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Faculty of Mathematics




  • 2015-    : Senior College Lecturer and DoS, Homerton College.
  • 2013-15: Stokes Fellow, Pembroke College.
  • 2010-13: Junior Research Fellow, Emmanuel College.
  • 2009-10: EPSRC Prize Fellow, DAMTP.
  • 2006-09: PhD Candidate, DAMTP.
  • 2005-06: Part III Mathematics, Peterhouse.
  • 2002-05: BA Mathematics, Peterhouse.


Anthony is a member of the Department of Applied Mathematics and Theoretical Physics and works in the Applied and Computational Analysis group. His research interests include: novel approaches  to elliptic boundary value problems, Lie groups in PDE, new approaches to rigorous problems in Linear PDE theory and certain aspects of mathematical physics.


I am lecturing the Part III course Distribution Theory & Applications, which focuses on classical distribution theory and application to the analysis of linear PDEs. Example sheets are here and the corresponding solutions will appear below

There are also some handouts

  1. Notation
  2. Distributions supported at a point
  3. Fundamental solution for heat operator

I've also lectured Part II Integrable Systems which includes topics such as: the Arnold-Liouville theorem, inverse scattering, infinite dimensional Hamiltonian systems and Lie group methods in PDE. Handouts were

  1.  Generating functions
  2.  Arnold-Liouville theorem and worked example
  3.  Evolution of scattering data
  4.  From Lax pairs to zero curvature
  5.  Painlevé equations

From 2017-2020 I lectured Part IA Vector Calculus, where we take a look extending our ideas about calculus to two and three dimensions. We go on to discuss some aspects of PDE and right towards the end we start to look at Cartesian tensors. Handouts are

  1.  Coordinate systems
  2.  Change of variables in 2D integrals
  3.  Divergence and curl formulae

I will update this set of notes as we progress through the course. They currently cover lectures 1-24. 


Selected Publications

  • Ashton, 2015. A new weak formulation of the Dirichlet-Neumann map on convex polyhedra with explicit coercivity constants. (in preparation).
  • Ashton & Crooks, 2014. Numerical Analysis of Fokas' Unified Method for Linear Elliptic PDEs, Appl. Num. Math. (in press).
  • Ashton, 2014. Laplace's Equation on Convex Polyhedra via the Unified Method, Proc. Roy. Soc. A 471(2176).
  • Ashton, 2014. Elliptic PDEs with Constant Coefficients on Convex Polyhedra via the Unified Method, J. Math. Anal. & Appl. 425(1).
  • Ashton & Fokas, 2014. Elliptic Equations with Low Regularity Boundary Data via the Unified Method, Complex Var. & Elliptic Eq. 60(5).
  • Ashton, 2013. The spectral Dirichlet-Neumann map for Laplace's equation in a convex polygon, SIAM J. Math. Anal. 45(6).
  • Ashton, 2012. On the rigorous foundations of the Fokas method for linear elliptic PDEs, Proc. Roy. Soc. A 468(2142).
  • Ashton & Fokas, 2011. A Nonlocal Formulation of  Rotational Water Waves, J. Fluid. Mech. 689(1).
  • Ashton, 2011. Regularity of Elliptic and Hypoelliptic Operators via the Global Relation, J. Part. Diff. Eq. 24(1).
  • Ashton, 2011. On The Non-Existence of Three Dimensional Water Waves with Finite Energy, Nonlin. Anal. B 12(4).
  • Ashton, 2010. Stability of Parallel Fluid Loaded Plates: A Nonlocal Approach, Stud. App. Math. 125(3).
  • Ashton & Fokas, 2009. A Novel Approach to the Fluid Loaded Plate, Proc. Roy. Soc. A 465(2112).
  • Ashton, 2008. Conservation Laws and Non-Lie Symmetries, J. Nonlin. Math. Phys. 15(3).
  • Ashton, 2008. The Fundamential k-form and Global Relations, SIGMA 4(33).



Relations among the Riemann zeta and Hurwitz zeta functions, as well as their products
ACL Ashton, AS Fokas
– Symmetry
Numerical analysis of Fokas' unified method for linear elliptic PDEs
ACL Ashton, KM Crooks
– Applied Numerical Mathematics
Elliptic PDEs with constant coefficients on convex polyhedra via the unified method
ACL Ashton
– Journal of Mathematical Analysis and Applications
Laplace's equation on convex polyhedra via the unified method.
ACL Ashton
– Proceedings. Mathematical, physical, and engineering sciences
Elliptic equations with low regularity boundary data via the unified method
ACL Ashton, AS Fokas
– Complex Variables and Elliptic Equations
The Spectral Dirichlet--Neumann Map for Laplace's Equation in a Convex Polygon
ACL Ashton
– SIAM Journal on Mathematical Analysis
Elliptic boundary value problems in convex polygons with low regularity boundary data via the unified method
ACL Ashton, AS Fokas
– Journal of Differential Equations
On the rigorous foundations of the Fokas method for linear elliptic partial differential equations
ACL Ashton
– Proceedings of the Royal Society A
A Nonlocal Formulation of Rotational Water Waves
ACL Ashton, AS Fokas
– Journal of Fluid Mechanics
A non-local formulation of rotational water waves
ACL Ashton, AS Fokas
– Journal of Fluid Mechanics
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Research Group

Nonlinear Dynamical Systems




01223 337904