We study a third order dispersive linear evolution equation on the finite interval subject to an initial condition and inhomogeneous boundary conditions but, in place of one of the three boundary conditions that would typically be imposed, we use a nonlocal condition, which specifies a weighted integral of the solution over the spatial interval. Via adaptations of the Fokas transform method (or unified transform method), we obtain a solution representation for this problem. We also study the time periodic analogue of this problem, and thereby obtain long time asymptotics for the original problem with time periodic boundary and nonlocal data.

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficient are thus included. Both homogeneous and inhomogeneous problems are solved.

The classical half line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $bq(0,t)+q_x(0,t)=0$ is replaced with a dynamic Robin condition; $b=b(t)$ is allowed to vary in time. We present a solution representation, and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation, and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half line, with arbitrary linear dynamic boundary conditions.

We study the long time behaviour of the solution of the Stokes equation on a finite interval when the prescribed boundary and non-local conditions are time-periodic. We adapt the Q-equation approach pioneered by A. S. Fokas and M. C. Van der Weele in 2021 to solve the problem in a time-periodic regime and then reconstruct the solution to the problem with an explicit initial condition. We show that the transient effect of the initial condition may be ignored in the long time limit under certain assumptions. Finally, we formulate a useful conjecture encompassing the main results found in this project.

In this paper, I present analytic formulae for the zeros of some cases of exponential sums, and a complex root finder algorithm implemented in a Julia package. In general, there is no analytic formula for the zeros of exponential sums. However, their zeros correspond to the eigenvalues of differential operators, so finding the zeros with good accuracy is crucial to solving differential equations. In the analytical part, I derived formulae for asymptotic loci of zeros based on the theory of Langer. I showed that if an exponential sum satisfies certain properties, then we can obtain asymptotic formulae for all zeros with the modulus being sufficiently large. In the numerical part, I designed a tail-recursive subdivision algorithm that finds approximations of zeros of an analytic function within a given rectangular domain on the complex plane. I implemented the algorithm in an open-source package in Julia, and designed unit tests and randomised testing to evaluate its effectiveness.

In this paper, we will study flapping propulsion: e.g. a fish that propels itself through a fluid by waving its tail. An important question to ask is how to maximise the thrust generated by the flapping motion. One of the key parameters in the design of the tail, which we will model as a thin plate, is the distribution of rigidity across it. Our goal is to determine the optimal distribution to maximise thrust. We will present a unique approach to this problem, and our work complements earlier work done by Moore and Wu. In particular, we will solve for the kinematics of the tail/plate using the Unified Transform Method (UTM). Through this process, we will also facilitate the application of the UTM to physical problems.

Interface problems for dispersive wave equations define a class of problems involving higher spatial order initial boundary value problems. Such problems are either impossible or infeasibly difficult to solve using classical methods such as Fourier transform pairs. In this paper, we study the use of a new method for solving such problems, the unified transform method, which involves the synthesis rather than separation of variables. We study a full implementation of the UTM on the half-line heat problem and partial implementations on simple interface problems for the linear Schrödinger and linearised Korteweg-de Vries equations.

Evolution partial differential equations (PDEs) relate a quantity to its rates of change with respect to both time and position. Evolution PDEs can model a variety of phenomena, such as heat flow, wave propagation, and particle motion. This project concerns implementing an algorithmic procedure to solve a class of initial-boundary value problems (IBVPs) for linear evolution PDEs of arbitrary spatial order in the finite interval based on the Fokas transform method. The implementation is developed in Julia, featuring symbolic computation along with numeric evaluation.

A D to N map provides the unknown boundary values given the boundary data of an initial boundary value problem. A novel method, known as the Q equation method, was introduced in 2021 by Fokas and van der Weele for constant coefficient linear evolution equations on the half line. The method was generalized by Fokas, Pelloni and Smith in 2022 to study problems on the finite interval. In the present work, we significantly expand upon these works by demonstrating applicability of the Q equation method to evolution equations with variable coefficients. Indeed, we argue that the spatial Fourier transform traditionally employed to derive the Q equation is unnecessary, and may be replaced by an an alternative integral transformation, whose inverse need not be readily discernible. To illustrate the extension, we present the D to N maps for the heat equation on a disc and an annulus.

The linearised KdV equation is solved on a finite interval, subject to Dirichlet conditions at each end, and a nonlocal condition. The unified transform method is implemented in full for this problem.

The linearised KdV equation is solved on a finite interval, subject to a Dirichlet and Neumann condition both at the same end, and a nonlocal condition. The unified transform method is implemented in full for this problem.

Systems of differential equations can be used to model how a collection of quantities are related to one another and their derivatives. Systems of partial differential equations admit derivatives in different variables, such as time and space. Exponentiation of matrices is a crucial step in solving linear systems of differential equations. Indeed, a first order system of linear ordinary differential equations can be solved just as easily as a nonsystem first order linear ordinary differential equation, provided the matrix representing the system is diagonalizable. Unfortunately, not all matrices can be diagonalized. But they can all be expressed in Jordan form, and given a Schur decomposition. Those approaches each yield exponential formulae much simpler than a naive approach, but still not as simple as the formula for a diagonalisable matrix. However, there are approximation methods, in which nondiagonalizable matrices are replaced by diagonalizable approximations, with simple matrix exponentials and controls on the error. One method for solving systems of linear partial differental equations was developed by Deconinck's group as an extension of the Fokas transform method. But that approach assumes the system is diagonalizable. The purposes of this report are to (1) adapt the method of Deconinck's group to admit nondiagonalizable systems and (2) investigate the utility of diagonalizable approximations in this setting.

The nonlinear Schrödinger equation is one of the most salient expressions governing, in general, systems exhibiting nonlinear, dispersive or energy-preserving properties. In this paper, we formulate a linearised model of the NLS equation. Our model linearises the NLS equation about a step function approximation of the "bright" soliton in such a way as to make the PDE constant coefficient whilst modelling the nonlinear properties of the NLS equation. Finally, we apply the Unified Transform Method to the interface linear model of the NLS equation to present fully explicit solutions for the problem. Deconinck and Sheils's implementation of the Unified Transform Method is instrumental in solving the interface IVP.

We use Transforms to make differential equations simpler to solve, and in the Transform Method that I describe in detail in this document, a linear first order ODE that is easily solved. The Transform method can be used on 2nd order linear partial differential equations. The examples I will use involve a PDE that is 2nd order with respect to x and 1st order with respect to time t. This method works for both constant- and variable-coefficients with roughly equal ease.

We construct the adjoints of general multipoint differential operators, by generalizing the argument of Coddington & Levinson (1955).

The aim of this document was to explain Langer's paper on the zeros of exponential sums. We followed the general structure of the original paper and explained the main theorems for the cases that are relevant to our project. These cases were illustrated with corresponding plots for arbitrary examples.

This report aims to derive the integral representation of the solution to the heat equation with dynamic Robin condition via the Fokas method. We begin with a construction of the solution representation via the Fokas method and show that the problem reduces to a fractional linear ordinary differential equation (FLODE). We then give a brief overview of Caputo fractional derivative and solve the FLODE via the Frobenius Method. Finally, we provide an error estimation via the generalised Taylor's Remainder Theorem.

Numerical integration, otherwise known as quadrature, denotes a set of algorithms in which a definite integral is approximated, rather than determined exactly. These algorithms are used when direct (exact) integration of the function is difficult or impossible. We experimented using the Newton-Cotes rules, Lagrange polynomial interpolation, Chebyshev polynomial interpolation, and Taylor polynomial approximation, to perform numerical integration. These algorithms approximate the integrand with an approximation polynomial. The approximation polynomial is integrated (which is trivially easy) to attain an approximation of the actual definite integral. While the Newton-Cotes rules appear to bypass the formation of an approximation polynomial, they are in fact equivalent to Lagrange polynomial interpolation.

Given an operator, its adjoint is a linear transformation that can be used to characterise a wide range of phenomena in mathematics and physics. In this project we seek to characterise an adjoint of an ordinary differential operator with multipoint homogeneous conditions. Extending the ideas of Linfan Xiao, we show that the operator and its adjoint satisfy the multipoint form formula. In doing so, we provide an explicit construction of adjoint multipoint conditions, and provide an explicit checking test to determine whether a given set of multipoint conditions satisfies the adjoint property. Finally, we implement a method of deriving the adjoint multipoint conditions in a programming language Julia, thus providing a fast way to compute the adjoint.

The heat equation is a partial differential equation that describes how heat is distributed in a region over time. In this project, we seek to solve the heat equation on the half line where the boundary condition at one end evolves with time using the Fokas method. We show that the problem reduces to a fractional linear ordinary differential equation (FLODE) with a variable coefficient. Drawing from ideas in fractional calculus, we obtain a solution to the FLODE through the Frobenius method, thus solving the heat equation.

Numerical integration, otherwise known as quadrature, denotes a set of algorithms in which a definite integral is approximated, rather than determined exactly. These algorithms are used when direct (exact) integration of the function is difficult or impossible. We experimented using the Newton-Cotes rules, Lagrange polynomial interpolation, Chebyshev polynomial interpolation, and Taylor polynomial approximation, to perform numerical integration. These algorithms approximate the integrand with an approximation polynomial. The approximation polynomial is integrated (which is trivially easy) to attain an approximation of the actual definite integral. While the Newton-Cotes rules appear to bypass the formation of an approximation polynomial, they are in fact equivalent to Lagrange polynomial interpolation.

We construct the adjoints of general multipoint differential operators, by generalizing the argument of Coddington & Levinson (1955).

Evolution partial differential equations (PDEs) relate a quantity to its rates of change with respect to both time and position. Evolution PDEs can model a variety of phenomena, such as heat flow, wave propagation, and particle motion. This project concerns implementing an algorithmic procedure to solve a class of initial-boundary value problems (IBVPs) for linear evolution PDEs of arbitrary spatial order in the finite interval based on the Fokas transform method. The implementation is developed in Julia, featuring symbolic computation along with numeric evaluation.

A julia library providing for the efficient description and algebra of exponential sums. Exponential sums are exponential polynomials in which each polynomial coefficient is a complex constant. We also survey Langer's (1931) paper on the asymptotic locus of zeros of such functions.