Applied Mathematics Seminar

It is shown that the new method for solving initial-boundary value problems for scalar evolution equations recently introduced by Fokas can also be applied to systems of evolution equations. The novel step needed in this case is the construction of a scalar Lax pair by using a suitable parametrisation of the dispersion relation as well as certain linear transformations. The simultaneous spectral analysis of the Lax pair yields the solution of a given initial-boundary value problem in terms of an integral in the complex spectral plane which involves an appropriate x-transform of the initial conditions and an appropriate t-transform of the boundary conditions. These transforms are neither the x-Fourier transform nor the t-Laplace transform, rather they are new transforms custom made for the given system of PDEs and the given domain. This method is illustrated by solving on the half-line the linearised equations governing infinitesimal deformations in a heat conducting bar.