Abstract

The unified transform is typically associated with the solution of integrable nonlinear PDEs. However, after an appropriate linearisation, one can also apply the method to linear PDEs and develop a spectral boundary-based method. I will discuss recent advances of this method, in particular, the application of the method to problems in unbounded domains with solutions having corner singularities. Consequently, a wide variety of mixed boundary condition problems can be solved without the need for the Wiener-Hopf technique. Such problems arise frequently in acoustic scattering or in the calculation of electric fields in geometries involving finite and/or multiple plates. The new approach constructs a global relation that relates known boundary data, such as the scattered normal velocity on a rigid plate, to unknown boundary values, such as the jump in pressure upstream of the plate. This can be viewed formally as a domain dependent Fourier transform of the boundary integral equations. By approximating the unknown boundary functions in a suitable basis expansion and evaluating the global relation at collocation points, one can accurately obtain the expansion coefficients of the unknown boundary values. The local choice of basis functions is flexible, allowing the user to deal with singularities and complicated boundary conditions such as those occurring in elasticity models or spatially variant Robin boundary conditions modelling porosity.