# 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].

### Dispersive wave equations on networks

Open project.

Supervisor: Dave Smith.

Dispersive wave equations on complex domains can be used to model information flow through social networks, pressure waves in vascular systems, and flow of light packets through branching fibre topographies. Typically, due to their complexity, these problems are studied numerically. Classical analytic Fourier methods would not be applicable to such problems in general, particularly for dispersive wave equations of high spatial order. The unified transform method is a modern Fourier method applicable to equations of arbitrary spatial order, but the only work on the unified transform method on network domains is restricted to the heat equation.

The purpose of this project is to extend the unified transform method to analysis of dispersive wave equations, such as the Schrödinger and linearized Korteweg-de Vries equations, on network domains. Particular applications may also be explored.

Required modules: Linear algebra, Proof, Real Analysis.

Useful modules: Foundations of applied mathematics, Ordinary & partial differential equations, Complex analysis.

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

Open project.

Supervisor: Dave Smith.

Initial boundary value problems for linear evolution equations are used to model physical systems. Some the equations that make up these problems, the boundary conditions, describe information about the system at its spatial boundaries, such as temperature at the surface or water elevation at the edge of a tank.

But certain problems require instead nonlocal conditions, specifying an average temperature over a region, or average water elevation. Such problems are known as initial nonlocal value problems. An analytic solution to such a problem for the heat equation was recently attained by Miller and Smith. They used the unified transform method, which has particular power in its applicability to problems of high spatial order, but there was no attempt to extend their results beyond the second order heat equation.

The purpose of this project is to extend the unified transform method to analysis of initial nonlocal value problems for evolution equations of high spatial order. A possible application is study of wellposedness of the linear part of the celebrated Korteweg-de Vries equation subject to nonlocal conditions.

Required modules: Linear algebra, Proof, Real analysis.

Useful modules: Foundations of applied mathematics, Ordinary & partial differential equations, Complex analysis.

### Initial interface value problems for the wave and beam equations

Open project.

Supervisor: Dave Smith.

Linear evolution partial differential equations can be used to model physical systems. In particular, the wave equation describes waves on a thin string and the beam equation models the displacement of a rigid beam. The unified transform method is a recently developed advanced analytic method for partial differential equations. Over the past 20 years, it has been successfully applied to a variety of linear evolution equations with arbitrary spatial derivatives, but typically with first order time derivative. A recent advance by Deconinck's group saw the generalization of the unified transform method to the wave equation, the prototypical linear evolution equation featuring a second order time derivative. The beam equation is a more complicated equation, also featuring a second order time derivative.

The primary aim of this project is to apply the unified transform method to analyze the beam equation. An application to wing flapping requires the analysis of interface problems for the wave and beam equations.

Required modules: Linear algebra, Proof, Real analysis.

Useful modules: Foundations of applied mathematics, Ordinary & partial differential equations, Complex analysis.

### Wave equation with nonlocal conditions

In progress 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.

## 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

In progress 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.

### 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.

### Interface linearization of KdV solitons

Open project.

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. The interface linearization and stage 1 of the unified transform method was already performed in project UTM stage 1 for interface linearization of KdV solitons, however that work only covers the crudest soliton approximation: the box.

The main aim of this project is to complete the solution of the reduced interface system. This project can be extended by using a finer soliton approximation for the interface linearization and solving the resulting more complicated linear problem.

Required modules: Linear algebra, Proof, Real analysis.

Useful modules: Foundations of applied mathematics, Ordinary & partial differential equations, Complex analysis.

### Interface linearization of NLS solitons

In progress 2020 summer research project.

Student: Upaasna Parankusam. Supervisor: Dave Smith.

We model solutions of the Nonlinear Schrödinger 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

In progress 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 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 interface ordinary differential operators.

### Boundary wellposedness of initial multipoint value problems

Open project.

Supervisor: Dave Smith.

A linear evolution partial differential equations, together with initial and boundary conditions, is called an initial boundary value problem. Such problems are often used to model physical systems, such as water waves or the distribution of heat in a solid. Hadamard gave specific criteria for such a problem to be "wellposed": existence and unicity of a solution, and that solution changing only slightly with slight changes in the data of the problem. The data of the problem are typically the initial state of the system and information about how the system evolves at the spatial boundaries. Therefore, Hadamard's third criterion is important, because it ensures that a small measurement error will not have ruinous effect on the output of the model.

If the differential equation is high order in space, then it is necessary to specify more than one condition at the spatial boundary. Typically, the value of the quantity under study and a spatial derivative are both prescribed at the boundary. But what if there were a small error in the measurement of *where* one of these were specified?

The unified transform method, as recently extended by Pelloni & Smith, provides a powerful tool for the analysis of high order initial boundary problems, and their generalizations relevant to the above question, initial multipoint value problems. The method exploits complex analysis to provide novel solutions to problems for which classical Fourier analysis is not applicable. Preliminary work on this project is the investigation of Hadamard's third criterion using solution representations obtained via the unified transform method.

The main aim of this project is to investigate how solutions of high order initial boundary value problems and initial multipoint value problems depend upon the points at which their conditions are specified.

Required modules: Linear algebra, Proof, Real analysis.

Useful modules: Foundations of applied mathematics, Ordinary & partial differential equations, Complex analysis.

## 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.

## 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.

### Numerical locus of zeros of exponential polynomials

In progress 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. We use this theorem to numerically locate the zeros of exponential polynomials within a given region, using a julia script.

### Asymptotic locus of zeros of exponential polynomials

In progress 2020 summer research project.

Student: Wang Yanhua. Supervisor: Dave Smith.

Langer (1931) uses geometric and asymptotic analysis to find asymptotic expressions for the zero locus of exponential polynomials. A range of theorems cover increasingly complicated classes of exponential polynomials, characterised by the distribution of exponents within the convex hull of those exponents. We implement these theorems computationally in julia.

### Analytic-geometric asymptotic analysis of exponential polynomials

In progress 2020 summer research project.

Student: Zhang Liu. Supervisor: Dave Smith.

Langer (1931) uses geometric and asymptotic analysis to find asymptotic expressions for the zero locus of exponential polynomials. Langer's arguments rely on asymptotic analysis of the relative dominance of each term in the exponential polynomial in different regions of the complex plane, informed by the geometry of the convex hull of the exponents. Here we apply similar relative dominance arguments to the numerical evaluation of exponential polynomials, disregarding terms of lesser importance to improve efficiency.

### Hybrid analytic numerical evaluation of oscillatory complex contour integrals

Open project.

Supervisor: Dave Smith.

An integral in which the integrand rapidly switches between positive and negative, almost but not exactly cancelling, are called oscillatory. Oscillatory integrals often arise in the solution of differential equations, making their evaluation an important but challenging problem. Some powerful tools for numerical evaluation of oscillatory integrals have recently been developed, including the julia package ApproxFun, which uses orthogonal polynomial approximation to efficiently evaluate integrals to high accuracy, but there is necessarily a time cost in the initial polynomial approximation step. Combining the tools of geometry with asymptotic analysis, it is possible to simplify certain integrands by discarding extremely small terms. Complex analysis also provides tools such as Cauchy's theorem, Jordan's lemma and the method of steepest descent to make integrals of analytic functions, including some oscillatory integrals, much easier to evaluate numerically.

The purpose of this project is to develop, and to implement as a julia package, a method for evaluating oscillatory contour integrals. Applications include the analysis of integrals appearing in the solutions representations of initial boundary value problems for linear evolution equations of high spatial order.

Required modules: Proof, Foundations of applied mathematics, Introduction to computer science.

Useful modules: Real analysis, Python, Data structures and algorithms, Linear algebra, Ordinary & partial differential equations, Complex analysis.