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Athanassios S. Fokas

Publications and source records attributed to Athanassios S. Fokas.

16 recordsLinked to original sources
arXiv2025

A complex spatial frequency approach to optimal control of finite-extent linear evolution systems

We consider the linear quadratic regulator (LQR) for one-dimensional linear evolution partial differential equations (PDEs) on a finite interval in space. The control is applied as an additive forcing term to PDEs. Existing methods for closed-form optimal control only apply to homogeneous (zero) boundary conditions, often resulting in series representations. In this paper, we consider general smooth boundary conditions. We use the unified transform, namely the Fourier transform restricted to the bounded spatial domain, to decouple PDEs into a family of ordinary differential equations (ODEs) parameterized by complex spatial frequency variables. Then, optimal control in the frequency domain is derived using LQR theory for ODEs. The inverse Fourier transform leads to non-causal terms in optimal control corresponding to integrals, over the real line, of future values of unspecified boundary conditions. To eliminate this non-causality, we deform the integrals to well-constructed contours in the complex plane along which the contribution of unknowns vanishes. For the reaction-diffusion equation, we show that the integral representation can be reformulated as a series representation, which leads to a state-feedback convolution form for optimal control, with the boundary conditions appearing as an additive term. In numerical experiments, we illustrate the computational advantages of the integral representation in comparison to the series representation and structural properties of the convolution kernel.

eess.SY↗
arXiv2020

Fokas method for linear boundary value problems involving mixed spatial derivatives

We obtain solution representation formulas 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 mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well-defined provided that certain analyticity issues are appropriately addressed.

math.AP↗
arXiv2019

A novel approach to the Lindelöf hypothesis

Lindel{ö}f's hypothesis, one of the most important open problems in the history of mathematics, states that for large $t$, Riemann's zeta function $ζ(1/2+it)$ 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 transcendental exponential sum. The usual approach to the Lindelöf hypothesis involves the use of ingenious techniques for the estimation of this sum. However, since such estimates can not yield an asymptotic formula for the above sum, it appears that this strategy cannot lead to the proof of Lindelöf's hypothesis. Here, a completely different approach is introduced. In particular, a novel linear integral equation is derived for $|ζ(σ+it)|^2, \ 0<σ<1$ whose asymptotic analysis yields asymptotic results for a certain Riemann zeta-type double exponential sum. This sum has the same structure as the sum describing the leading asymptotics of $|ζ(σ+it)|^2$, namely it involves $m_1^{-σ-it}m_2^{-σ-it}$, but its summation limits are different than those of the sum corresponding to $|ζ(σ+it)|^2$. The analysis of the above integral equation requires the asymptotic estimation of four different integrals denoted by $I_1,I_2,\tilde{I}_3,\tilde{I}_4$, as well as the derivation of an exact relation between certain double exponential sums. Here, the latter relation is derived, and also the rigorous analysis of the first two integrals $I_1$ and $I_2$ is presented. For the remaining two integrals, formal results are only derived that suggest a possible roadmap for the derivation of rigorous asymptotic results of the above double exponential sum, as well as for other sums associated with $|ζ(σ+it)|^2$. Additional developments suggested by the above novel approach are also discussed.

math.CA↗
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↗
arXiv2018

Solving PDEs of fractional order using the unified transform method

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 implementing it to solve a model fractional problem.

math.AP↗
arXiv2018

Equations of motion of self-gravitating $N$-body systems in the first post-Minkowskian approximation

We revisit the problem of the equations of motion of a system of $N$ self-interacting massive particles (without spins) in the first post-Minkowskian (1PM) approximation of general relativity. We write the equations of motion, gravitational field and associated conserved integrals of the motion in a form suitable for comparison with recently published post-Newtonian (PN) results at the 4PN order. We show that the Lagrangian associated with the equations of motion in harmonic coordinates is a generalized one, and compute all the terms linear in $G$ up to 5PN order. We discuss the Hamiltonian in the frame of the center of mass and exhibit a canonical transformation connecting it to previous results directly obtained with the Hamiltonian formalism of general relativity. Finally we recover the known result for the gravitational scattering angle of two particles at the 1PM order.

gr-qc↗
arXiv2018

Uniform asymptotics as a stationary point approaches an endpoint

We obtain the rigorous uniform asymptotics of a particular integral where a stationary point is close to an endpoint. There exists a general method introduced by Bleistein for obtaining uniform asymptotics in this situation. However, this method does not provide rigorous estimates for the error. Indeed, the method of Bleistein starts with a change of variables, which implies that the parameter governing how close the stationary point is to the endpoint appears in several parts of the integrand, and this means that one cannot obtain general error bounds. By adapting the above method to our particular integral, we obtain rigorous uniform leading-order asymptotics. We also give a rigorous derivation of the asymptotics to all orders of the same integral; the novelty of this second approach is that it does not involve a global change of variables.

math.CA↗
arXiv2017

Asymptotics to all orders of the Hurwitz zeta function

We present several formulae for the large-$t$ asymptotics of the modified Hurwitz zeta function $ζ_1(x,s),x>0,s=σ+it,0<σ\leq1,t>0,$ which are valid to all orders. In the case of $x=0$, these formulae reduce to the asymptotic expressions recently obtained for the Riemann zeta function, which include the classical results of Siegel as a particular case.

math.NT↗
arXiv2017

On the proof of a variant of Lindelöf's hypothesis

The leading asymptotic behaviour as $t\to \infty$ of the celebrated Riemann zeta function $ζ(s), \ s = σ+ it, \quad 0<σ<1, \quad t>0 , \ t\to\infty,$ can be expressed in terms of a transcendental sum. The sharp estimation of this sum remains one of the most important open problems in mathematics with a long and illustrious history. Lindelöf's hypothesis states that for $σ=1/2$, this sum is of order $O\left(t^\varepsilon\right)$ for every $\varepsilon>0$. We have recently introduced a novel approach for estimating such transcendental sums: we have first embedded the Riemann zeta function in a certain Riemann-Hilbert problem and we have began the analysis of the large $t$-asymptotics of the associated integral equation. The asymptotic analysis of the resulting integral equation requires the further splitting of the relevant interval of integration into four subintervals which are defined in terms of the small positive numbers $\{δ_j\}_1^4$. The rigorous asymptotic analysis of the first two relevant integrals, $I_1$ and $I_2$, was performed in [F]. Here, the rigorous analysis is performed of the last two integrals, $I_3$ and $I_4$. The combination of the above results yields a proof for the analogue of Lindel{ö}f's hypothesis for a slight variant of the transcendental sum characterising the large $t$-asymptotics of $|ζ(s)|^2$, namely for a sum which differs from the latter sum only in the occurrence of a logarithmic term which is larger than $\frac{1}{2}\ln t$ and smaller than $t^{\varepsilon}$. Interestingly, the parameter $\varepsilon$ in Lindelöf's hypothesis is explicitly defined in terms of $δ_3$.

math.CA↗
arXiv2016

Analytical computation of the Mercury perihelion precession via the relativistic gravitational law and comparison with general relativity

Let $r(φ)$ denote the orbit of Mercury. We compare the formulae obtained via general relativity for $r(φ)$ and for the corresponding perihelion precession angle $Δφ$, with the formulae obtained via the relativistic gravitational law, $F=GMmγ^6/r^2$. The latter law can be derived from Newton's gravitational law by employing the gravitational rather than the rest masses of the Sun and Mercury. Remarkably, it is found that the two expressions for the dominant part of $r(φ)$ are identical and that the two expressions for $Δφ$ are also identical.

physics.gen-ph↗
arXiv2016

Generalized Dirichlet to Neumann Maps for Linear Dispersive Equations on Half-Line

A large class of initial-boundary value problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit formulae are presented for the generalized Dirichlet to Neumann maps. Namely, the determination of the unknown boundary values in terms of an essential set of given data.

math.AP↗
arXiv2016

Evolution PDEs and augmented eigenfunctions. Finite interval

The so-called unified method expresses the solution of an initial-boundary value problem (IBVP) for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple IBVP, which will be referred to as problems of type~I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated IBVP, which will be referred to as problems of type~II, there does \emph{not} exist a classical transform pair and the solution \emph{cannot} be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type~II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalised eigenfunctions described in the sixties by Gel'fand and his co-authors.

math.SP↗
arXiv2011

The Kadomtsev-Petviashvili I Equation on the Half-Plane

A new method for the solution of initial-boundary value problems for \textit{linear} and \textit{integrable nonlinear} evolution PDEs in one spatial dimension was introduced by one of the authors in 1997 \cite{F1997}. This approach was subsequently extended to initial-boundary value problems for evolution PDEs in two spatial dimensions, first in the case of linear PDEs \cite{F2002b} and, more recently, in the case of integrable nonlinear PDEs, for the Davey-Stewartson and the Kadomtsev-Petviashvili II equations on the half-plane (see \cite{FDS2009} and \cite{MF2011} respectively). In this work, we study the analogous problem for the Kadomtsev-Petviashvili I equation; in particular, through the simultaneous spectral analysis of the associated Lax pair via a d-bar formalism, we are able to obtain an integral representation for the solution, which involves certain transforms of all the initial and the boundary values, as well as an identity, the so-called global relation, which relates these transforms in appropriate regions of the complex spectral plane.

math.AP↗
arXiv2003

A Generalization of both the Method of Images and of the Classical Integral Transforms

A new method for the solution of initial-boundary value problems for evolution PDEs recently introduced by Fokas is generalised to multidimensions. Also the relation of this method with the method of images and with the classical integral transforms is discussed. The new method is easy to implement, yet it is applicable to problems for which the classical approaches apparently fail. As illustrative examples, initial-boundary value problems for the diffusion-convection equation in one and higher dimensions, as well as for the linearised Korteweg-de Vries equation with the space variables on the half-line are solved. The suitability of the new method for the analysis of the the long-time asymptotics is ellucidated.

cond-mat↗
arXiv2001

The Solution of the Modified Helmholtz Equation in a Triangular Domain and an Application to Diffusion-Limited Coalescence

A new transform method for solving boundary value problems for linear and integrable nonlinear PDEs recently introduced in the literature is used here to obtain the solution of the modified Helmholtz equation q_{xx}(x,y)+q_{yy}(x,y)-4b^2q(x,y)=0 in the triangular domain 0 < x < L-y < L, with mixed boundary conditions. This solution is applied to the problem of diffusion-limited coalescence, A + A <--> A, in the segment (-L/2,L/2), with traps at the edges.

cond-mat.stat-mech↗