I am a Postdoctoral Research Associate in the Numerical Analysis of Nonlocal PDEs in the group of José A. Carrillo at the University of Oxford. My research focus is on numerical analysis and the development of computational tools for integral equations, PDEs and fractional differential equations with applications in the natural sciences. I am also more generally interested in computational orthogonal polynomials and their applications in the development of multivariate spectral methods.


See also the automated listings at Google Scholar or Semantic Scholar.

Wave equation on a disk dampened by a time-fractional Caputo derivative term.

Fractional Differential Equations

Fractional differential equations generalize ODEs and PDEs to include derivative operators of non-integer order. The resulting operators are generally nonlocal and thus pose several challenges to conventional numerical approaches.

Fractional PDEs appear in natural science applications featuring memory effects, e.g. photoacoustic imaging in medical physics.

  1. A static memory sparse spectral method for time-fractional PDEs in arbitrary dimensions
    T. S. Gutleb, J. A. Carrillo.  arXiv
Degree 500 orthogonal polynomial obtained from modifying Jacobi polynomials with a certain rational weight, including the Szegő envelope.

Computational Orthogonal Polynomials

Orthogonal polynomials can be used to accurately and efficiently approximate functions.

They further enable the construction of spectral methods for solving integral and differential equations as well as quadrature methods.

  1. Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations
    T. S. Gutleb, S. Olver, R. M. Slevinsky.  arXiv
Convergence of histograms associated with attractive-repulsive particle simulation to the equilibrium measure.

Computing Equilibrium Measures

Equilibrium measure problems arise in the continuous mean-field limit of attractive-repulsive particle systems and have diverse applications ranging from swarming behavior in animals to self-assembly properties of particle systems.

As power law interactions involve Riesz potentials, this special case also relates to fractional differential equations.

  1. Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension
    T. S. Gutleb, J. A. Carrillo, S. Olver. Constructive Approximation (2022).   arXiv  DOI
  2. Computing Equilibrium Measures with Power Law Kernels
    T. S. Gutleb, J. A. Carrillo, S. Olver. Mathematics of Computation (2022).   arXiv   DOI
Spy plot of a banded Volterra integral operator acting on a basis of Jacobi polynomials.

Spectral Methods for Integral Equations

By using orthogonal polynomial approximation bases, one can turn very general integral equations into banded sparse linear systems which can be solved with high efficiency and exceptional convergence properties.

We used this to develop solvers for linear and nonlinear integral and integro-differential equations which have applications in the natural sciences and finance.

  1. A Fast Sparse Spectral Method for Nonlinear Integro-differential Volterra Equations with General Kernels
    T. S. Gutleb. Advances in Computational Mathematics (2021).   arXiv   DOI
  2. A Sparse Spectral Method for Volterra Integral Equations Using Orthogonal Polynomials on the Triangle
    T. S. Gutleb, S. Olver. SIAM Journal on Numerical Analysis (2020).   arXiv   DOI
Isosurface evolution of spin up (red) and spin down (blue) states as solutions of the Pauli equation coupled to an external magnetic field.

Numerics for magnetic Schrödinger equations

Strang splitting methods have been successfully used for the numerical treatment of non-relativistic magnetic Schrödinger equations in their appropriate 3D context.

I have worked towards extending these results to also work for relativistic equations which account for quantum mechanical spin.

  1. A time splitting method for the three-dimensional linear Pauli equation
    T. S. Gutleb, N. J. Mauser, M. Ruggeri, H. P. Stimming.   arXiv

Selected Talks

Below is a selection of my conference talks. I'll update the website with a more complete list soon.


  • Intercollegiate Tutor for Numerical Linear Algebra
    Oct 2022 - Jan 2023, University of Oxford, UK
  • Senior GTA for Numerical Analysis
    Jan 2022 - Apr 2022, Imperial College London, UK
  • GTA and student project mentor for Mathematical Modeling
    Mar 2021 - Jun 2021 & May 2022 - August 2022, Wolfgang Pauli Institute, University of Vienna, Austria
  • Graduate Teaching Assistant (GTA) in Analysis 1 and 2 for Physicists
    Oct 2015 - Jul 2018 , University of Vienna, Austria