| Abstract: |
| We survey some of our recent research results relating to construction of
explicit solutions (closed-form integral representations), in the
classical sense, as well as to qualitative theory for non-homogeneous
initial-boundary-value and interface problems for a variety of linear
(systems of) evolution partial differential equations (PDE) with constant
and with variable coefficients. Such PDE emerge in connection to a
plethora of natural phenomena and mathematical models in physics, biology,
chemical engineering, finance and other applied sciences; examples include
continuum mechanics, heat transfer, biomedicine, electron optics, and
battery technology. Our research program relies on, explores and extends
the applicability of the celebrated complex-analytic unified transform
method of Fokas. This is joint work with a large global network of
collaborators. Notable findings include: long-range instabilities (see
e.g. [A. Chatziafratis, T. Ozawa, S.-F. Tian, “Rigorous analysis of the
unified transform method and long-range instabilities for the
inhomogeneous time-dependent Schrödinger equation on the quarter-plane”,
Math. Annalen, 2023] and [A. Chatziafratis, L. Grafakos, S. Kamvissis,
Long-range instabilities for linear evolution PDE on semi-bounded domains
via the Fokas method, Dyn. PDE, 2024]), time-asymptotic break-down effects
(e.g. [J.L. Bona, A. Chatziafratis, H. Chen, S. Kamvissis, “The linear
BBM-equation on the half-line revisited”, Lett. Math. Phys., 2024]), and
counter-examples to solution uniqueness (e.g. [A. Chatziafratis, A.
Miranville, G. Karali, A.S. Fokas, E.C. Aifantis, "Higher-order diffusion
and Cahn-Hilliard-type models revisited on the half-line", Math. Models
Methods Appl. Sci., 2025] and [A. Chatziafratis, S. Kamvissis, “Infinity
of solutions to initial-boundary value problems for linear
constant-coefficient evolution PDEs on semi-infinite intervals”, Bull.
London Math. Soc., 2025]) |
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