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.
The drug diffusion problem: comparison of analytic methods
In progress 2025–2026 AMSI summer research project and Honours project
Student: Noah Cresp. Supervisor: Dave Smith.
We consider a two-layer diffusion problem representing transdermal drug delivery as a one-dimensional model. The model describes Fickian diffusion in each layer, coupled by continuity of flux and a partition condition at the interface, together with appropriate boundary conditions. It is solved analytically using two distinct methods, the Laplace transform approach and the Unified Transform Method (Fokas’ Method). We compare the resulting solution representations and discuss practical considerations for computation, including inversion and numerical evaluation. This highlights strengths and weaknesses of each method, especially in the context of extending the model to more layers or modified boundary and interface conditions.
Outputs: [Cre2026a], [Cre2026b].
The unified transform method on metric graph domains
A metric graph is a graph where each edge is associated with a spatial interval and each vertex has some "vertex conditions" which specify how the boundary values of each incident edge interact. Inital velue problems posed on such metric graphs can be used to model waves in metamaterials and blood pressure waves in arterial networks. Early work on adapting the unified transform method to solve such problems was [SS2015a] for the heat equation, and the more recent [Smi2026a] demonstrated validity for an equation of higher spatial order.
These projects demonstrate the broad applicability of the method to other equations and on more complicated metric graphs.
The linearized Benjamin-Bona-Mahoney equation on metric star graphs
In progress 2025–2026 AMSI summer research project and Winter research project.
Student: Sam Walsh. Supervisor: Dave Smith.
The Benjamin-Bona-Mahoney equation models blood pressure waves in arterial networks. Here, we solve the small amplitude linearization of this equation on a metric star graph with arbitrarily many incoming and outgoing semiinfinite edges meeting at a single vertex.
Outputs: [Wal2026a], [Wal2026b].
The linearized KdV equation on metric star graphs
In progress 2025–2026 SCIS summer research project and SCIS Winter research project.
Student: Aidan R de Luzuriaga. Supervisor: Dave Smith.
The Korteweg-de Vries equation models water waves in narrow channels of shallow water. Here, we solve the small amplitude linearization of this equation on a metric star graph with arbitrarily many incoming and outgoing semiinfinite edges meeting at a single vertex.
Dispersive revivals
If a dispersive evolution equation on a periodic domain is given a step function as an initial datum then, at almost all later times, the solution will be continuous but nowhere differentiable, appearing like a fractal. However, at certain "rational" times, the solution appears to jump to a step function. It turns out that the solution at these rational times is a linear combination of finitely many shifted copies of the initial datum. This visually striking effect, first discovered by pioneer of photography Talbot in 1836, has since been rediscovered by physicists and mathematicians several times, most recently by Peter Olver. Olver showed the remarkable generality of this phenomenon in dispersive equations, and it has since been shown to survive in the presence of a potential, nonperiodic boundary conditions, and even nonlinearities.
Revivals in the linear Schrödinger equation on the disc
In progress 2025–2026 honours project.
Student: Noah Castle. Supervisor: Dave Smith.
We study the linear Schrödinger equation on the disc with a homogeneous Dirichlet condition at the boundary. In numerical experiments, we observe structures that appear to be revivals in the radial variable, except with logarithmic cusps superimposed. We explain this phenomenon as weak cusped revivals: continuous perturations of revivals of the initial datum and its Fourier transform.
Dispersive-dissipative interface problems on the ring
In progress 2026 SCIS winter research project.
Student: Isabella Swadling. Supervisor: Dave Smith.
Partial differential equations may be used to model a wide variety of physical phenomena with categorically different behaviours. For example, while the heat equation models the spread of impurities in a fluid or heat through a solid, described by a diffusive relaxation of the profile, the linear Schrodinger equation represents waves travelling through complex media, with different frequencies propagated at different speeds. These fundamentally different effects are known as diffusion and dispersion, respectively, and the heat and linear Schrodinger are merely the simplest prototypical examples of broad classes of differential equations, each exhibiting categorically similar evolution.
Despite this difference in behaviour, the heat and Schrodinger equations may be solved via remarkably similar mathematical techniques, such as Fourier series and transforms. More modern techniques, such as the Unified Transform Method may also be applied to both these equations, enabling their efficient solution even in problems with complex boundary conditions. Significant contributions were made by Sheils and collaborators (see, for example, [Deconinck, Pelloni & Sheils 2014], [Sheils & Smith 2015], [Deconinck & Sheils 2022]) to the solution of complex interface problems for both dissipative and dispersive evolution equations via the Unified Transform Method. In contrast with other spectral methods for finite interval problems, this method produces a complex contour integral representation of the solution, rather than a series expansion in the eigenfunctions. This at once obviates the explicit calculation of eigenvalues and enables the application of the powerful tools of complex analysis to the efficient analysis of the solution. In particular, asymptotic analysis of these Fourier type integrals is typically more efficient than would be an analysis of the Fourier series.
Recent works [Deconinck & Farkas 2024] and [Mantzavinos, Pethiyagoda & Smith 2026] have built upon Sheils's results to enable the solution of interface problems between partial differential equations that may be of different type (dissipative vs dispersive) or even of different spatial order. In the presently proposed research, these methods will be extended further to the study of dissipative-dispersive interface problems on finite domains. It is expected that complex analytic methods may be used to solve these equations and their solutions plotted using a mathematical programming language such as julia.
An interesting case is the dispersive-dissipative problem on a ring, and specifically its limit as the dissipative interval contracts. It is known that the purely dispersive problem features quantum revivals, visually striking differences between the solution at rational and irrational times, while the purely dissipative problem does not. But it is not known how the dispersive-dissipative interface problem will produce such revivals in the limit.
D to N maps for evolution equations
An initial boundary value problem for an evolution partial differential equation is the problem of finding the full spatial and temporal dependence of a function based on information at the edges of the spacetime domain. In a D to N map, one seeks to map only from certain known boundary data to other information at the spatial boundaries. For example, one might seek to reconstruct a function at the boundary if one knows only the value of its derivative at the boundary and its initial value.
D to N maps are useful because, in certain situations, one does not really care about the value of a function in the body, only at the edges of its domain. In other settings, the construction of a D to N map is a crucial step in solving the full initial boundary value problem.
Many of these projects extend the applicability of the "
Time periodic nonlocal value problems for the Airy equation
In progress 2026 SCIS winter research project.
Student: Cameron West. Supervisor: Dave Smith.
In [Normatov & Smith 2024] it was shown that this method is readily extended to a time periodic nonlocal value problem for the Airy equation with specific nonlocal and boudnary conditions. In this project, we examine a more general class of boundary and nonlocal conditions for the time periodic Airy problem, aiming to determine wellposedness criteria.