The Unified Transform, Imaging and Asymptotics

A plethora of physical, chemical, biological and even social processes, can be modelled by mathematical equations. Many of these processes involve continuous change, and then the relevant equations take the form of differential equations. In models containing more than one variable, which is the great majority of situations, the relevant equations are called partial differential equations (PDEs). Given that these equations are instrumental in modelling the world around us, it is crucial that appropriate tools are developed for solving PDEs so that the associated models can be properly analysed. PDEs come into two broad categories: linear and non-linear. A general technique for solving linear PDEs was developed by the great French mathematician Fourier in the early 1800s. Non-linear PDEs are much more difficult to solve analytically. In 1997 the PI introduced a new method for solving a large class of non-linear PDEs. In an unexpected development, these results have motivated the development of a completely new method for solving linear PDEs in two dimensions. This is remarkable, since until then it was thought that the methods developed by Fourier and others in the 18th century could not be improved. This method is reviewed by three authors in the March 2014 issue of the Journal SIAM Review in the article titled "The Method of Fokas for solving linear PDEs", and in an accompanied editorial the importance of this method for solving linear PDEs is compared with the importance of the "Fosbury flop" in the high jump. The first part of this project involves completing the implementation of the above method to some important linear and non-linear problems in two dimensions, and then extending this method from 2 to 3 dimensions.

Several medical imaging techniques, including Computed Tomography, Positron Emission Tomography (PET) and Single Photon Emission Computed Tomography (SPECT) are based on the solution of a particular class of mathematical problems, called inverse problems. In the second part of this project, new numerical and analytical techniques will be implemented for PET and SPECT.

The Riemann function occurs in many different areas of mathematics. Several conjectures related to the Riemann function remain open, including the famous Riemann hypothesis and the Lindeloef hypothesis. The third part of the project involves the analysis of the asymptotics of the Riemann and related functions, which is expected to enhance our understanding of the relevant, most important mathematical structures.

This proposal consists of three parts. The first part is concerned with aspects of the so-called "unified transform method". This method was introduced by the PI in the late nineties for the analysis of boundary value problems (BVPs) of integrable nonlinear PDEs. In a highly unexpected development the unified method has led to the emergence of a novel and powerful approach for studying, both analytically and numerically, boundary value problems for linear PDEs. We propose to employ this method in order to investigate: (a) BVPs with x-periodic boundary conditions (the general case as opposed to the case of N-gap solutions) for integrable nonlinear evolution PDEs; (b) numerical techniques for linear elliptic PDEs in the interior and exterior of polygons; (c) the extension of the unified method to evolution PDEs in two spatial dimensions and to elliptic PDEs in three dimensions.

The second part is concerned with an analytical approach to certain inverse problems in medical imaging. In particular, we will implement the spline reconstruction technique (SRT) to the Single Photon Emission Computed Tomography (SPECT) with the aim of supplementing the commercially used iterative algorithm OSEM with the above analytical algorithm. In addition, we will implement the SRT to a variety of situations requiring non-uniform geometry, and we will perform 3D inversions in both PET and SPECT via 2D projections.

The third part is concerned with the investigation of the large t asymptotics of the Riemann zeta and of related functions. In particular, we will explore the connection we have established of the Riemann function with the Laplace equation, in order to obtain useful results for the large t asymptotics of the Riemann function by employing the so-called "global relation", which plays a crucial role in the unified transform method. In addition, we will utilize novel relations of the Riemann zeta function with the Lerch and the Hurwitz functions, in order to investigate the asymptotics of certain interesting sums and to make progress towards the Lindeloef hypothesis.