# Completed projects

## IBVP with complicated boundary conditions

Initial boundary value problems (IBVP) are typically specified as a partial differential equation (PDE) describing the evolution of a physical system, an initial condition (IC) describing the initial state of the system and some collection of boundary conditions (BC) constraining the behaviour of the system at the spatial boundaries. Such IBVP are extremely successful at describing the real world, representing heat flow through a solid, water waves in a canal, deformation of a flexible beam, and countless other examples. But there are situations where classical two point or one point boundary conditions are not sufficient to model the physics.

These projects describe physical and mathematical extensions of IBVP and extensions of solution methods using the unified transform method.

### Linear evolution equations on the half line with dynamic boundary conditions

Completed 2018 summer research project and semester research in 2019.

Student: Toh Wei Yang. Supervisor: Dave Smith.

If one end of a long metal rod is placed in a water bath, then the distribution of heat in the solid may be represented as the solution of an initial boundary value problem involving a Robin boundary condition at the end. The Robin coupling coefficient betwee Dirichlet and Neumann boundary values is determined by the physical properties of the contact between metal and water. If the water were replaced by a corrosive substance, then the coupling coefficient must evolve in time, producing a dynamic Robin boundary condition. This project aims to solve such problems, and generalizations to other PDE such as the linear free space Schrödinger and linearized Korteweg-de Vries equations.

Outputs: [Toh2018a], [Toh2019a], [ST2019a].

### Wave equation with nonlocal conditions

Completed 2020 summer research project.

Student: Dion Ho. Supervisor: Dave Smith.

Boundary conditions specify the value of the solution at precisely the edges of the spatial domain. Nonlocal conditions specify instead a weighted integral (an average) of the solution over some interval subset of the spatial domain. Such problems were solved for the heat equation by Miller and Smith (2018) via an extension of the unified transform method. In this project, we seek solutions of the wave equation subject to nonlocal conditions.

Outputs: [Ho2020a],

### Optimal distribution of flexural rigidity across a flapping plate to maximise thrust

Completed 2020–2021 capstone project.

Student: Dion Ho. Supervisor: Dave Smith.

In this project, 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 investigate 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.

Outputs: [Ho2021a],

### Dispersive wave equations on networks

Completed 2020–2021 capstone project.

Student: Aditya Karkera. Supervisor: Dave Smith.

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 project, 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.

Outputs: [Kar2021a],

### Transform method for variable coefficient PDE

Student: Kshitij Bansal (Stitch). Supervisor: Dave Smith.

The usual approach for solving the heat equation on the full line is to apply the exponential Fourier transform. On the half line, the boundary conditions interact unfavourably with the exponential Fourier transform. Therefore, one selects the Fourier sine transform for the half line Dirichlet heat problem, or the Fourier cosine transform for the half line Neumann heat problem. For problems on the finite interval, one typically begins with separation of variables to derive and solve a Sturm-Liouville problem. The eigenfunctions that selfadjoint eigenvalue problem are then used as a basis for the solution space and the coefficients of the expansion are easily obtained as inner products of the initial datum with the basis functions.

Although they appear to be different approaches, the latter method may be reexpressed using the language of the former: the forward transform is the construction of the coefficients and the inverse transform is the evaluation of the basis expansion series. One can reduce the problem "find the right transform pair" to the familiar Sturm-Liouville problem formulation. At least, this can be done for the ordinary Fourier sine, cosine and periodic bases. The purpose of this project is to extend this approach to variable coefficient partial differential operators such as Bessel's equation.

Outputs: [Ban2021a],

### Approximate diagonalizations in Fokas transform method for systems

Completed 2021 summer research project.

Student: Naman Kedia. Supervisor: Dave Smith.

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 differential 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 project 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.

### Initial nonlocal value problems for evolution equations of high spatial order

Completed 2021 summer research project.

Student: Bekzod Sanjarovich. Supervisor: Dave Smith.

Initial boundary value problems for evolution equations are the problems of finding solutions to evolution partial differential equations satisfying an initial condition and various boundary conditions. Such evolution equations often describe physical systems, with the boundary and initial conditions representing measurements taken at the edges of the spacetime domain. In some circumstances, it is impractical to take measurements at exact points, but easy to measure an average, or weighted average, over an interval. For example, a pressure sensor can only measure the average pressure over its width. Similarly, an opacity sensor measures the average opacity of a substance over the width of its photosensitve cell. In such a circumstance, the exact boundary condition is replaced with a nonlocal condition, representing a weighted average of the pressure, opacity, etc. The resulting problem is known as an initial nonlocal value problem.

While solution of initial boundary value problems for simple evolution equations is well studied, there has been relatively little work on initial nonlocal value problems. The unified transform method was developed around the turn of the millenium, and is uniquely positioned among spectral methods to solve initial boundary value problems of high spatial order. The method was recently adapted (Miller & Smith 2018) to treat an initial nonlocal value problem for the heat equation.

The purpose of this project is to further develop the nonlocal formulation of the unified transform method so that it may be applied to problems for high order evolution equations such as the linearized KdV equation. A possible application is study of wellposedness of the linear part of the celebrated Korteweg-de Vries equation subject to nonlocal conditions.

## Algorithmetizing the unified transform method

The unified transform method can be used to solve two point initial boundary value problems for linear evolution equations of arbitrary spatial order. It was originally understood as a 3 stage method, in which a candidate solution is constructed, then shown to satisfy the original problem. The solution construction involves the solution of certain linear systems, analysis of zeros of exponential polynomials and complex asymptotic analysis of certain meromorphic functions; each step is specialized to a particular set of boundary conditions. Despite general results on its applicability, the method remained ad hoc until [FS2016a], where the method was reformulated as the explicit and general construction of a transform pair, directly from the ordinary differential operator which describes the spatial part of the initial boundary value problem.

These projects extend this work, by implementing such constructions in computer code, and extending the mathematical construction to other classes of boundary conditions.

### Algorithmic solution of high order PDE in julia via the Fokas transform method

Completed 2018–2019 capstone project. Winner of the Yale-NUS College MCS Capstone prize 2019.

Student: Xiao Linfan. Supervisor: Dave Smith.

The algorithmic construction of a transform pair for a two point initial boundary value problem described in [FS2016a] is partially implemented as a computer code. The implementation is developed in Julia, featuring symbolic computation along with numeric evaluation.

Outputs: [Xia2019a], [Xia2019b].

### UTM as diagonalization of multipoint differential operators

Completed 2020 summer project.

Student: Sultan Aitzhan. Supervisor: Dave Smith.

The algorithmic version of the unified transform method for initial boundary value probelms was first presented in [FS2016a]. Here, using results from project Adjoints of general multipoint differential operators, we aim to achieve the same algorithmetization results for initial multipoint value problems.

## Interface linearizations to model nonlinear effects

Nonlinear partial differential equations are typically extremely difficult to solve analytically. They usually require complicated ad hoc arguments, such as inverse scattering, or are impossible to solve using known methods. Therefore, nonlinear partial differential equations are often "linearized", meaning that their nonlinear terms are replaced with linear approximations, typically using a known solution such as the zero solution. If a nonconstant solution, such as the famous soliton or dispersive shock solutions, is selected to linearize around, then the linearized system is necessarily a variable coefficient partial differential equation. Variable coefficient linear partial differential equations are usually easier to solve than nonlinear equations, but still may be very difficult. In contrast, the unified transform method is an efficient tool available for constant coefficient linear partial differential equations.

In a recent work of Smith, Trogdon & Vasan, the Korteweg-de Vries equation was linearized about a piecewise constant approximation of its dispersive shock wave solution. This reduced the nonlinear partial differential equation to an interface problem, with interface translating at constant velocity, for a linear third order dispersive equation. They refer to the reduction as an "interface linearization". The unified transform method, as extended by Sheils to interface problems, was then applied to solve the approximate problem, and similarities to the full dispersive shockwave problem were noted.

The aim of these projects is to implement an interface linearization of classical nonlinear equations, then solve the resulting system using the unified transform method. The purpose is to cheaply model the original more complicated systems while preserving the qualitative features of the nonlinear effects.

### Interface linearization of NLS solitons

Completed 2021 summer research project.

Student: Ian Masila. Supervisor: Dave Smith.

There are certain features of some nonlinear equations that can never appear in linear equations. Examples include dispersive shock waves and solitons. We aim to find linear equations that model the nonlinear effects. The unified transform method can be used to solve initial-interface value problems, for constant coefficient linear evolution equations, even where the interface separates between intervals where the dispersion relation is different. The first works on this are mostly by Sheils and collaborators, in which the dispersion relation is always monomial, and the change across a boundary is simply a (possibly complex) scaling of that monomial. The more recent work of Sheils & Deconinck assumes that the dispersion relation is polynomial, with only lower order terms changing across the interval. It is possible, as Sheils & Deconinck describe, to reformulate a variable coefficient problem as such an interface problem, provided the variable coefficients are step functions.

We take the approach of Sheils & Deconinck, but extend in two ways. Firstly, we model a nonlinear equation as a variable coefficient problem, in which the variable coefficient is determined by the (assumed steplike) initial datum. We go further, by allowing the interfaces to move in space at a constant speed, carrying with them the discontinuities in the variable coefficients. This allows us to attempt to model nonlinear phenomena which have features translating at fixed speeds. We assume that all interfaces are translating at the same speed and in the same direction.

The focusing nonlinear Schrödinger equation iut = uxx+2|u|2u = 0 has bright solitons ηsech[η(x+2ξ t-x0)]e-iΘ, where Θ = ξ x + (ξ22)t + Θ0. We aim to approximate such a soliton by a step function (in the first instance, we will try a box) translating at constant speed 2ξ and substitute that for the |u|2 in the equation, to obtain a linear equation on each subinterval that can be solved using Deconinck & Sheils's implementation of the unified transform method.

Outputs: [Mas2021a],

### UTM stage 1 for interface linearization of KdV solitons

Completed 2019 summer research project.

Student: James Pierog. Supervisor: Dave Smith.

We model solutions of the Korteweg-de Vries equation with a constant coefficient linear equation by linearizing about a translating step function approximation of the soliton.

## Spectral theory of ordinary differential operators

The spectral theory of ordinary differential operators is crucial to the analysis of partial differential equations representing evolution with spatial constraints. The classical theory of two point differential operators with constant coefficients has been quite well understood for more than a century, although advances such as [FS2016a] have been made more recently. In particular, the extent of failure of selfadjointness caused by particular classes of boundary conditions, and the effects of such failure on the spectral theory is still not concluded.

These projects extend classical results, such as the construction of adjoint operators and spectral analysis, from two point differential operators to differential operators subject to more complicated classes of boundary conditions.

### Adjoints of general multipoint differential operators

Completed 2019 summer research project.

Student: Sultan Aitzhan. Supervisor: Dave Smith.

Coddington and Levinson's 1955 textbook shows how to construct the adjoint of a formal ordinary differential operator and the adjoint of atwo point ordinary differential operator. In [Xia2019b] this algorithm was implemented as a collection of julia scripts. This project extends both the theoretical and computational results to the case of multipoint ordinary differential operators.

Outputs: [AS2019a], [AS2019b], [Ait2019a].

### Adjoints of interface differential operators

Completed 2019 summer research project.

Student: Peeranat Tokaeo. Supervisor: Dave Smith.

The adjoints are constructed for some classes of interface ordinary differential operators.

### Adjoints of general interface differential operators

Completed 2020 summer research project.

Student: Sambhav Bhandari. Supervisor: Dave Smith.

Coddington and Levinson's 1955 textbook shows how to construct the adjoint of a formal ordinary differential operator and the adjoint of a two point ordinary differential operator. In [Xia2019b] this algorithm was implemented as a collection of julia scripts. This project extends both the theoretical and computational results to the case of interface ordinary differential operators.

## Orthogonal polynomial bases

Orthogonal polynomials form convenient basis systems for representation of nonelementary functions. The choice of class of orthogonal polynomials depends upon the kind of function one wishes to represent, with Chebyshev polynomials a natural choice for functions defined on a finite interval. The ApproxFun package for julia offers a suite of tools for working efficiently with orthogonal polynomial expansions.

These projects explore applications of orthogonal polynomial expansions, or study associated algorithms.

### Efficient representation of functions and numerical integration: julia & ApproxFun

Completed 2018 summer project.

Student: Dion Ho. Supervisor: Dave Smith.

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.

Outputs: [Ho2018a], [Ho2018b].

### Generalized Fourier series methods using ApproxFun

Completed 2019–2020 semester project.

Student: Sultan Aitzhan. Supervisor: Dave Smith.

The classical separation of variables and Sturm-Liouville theory shows how eigenfunction expansions can be used to solve initial boundary value problems for second order evolution partial differential equations, and express the solutions as generalized Fourier series. Similar solution and expansions are possible for some equations of higher spatial order. However, finding the eigenfunctions of variable coefficient differential operators is a very difficult problem. In this project, we study differential operators restricted to domains of Chebyshev functions so that their (approximate) eigenfunctions may be readily determined using ApproxFun. The adjoints operator is constructed using methods developed in Algorithmic solution of high order PDE in julia via the Fokas transform method and refined in Adjoints of general multipoint differential operators, so that even (certain classes of) nonselfadjoint equations initial boundary value problems can be solved using this code.

## Long time and semiclassical asymptotics

Asymptotics is the study of how a mathematical object behaves in some kind of limit. This field studies questions such as "using the laws of thermodynamics, how does a physical system relax given enough time?" and "how can a small high order derivative prevent wave breaking?" Asymptotic analysis synthesizes the tools of real and complex analysis to quantify the dominant effects of parameters.

### Long time asymptotics for periodic boundary data

Completed 2021 summer research project.

Student: Neha Gupta. Supervisor: Dave Smith.

Initial boundary value problems for partial differential equations are often used to model physical systems. For some systems, such as the distribution of heat in an insulated metal body, it makes sense to think about a "steady state solution", towards which the solution of the full problem relaxes given sufficient time. By definition, a steady state solution must have temporal derivative 0. But for some systems it is not reasonable to expect that such a state is ever achieved. For example, if the metal body is left outside (with perfect weather) then the diurnal fluctuations in temperature mean that no steady state can be arrived at.

When there is no simple, one must study instead "long time asymptotics". The solution is broken down into components which behave differently for different time scales. For example, the diurnal effects of the sun heating the metal body are a periodic component of the solution. The initial distribution of heat in the metal (before it was brought outside) will also affect the solution, but as time increases that effect reduces exponentially.

Different representations of the same solution of an initial boundary value problem can be more or less amenable to long time asymptotics. In this project, we will investigate long time asymptotics for various initial boundary value problems for the heat equation using the solution representation obtained via the unified transform method.

### Comparison of derivations of the Q equation

Completed 2021 summer research project.

Student: Niall Shah. Supervisor: Dave Smith.

If an initial boundary value problem has all boundary data periodic with the same period, then it may be reasonable to assume that the solution is also periodic with that period, or at least that the "nonperiodic bit" decays in time. It was recently shown by Fokas and van der Weele that, under such an assumption, the DtoN map may be obtained via a very simple argument. The argument relies on the beginning of the argument for Stage 1 of the unified transform method to derive "the Q equation", which is really just an equation for the Fourier transform of the solution of the initial boundary value problem. Fokas and van der Weele study the advection-diffusion equation (heat with drift) on the half line. A more recent derivation of the Q equation by Fokas, Pelloni & Smith follows a different approach.

In this project, we compare and contrast the derivations of the Q equation, and give a detailed explanation the full derivation for the heat equation.

## Computational analysis of exponential polynomials

Exponential polynomials are a special class of holomorphic (complex analytic) functions which can be expressed as a finite linear combination of pure exponential functions, in which each coefficient is a polynomial function. Examples include the trigonometric and hyperbolic sine and cosine functions, but there are others that are rather more complicated. These functions, especially the locus of their zeros and their efficient evaluation, are crucial for computational and numerical implementation of the unified transform method. As soon as a trigonometric polynomial has more than two terms, it is rare to be able to find its zeros exactly in closed form. Instead one can use tools of asymptotic analysis and geometry to construct approximate zeros, and complex analysis to find zeros to arbitrary precision.

These projects are related to the computational and numerical treatment of zero locus problems for exponential polynomials, and efficient evaluation of related oscillatory integrals.

### Julia library for exponential sums and survey of Langer's work

Completed 2019 summer research project.

Student: Zhang Liu. Supervisor: Dave Smith.

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.

Outputs: [Zha2019a], [Zha2019b].

### Argument principle for root counting

Completed 2019 summer research project.

Student: Ahmed Fedi Lassoued. Supervisor: Dave Smith.

The argument principle can be used to find the number of zeros of an analytic function, counted according to multiplicity, that are contained within a simple closed contour. This application of the argument principle is implemented as a numerical method to count the zeros of an analytic function.

### Roots of holomorphic functions and the argument principle

Completed 2020 summer research project.

Student: Juwon Lee. Supervisor: Dave Smith.

The argument principle can be used to find the number of zeros of an analytic function, counted according to multiplicity, that are contained within a simple closed contour. In this way, we implement an algorithm to count the zeros of analytic functions.

### Julia library for exponential polynomials

Completed 2020 summer research project.

Student: Wang Yanhua. Supervisor: Dave Smith.

A julia library providing for the efficient description and algebra of exponential polynomials.

### Relative growth of terms in exponential polynomials

Completed 2020 summer research project.

Student: Zhang Liu. Supervisor: Dave Smith.

In different regions of the complex plane, different terms of an exponential polynomial dominate others. We use analytic and geometric arguments to compare the relative dominance of terms.

## Other projects

Some projects don't exactly fit into the main topics studied in the unified transform lab. These projects can explore topics as diverse as complex analysis, numerical analysis, and numerical computation.

### Complex analytic arguments in the unified transform method

Completed 2020 summer research project.

Student: Upaasna Parankusam. Supervisor: Dave Smith.

We study topics from complex analysis required for justification of the unified transform method.