Athanassios S. Fokas

• Doctor of Philosophy
• University of Cambridge

The process of ‘comprehending how we comprehend’, requires elucidating fundamental cognitive mechanisms used by the brain. In these regards, this article discusses two basic hypotheses. First, unconscious and conscious processes form a continuum. Second, humans have a predisposition to construct real versions of mental images and of unconscious str...

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mec...

We consider the linear quadratic regulator of the heat equation on a finite interval. Previous frequency-domain methods for this problem rely on discrete Fourier transform and require symmetric boundary conditions. We use the Fokas method to derive the optimal control law for general Dirichlet and Neumann boundary conditions. The Fokas method uses...

The methodology based on the so‐called global relation, introduced by the first author, has recently led to the derivation of a novel nonlinear integral‐differential equation characterizing the classical problem of the Saffman–Taylor fingers with nonzero surface tension. In the particular case of zero surface tension, this equation is satisfied by...

A novel method is presented for explicitly solving inhomogeneous initial-boundary-value problems (IBVPs) on the half-line for a well-known coupled system of evolution partial differential equations. The so-called double-diffusion model, which is based on a simple, yet general, inhomogeneous diffusion configuration, describes accurately several impo...

A novel technique is presented for explicitly solving inhomogeneous initial‐boundary‐value problems (IBVPs) (Dirichlet, Neumann and Robin) on the half‐line, for a well‐known pseudo‐parabolic partial differential equation. This so‐called Barenblatt's equation arises in a plethora of important applications, ranging from heat‐mass transfer, solid‐flui...

The transformative achievements of deep learning have led several scholars to raise the question of whether artificial intelligence (AI) can reach and then surpass the level of human thought. Here, after addressing methodological problems regarding the possible answer to this question, it is argued that the definition of intelligence proposed by pr...

Since their introduction, Chebyshev polynomials of the first kind have been extensively investigated, especially in the context of approximation and interpolation. Although standard interpolation methods usually employ equally spaced points, this is not the case in Chebyshev interpolation. Instead of equally spaced points along a line, Chebyshev in...

The Korteweg-de Vries equation (KdV) is a classic representative of one-dimensional integrable systems, while the Kadomtsev-Petviashvili (KP) equation is a representative of two-dimensional integrable systems, which is an extension of the KdV equation in two dimensions. However, constructing three-dimensional integrable nonlinear equations has alwa...

Mental processes are associated with brain activation, which in turn gives rise to neuronal electric currents, generating electric fields. Electroencephalography (EEG) is based on the measurements of the electric potential on the scalp and has a variety of neurophysiological and clinical applications. In recent years, there have been efforts to use...

In 2009 Fokas began a program of study of the investigation of the large t-asymptotics of the Riemann zeta function, \(\zeta (\sigma +it)\). In the current work we present a novel difference-integral equation which is satisfied asymptotically by \(\zeta (1/2+it)\). This equation is obtained starting with a singular integral equation presented for t...

We investigate the Cauchy problem on the cylinder, namely the semi-periodic problem where there is periodicity in the x -direction and decay in the y -direction, for the Kadomtsev–Petviashvili II equation by the inverse spectral transform method. For initial data with small L ¹ and L ² norms, assuming the zero mass constraint, this initial-value pr...

The most extensively used mathematical models in epidemiology are the susceptible-exposed-infectious-recovered (SEIR) type models with constant coefficients. For the first wave of the COVID-19 epidemic, such models predict that at large times equilibrium is reached exponentially. However, epidemiological data from Europe suggest that this approach...

We investigate the Cauchy problem on the cylinder, namely the semi-periodic problem where there is periodicity in the $x$-direction and decay in the $y$-direction, for the Kadomtsev-Petviashvili II equation by the inverse spectral transform method. For initial data with small $L^1$ and $L^2$ norms, assuming the zero mass constraint, this initial-va...

We derive the solution of the one dimensional wave equation for the Dirichlet and Robin initial-boundary value problems (IBVPs) formulated on the half line and the finite interval, with nonhomogeneous boundary conditions. Although explicit formulas already exist for these problems, the unified transform method provides a convenient framework for de...

Activation functions play a key role in neural networks, as they significantly affect the training process and the network’s performance. Based on the solution of a certain ordinary differential equation of the Riccati type, this work proposes an alternative generalized adaptive solution to the fixed sigmoid, which is called “generalized Riccati ac...

We elaborate on a new methodology, which starting with an integrable evolution equation in one spatial dimension, constructs an integrable forced version of this equation. The forcing consists of terms involving quadratic products of certain eigenfunctions of the associated Lax pair. Remarkably, some of these forced equations arise in the modelling...

The celebrated Korteweg–de Vries and Kadomtsev–Petviashvili (KP) equations are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. The question of constructing integrable evolution equations in three-spatial dimensions has been one of the most important open problems in the history of integrabili...

There are integrable nonlinear evolution equations in two spatial variables. The solution of the initial value problem of these equations necessitated the introduction of novel mathematical formalisms. Indeed, the classical Riemann–Hilbert problem used for the solution of integrable equations in one spatial variable was replaced by a non-local Riem...

We elaborate on a new methodology, which starting with an integrable evolution equation in one spatial dimension, constructs an integrable forced version of this equation. The forcing consists of terms involving quadratic products of certain eigenfunctions of the associated Lax pair. Remarkably, some of these forced equations arise in the modelling...

We elaborate on a new methodology, which starting with an integrable evolution equation in one spatial dimension, constructs an integrable forced version of this equation. The forcing consists of terms involving quadratic products of certain eigenfunctions of the associated Lax pair. Remarkably, some of these forced equations arise in the modelling...

We present a general methodology, which starting with an integrable evolution equation in one or two spatial dimensions, constructs an integrable forced version of this equation and solves the associated initial value problem of an integrable forced version of the given equation. The forcing consists of nonlinear terms involving the eigenfunctions...

A central feature of pandemics is the emergence and decay of localized infection waves. While traditional SIR models for infectious diseases can reproduce such waves, they fail to capture two key features. First, SIR models are unable to represent short-duration super-spreader events which often trigger infection waves in a community. Second, SIR m...

Predictive modelling of infectious diseases is very important in planning public health policies, particularly during outbreaks. This work reviews the forecasting and mechanistic models published earlier. It is emphasized that researchers’ forecasting models exhibit, for large t, algebraic behavior, as opposed to the exponential behavior of the cla...

Over the last decades, there has been an increasing interest in dedicated preclinical imaging modalities for research in biomedicine. Especially in the case of positron emission tomography (PET), reconstructed images provide useful information of the morphology and function of an internal organ. PET data, stored as sinograms, involve the Radon tran...

A B S T R A C T Background and Objective The Spline Reconstruction Technique (SRT) is a fast algorithm based on a novel numerical implementation of an analytic representation of the inverse Radon transform. The purpose of this study is to provide a comparison between SRT, Filtered Back-Projection (FBP), Ordered Subset Expectation Maximization 2D (...

We present several formulae for the large t t asymptotics of the Riemann zeta function ζ ( s ) \zeta (s) , s = σ + i t s=\sigma +i t , 0 ≤ σ ≤ 1 0\leq \sigma \leq 1 , t > 0 t>0 , which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations...

This paper elaborates on a new approach for solving the generalized Dirichlet‐to‐Neumann map, in the large time limit, for linear evolution PDEs formulated on the half‐line with time‐periodic boundary conditions. First, by employing the unified transform (also known as the Fokas method) it can be shown that the solution becomes time‐periodic for la...

We study the large time behaviour of the solution of linear dispersive partial differential equations posed on a finite interval, when at least one of the prescribed boundary conditions is time periodic. We use the Q equation approach, pioneered in Fokas & Lenells 2012 and applied to linear problems on the half-line in Fokas & van der Weele 2021, t...

The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based...

The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based...

The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based...

The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based...

The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based...

The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based...

The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based...

Doubly localized two-dimensional rogue waves for the Davey–Stewartson I equation in the background of dark solitons or a constant, are investigated by employing the Kadomtsev–Petviashvili hierarchy reduction method in conjunction with the Hirota’s bilinear technique. These two-dimensional rogue waves, described by semi-rational type solutions, illu...

In a recent article, we introduced two novel mathematical expressions and a deep learning algorithm for characterizing the dynamics of the number of reported infected cases with SARS-CoV-2. Here, we show that such formulae can also be used for determining the time evolution of the associated number of deaths: for the epidemics in Spain, Germany, It...

The unified transform method (UTM) provides a novel approach to the analysis of initial-boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM express...

Quantitative magnetic resonance imaging (MRI) estimates magnetic parameters related to tissue, such as T1, T2 relaxation times and proton density. MR fingerprinting (MRF) is a new concept that uses pseudo-random, incoherent measurements to create a unique fingerprint for each tissue type to quantify magnet parameters. This paper aims to enhance MRF...

The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based...

Guided by a rigorous mathematical result, we have earlier introduced a numerical algorithm, which using as input the cumulative number of deaths caused by COVID-19, can estimate the effect of easing of the lockdown conditions. Applying this algorithm to data from Greece, we extend it to the case of two subpopulations, namely, those consisting of in...

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schr\"odinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann-Hil...

Starting from the 3-wave interaction equations in 2+1 dimensions (i.e., two space dimensions and one time dimension), we complexify the independent variables, thus doubling the number of real variables, and hence we work in 4+2 dimensions: x1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{am...

A correction to this paper has been published: https://doi.org/10.1007/s40315-021-00387-4

The inversion of the celebrated Radon transform in three dimensions involves two-dimensional plane integration. This inversion provides the mathematical foundation of the important field of medical imaging, known as three-dimensional positron emission tomography (3D PET). In this chapter, we present an analytical expression for the inversion of the...

The unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM express...

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann–Hilbe...

Humans are equipped with the so-called Mental Time Travel (MTT) ability, which allows them to consciously construct and elaborate past or future scenes. The mechanisms underlying MTT remain elusive. This study focused on the late positive potential (LPP) and alpha oscillations, considering that LPP covaries with the temporal continuity whereas the...

This chapter considers the so-called four-shell model of the brain and assumes that a continuously distributed primary neuronal current is supported either within the cerebral cortex or only on the surface of the cortex, S c , and is normal to this surface. The authors show that in the former case, electroencephalogram recordings are affected only...

In the 50 years since magnetoencephalography (MEG) was invented, various clinical and research applications of it have been attempted with considerable success. This is most notable in the area of epilepsy and presurgical functioning mapping. However, the best ways to apply MEG and interpret the findings still remain conjectural. As such, this book...

We introduce a novel methodology for predicting the time evolution of the number of individuals in a given country reported to be infected with SARS-CoV-2. This methodology, which is based on the synergy of explicit mathematical formulae and deep learning networks, yields algorithms whose input is only the existing data in the given country of the...

Following the highly restrictive measures adopted by many countries for combating the current pandemic, the number of individuals infected by SARS-CoV-2 and the associated number of deaths steadily decreased. This fact, together with the impossibility of maintaining the lockdown indefinitely, raises the crucial question of whether it is possible to...

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixe...

Using the unified transform, also known as the Fokas method, we analyse the modified Helmholtz equation in the regular hexagon with symmetric Dirichlet boundary conditions; namely, the boundary value problem where the trace of the solution is given by the same function on each side of the hexagon. We show that if this function is odd, then this pro...

UNSTRUCTURED Using rigorous mathematical results, evidence is presented which suggests that the easing of the lockdown measures adopted by several European counties may not cause a substantial ‘second wave’, which would then imply that the Covid-19 epidemics in these European countries may be coming to an end. This evidence is based on the review o...

Unstructured: Although the SARS CoV-2 virus has already undergone several mutations, the impact of these mutations on infectivity and virulence remains controversial. In this viewpoint we present arguments suggesting less virulence but much higher infectivity. This suggestion is based on the results of the forecasting and mechanistic models develo...

Following the highly restrictive measures adopted by many countries for combating the current pandemic, the number of individuals infected by SARS-CoV-2 and the associated number of deaths is steadily decreasing. This fact, together with the impossibility of maintaining the lockdown indefinitely, raises the crucial question of whether it is possibl...

We consider two sub-populations consisting of individuals below or above 40 years of age, which will be referred to as "young" and "older". A person infected with SARS-CoV-2, following an incubation period, will become either sick (with COVID-19) or will be asymptomatic; the latter will recover, whereas a sick person will either recover or will be...

We have recently introduced two novel mathematical models for characterizing the dynamics of the cumulative number of individuals in a given country reported to be infected with COVID-19. Here we show that these models can also be used for determining the time-evolution of the associated number of deaths. In particular, using data up to around the...

We model the time-evolution of the number N(t) of individuals reported to be infected in a given country with a specific virus, in terms of a Riccati equation. Although this equation is nonlinear and it contains time-dependent coefficients, it can be solved in closed form, yielding an expression for N(t) that depends on a function α(t). For the par...

By employing a novel generalization of the inverse scattering transform method known as the unified transform or Fokas method, it can be shown that the solution of certain physically significant boundary value problems for the elliptic sine-Gordon equation, as well as for the elliptic version of the Ernst equation, can be expressed in terms of the...

Specific mental processes are associated with brain activation of a unique form, which are, in turn, expressed via the generation of specific neuronal electric currents. Electroencephalography (EEG) is based on measurements on the scalp of the electric potential generated by the neuronal current flowing in the cortex. This specific form of EEG data...

By employing a novel generalization of the inverse scattering transform method known as the unified transform or Fokas method, it can be shown that the solution of certain physically significant boundary value problems for the elliptic sine-Gordon equation, as well as for the elliptic version of the Ernst equation, can be expressed in terms of the...

The inverse magnetoencephalography and electroencephalography problems for spherical models have been extensively discussed in the literature. Using the spherical multiple-shell model, we derive novel vector-valued and singularity-free integral equations for both problems based on the quasi-static Maxwell equations. These equations are solved via a...

We obtain solution representation formulas for some initial boundary value problems posed on the half space that involve mixed derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method in the case of second order parabolic pdes; in this case one can alternatively eliminate the mixed deriv...

One of the most well-known generalizations of the celebrated Radon transform is the so-called attenuated Radon transform. The inversion of the attenuated Radon transform provides the mathematical foundation of the important field of medical imaging, known as single photon emission computed tomography (SPECT). In this chapter, we present a novel mat...

The unified transform, also known as the Fokas method, was introduced in 1997 by one of the authors Fokas (Proc R Soc Lond A: Math Phys Eng Sci 453(1962):1411–1443, 1997 ) for the analysis of nonlinear initial-boundary value problems. Later, it was realised that this method also yields novel results for linear problems. In 2006, the classical water...

We study initial boundary value problems for linear evolution partial differential equations posed on a time-dependent interval $l_1(t)<x<l_2(t)$, $0<t<T$, where $l_1(t)$ and $l_2(t)$ are given, real, differentiable functions, and $T$ is an arbitrary constant. For such problems, we show how to characterize the unknown boundary values in terms of th...

Lindelöf’s hypothesis, one of the most important open problems in the history of mathematics, states that for large $t$, Riemann’s zeta function $\zeta (1/2+it)$ is of order $O(t^{\varepsilon })$ for any $\varepsilon>0$. It is well known that for large $t$, the leading order asymptotics of the Riemann zeta function can be expressed in terms of a tr...

We study initial boundary value problems for linear evolution partial differential equations (PDEs) posed on a time-dependent interval $l_1(t)<x<l_2(t)$, $0<t<T$, where $l_1(t)$ and $l_2(t)$ are given, real, differentiable functions, and $T$ is an arbitrary constant. For such problems, we show how to characterise the unknown boundary values in term...

A hybrid approach for the solution of linear elliptic PDEs, based on the unified transform method in conjunction with domain decomposition techniques, is introduced. Given a well-posed boundary value problem, the proposed methodology relies on the derivation of an approximate global relation, which is an equation that couples the finite Fourier tra...

Based on the new approach to Lindelöf hypothesis recently introduced by one of the authors, we first derive a novel integral equation for the square of the absolute value of the Riemann zeta function. Then, we introduce the machinery needed to obtain an estimate for the solution of this equation. This approach suggests a substantial improvement of...

This paper employs the unified transform, also known as the Fokas method, to solve the advection-dispersion equation on the half-line. This method combines complex analysis with numerics. Compared to classical approaches used to solve linear partial differential equations (PDEs), the unified transform avoids the solution of ordinary differential eq...

In this paper, several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation of the important results of Katsurada and Matsumoto on the mean square of the Hurwitz zeta function. Also, a relation derived here provides the starting...

In this study, the neuronal current in the brain is represented using Helmholtz decomposition. It was shown in earlier work that data obtained via electroencephalography (EEG) are affected only by the irrotational component of the current. The irrotational component is denoted by $\Psi$ and has support in the cerebrum. This inverse problem is sever...

[This corrects the article DOI: 10.1098/rspa.2018.0605.].

The equations of motion, as well as the potential energy V of a self-gravitating N-body system in the first post-Minkowskian approximation have recently been derived. Here, for the particular case of two equal masses, the ultra-relativistic limit of these equations is analysed. It is shown that the requirement that the component of the gravitationa...

This paper implements the unified transform to problems in unbounded domains with solutions having corner singularities. Consequently a wide variety of mixed-boundary condition problems can be solved without the need for the Wiener-Hopf technique. Such problems arise frequently in acoustic scattering or in the calculation of electric fields in geom...

We propose an efficient numerical approach to evaluate the infinite oscillatory integral for the heat equation’s transform solution on the half-line that was recently introduced by Fokas. The proposed approach consists of first bounding the tail of the integral by making use of the partitioning-extrapolation method, then applying the trapezoidal ru...

Recent work has given rise to a novel and simple numerical technique for solving elliptic boundary value problems formulated in convex polygons in two dimensions. The method, based on the unified transform, involves expanding the unknown boundary values in a Legendre basis and determining the expansion coefficients by evaluating the so-called globa...

We consider the unified transform method, also known as the Fokas method, for solving partial differential equations. We adapt and modify the methodology, incorporating new ideas where necessary, in order to apply it to solve a large class of partial differential equations of fractional order. We demonstrate the applicability of the method by imple...

In earlier work, the neuronal primary current was expressed via the Helmholtz decomposition in terms of its irrotational part characterised by a scalar function and its solenoidal part characterised by a vectorial function. Furthermore, it was shown that EEG data is affected only by the irrotational part of the current, whereas MEG data is affected...

A method for solving boundary value problems usually referred to as the unified transform or the Fokas method was introduced in the late nineties. A crucial role in this method is played by the so-called global relation, which characterizes the associated generalized Dirichlet-to-Neumann map, which can be used to determine the unknown boundary valu...

We present the attenuated spline reconstruction technique (aSRT) which provides an innovative algorithm for single photon emission computed tomography (SPECT) image reconstruction. aSRT is based on an analytic formula of the inverse attenuated Radon transform. It involves the computation of the Hilbert transforms of the linear attenuation function...