The numerical unified transform method for initial-boundary value problems on the half-line

Abstract We implement the unified transform method of Fokas as a numerical to solve linear evolution partial differential equations on half-line. The computes solution at any $x$ and $t$ without spatial discretization or time stepping. With help contour deformations oscillatory integration techniques, method’s complexity does not increase for large $x,t$ is more accurate (absolute errors are smaller, relative bounded). Our goal make no assumptions functional form initial boundary functions beyond some decay smoothness, while maintaining high accuracy in region $(x,t)$ plane.