Abstract

The Unified Transform maps the Helmholtz equation to spectral space not dissimilarly to the Fourier Transform. However, the resulting equations from the Unified Transform can be manipulated, through considering regions of analyticity, into a so-called global relation which relates known (transformed) boundary conditions to unknown (transformed) boundary data. By applying a spectral collocation method, the unknown boundary data can be recovered and the solution obtained throughout the whole domain. In this talk we extend previous theory for the Unified Transform in bounded domains to unbounded domains so that acoustic scattering problems can be tackled for a range of geometries typically approached via a matrix Wiener Hopf method. We also explain some of the links between the Unified Transform, Fourier Transform and Boundary Integral Equations. Numerical results are demonstrated for problems with corner singularities and comparisons made to other approaches.