IFSintegrals

A toolbox for solving problems defined on attractors of iterated function systems (IFSs). This project is an implementation of multiple research projects, with Andrea Moiola, David Hewett, Simon Chandler-Wilde, António Caetano, Joshua Bannister, Botond Major and Jeevon Greewal. In particular, the following project students have contributed to the development of the code: Joshua Bannister, Jeevon Greewal.

See Quadrature example.ipynb, BEM Cantor set example.ipynb and BEM 3D example.ipynb for examples. Click here to load these interactive examples in your browser (without the need to install any software). I would strongly recommend playing with these notebook files to understand what the code can do. Quadrature example.ipynb contains a sufficient introduction to the toolbox, and the introductory section of this should be understood first, even if you are not interested in understanding how the quadrature works.

Installation:

To install, type the following into Julia:

using Pkg

Pkg.add(url="https://github.com/AndrewGibbs/IFSintegrals.git")

Quadrature:

Weights and nodes for the evaluation of integrals with respect to Hausdorff (or equivalent) measure can be obtained using barycentre_rule, which is a generalisation of the midpoint rule to IFS attractors.

For IFS attractors which are subsets of $\mathbb{R}$, Gaussian quadrature is available using gauss_quad.

Certain classes of singular integrals can be evaluated using eval_green_double_integral and eval_green_single_integral_fixed_point.

BEM:

Boundary Integral operators can be defined and discretised on attractors, using the types BIO and DiscreteBIO. There are examples of these problems being solved in the notebook files, where the boundary integral equation for the Helmholtz equation is solved.

Bibliography

• Numerical Quadrature for Singular Integrals on Fractals, A. Gibbs, D. P. Hewett, A. Moiola, published article, arxiv preprint.
• Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets, Andrew Gibbs, David P. Hewett, Botond Major, published article, arxiv preprint
• Integral equation methods for acoustic scattering by fractals, A. M. Caetano, S. N. Chandler-Wilde, X. Claeys, A. Gibbs, D. P. Hewett, A. Moiola, arxiv preprint
• A Hausdorff-measure boundary element method for acoustic scattering by fractal screens, António M. Caetano, Simon N. Chandler-Wilde, Andrew Gibbs, David P. Hewett, Andrea Moiola, arxiv preprint
• A Hausdorff-measure boundary element method for acoustic scattering by fractal screens, António M. Caetano, Simon N. Chandler-Wilde, Andrew Gibbs, David P. Hewett, Andrea Moiola, arxiv preprint
• A Stable Stieltjes Technique for Computing Orthogonal Polynomials and Jacobi Matrices Associated with a Class of Singular Measures, G. Mantica, published article.