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Let $\mathcal{A}$ denote a real, $n$-dimensional, unital, associative algebra.This paper provides an introductory exposition of calculus over $\mathcal{A}$. An $\mathcal{A}$-differentiable function is one for which the differential is…
Nowadays the geometric approach in optics is often used to find out media parameters based on propagation paths of the rays because in this case it is a direct problem. However inverse problem in the framework of geometrical optics is…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
In this note we analyse the propagation of a small density perturbation in a one-dimensional compressible fluid by means of fractional calculus modelling, replacing thus the ordinary time derivative with the Caputo fractional derivative in…
We present a study of geometric phases in classical wave and polarisation optics using the basic mathematical framework of quantum mechanics. Important physical situations taken from scalar wave optics, pure polarisation optics, and the…
In order to improve the frequency dispersion effects of irrotational shallow water models in coastal oceanography, several full dispersion versions of classical models were formally derived in the literature. The idea, coming from G.…
In this paper, a multi-dimensional fractional wave equation that describes propagation of the damped waves is introduced and analyzed. In contrast to the fractional diffusion-wave equation, the fractional wave equation contains fractional…
In this paper, we construct the transport equation and the wave equation with specular derivatives and solve these equations in one-dimension. To solve these equations, we introduce new function spaces, which we term specular spaces,…
Deformations of the computational mesh arising from optimization routines usually lead to decrease of mesh quality or even destruction of the mesh. We propose a theoretical framework using pre-shapes to generalize classical shape…
In the present work, we investigate the potential of fractional derivatives to model atmospheric dispersion of pollutants. We propose simple fractional differential equation models for the steady state spatial distribution of concentration…
Cumulants represent a natural language for expressing macroscopic properties of a solid. We show that cumulants are subject to a nontrivial geometry. This geometry provides an intuitive understanding of a number of cumulant relations which…
The procedure of evaluating of the spectrum for discrete periodic operators perturbed by operators of smaller dimensions is obtained. This result allows to obtain propagative, guided, localised spectra for different kind of physical…
A general linear gauge-invariant equation for dispersive gravitational waves (GWs) propagating in matter is derived. This equation describes, on the same footing, both the usual tensor modes and the gravitational modes strongly coupled with…
It is standard practice to study the lensing of gravitational waves (GW) using the geometric optics regime. However, in many astrophysical configurations this regime breaks down as the wavelength becomes comparable to the Schwarzschild…
The laws of geometric optics and their corrections are derived for scalar, electromagnetic, and gravitational waves propagating in generic curved spacetimes. Local peeling-type results are obtained, where different components of…
For over 70 years it has been assumed that scalar wave propagation in (ensemble-averaged) random particulate materials can be characterised by a single effective wavenumber. Here, however, we show that there exist many effective…
Partial differential equations (PDEs) are central to computational electromagnetics (CEM) and photonic design, but classical solvers face high costs for large or complex structures. Quantum Hamiltonian simulation provides a framework to…
The classical numerical methods play important roles in solving wave equation, e.g. finite difference time domain method. However, their computational domain are limited to flat space and the time. This paper deals with the description of…
Electromagnetic waves are an inherent part of all plasmas -- laboratory fusion plasmas or astrophysical plasmas. The conventional methods for studying properties of electromagnetic waves rely on discretization of Maxwell equations suitable…
Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature…