface problems for the linearized Cahn-Hilliard equation with arbitrary initial data on the line and general interface conditions prescribed at the origin. Cahn-Hilliard-type models emerge in applied mathematics in connection to a spectacular variety of phenomena of mathematical physics, continuum mechanics, chemistry and biology. A novel implementation of Fokas' unified transform method is in force herein for a fourth-order operator for the first time, with particular challenges arising due to the nature and the generality of the problems under consideration. Our explicit formulae directly lend themselves to exploration of the solution's qualitative properties such as regularity and asymptotic behavior. This work is also useful in the investigation of well-posedness for nonlinear counterparts as well as in the study of free-boundary and diffuse-interface problems.

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arXiv2025

Fokas-type closed-form solution formulae for Sobolev-type equations with time-dependent coefficients

We analytically derive novel explicit integral representations for the solution of nonhomogeneous initial-boundary-value problems for a large category of evolution partial differential equations of Sobolev-Galpern type with generic temporally variable coefficients, satisfying suitable mild conditions, and with arbitrary data in classical function spaces. This work is based on the careful implementation of the pioneering Fokas unified transform methodology alongside its recently-proposed extension for solving a class of linear evolution equations with dispersion relation of specific polynomial type and time-dependent coefficients. We herein effectively extend those techniques to a special collection of evolution equations with time-dependent coefficients and mixed spatiotemporal derivatives, which induce rational dispersion relations. The new approach is exhibited in detail through illustrative generation of closed-form solutions for a multitude of such equations (such as Milne-Taylor-Barenblatt-Coleman-Ting-Chen-type, Benjamin-Bona-Mahony-type, as well as numerous higher-order variants) posed on the half-line. Challenging technical difficulties of complex-analytic and of algebraic flavour naturally emerge due the presence of mixed-derivative terms, and these are appropriately resolved in each case. The new formulas are of utility in subsequent investigation of qualitative properties and analysis of nonlinear counterparts too. Further extensions, generalizations, rigorous aspects and implementations to other types of problems as well will soon be reported in forthcoming publications.

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arXiv2025

Infinity of solutions to initial-boundary value problems for linear constant-coefficient evolution PDEs on semi-infinite intervals

In this short communication, we announce an algorithmic procedure for constructing non-uniqueness counter-examples of classical solutions to initial-boundary-value problems for a wide class of linear evolution partial differential equations, of any order and with constant coefficients, formulated in a quarter-plane. Our approach relies on analysis of regularity and asymptotic properties, near the boundary of the spatio-temporal domain, of closed-form integral-representation formulae derived via complex-analytic techniques and rigorous implementation of the modern PDE technique known as Fokas unified transform method. In order to elucidate the novel idea and demonstrate the proposed technique in a self-contained fashion, we explicitly present its application to two concrete examples, namely the heat equation and the linear KdV equation with Dirichlet data. New uniqueness theorems for these two models are also presented herein.

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arXiv2024

Continuous dependence on data for linear evolution PDEs on the quarter-plane

In this note, we announce a systematic analysis of continuous dependence on the data in classical spaces for the initial-boundary-value problem of the diffusion equation on the half-line, with data that are not necessarily compatible at the quadrant corner. This is based on a recent approach to rigorously analyzing integral representations derived via the unified transform method of Fokas. No exotic phenomena were discovered in this case, yet our findings appear to be new in the pertinent literature. These results supplement our previous investigations on existence and (non)uniqueness within the framework of well-posedness. The present detailed exposition elucidates the subtleties involved while also demonstrating a generic technique. Applications of the latter to several other IBVPs and PDEs will be reported elsewhere.

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arXiv2024

Boundary behaviour of the solution of the heat equation on the half line via the Fokas unified transform method

We consider the Fokas method expression for the solution of the heat equation on the half line with Dirichlet data and we study in detail its boundary behaviour near the spatiotemporal domain boundaries, i.e., the semi-axes, infinity and the origin, by analyzing the integrals involved. We also study the boundary behaviour of the derivatives of the solution. In particular we give conditions on the data which guarantee the extension of the solution to a smooth function up to the semi-infinite boundaries.

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arXiv2024

A note on uniqueness for linear evolution PDEs posed on the quarter-plane

In this paper, we announce a rigorous approach to establishing uniqueness results, under certain conditions, for initial-boundary-value problems for a class of linear evolution partial differential equations (PDEs) formulated in a quarter-plane. We also effectively propose an algorithm for constructing non-uniqueness counter-examples which do not satisfy the said conditions. Our approach relies crucially on the rigorous analysis of regularity and asymptotic properties of integral representations derived formally via the celebrated Unified Transform Method for each such PDE. For uniqueness, this boundary behavior analysis allows for a careful implementation of an energy-estimate argument on the semi-unbounded domain. Our ideas are elucidated via application of the present technique to two concrete examples, namely the heat equation and the linear KdV equation with Dirichlet data on the positive quadrant, under a particular set of conditions at the domain boundary and at infinity. Importantly, this is facilitated by delicate refinement of previous results concerning the boundary behavior analysis of these two celebrated models. In addition, we announce a uniqueness theorem for the linearized BBM equation, whose proof, in a similar spirit, will appear in a forthcoming paper. Finally, we briefly demonstrate how the general case of oblique Robin data can be recast as a Dirichlet problem. To the best of our knowledge, such well-posedness results appear for the first time in either the classical or the weak sense. Extensions to other classes of equations are underway and will appear elsewhere.

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