Interface problems of mixed spatial order

We solve interface problems on the line between various constant coefficient linear evolution partial differential equations. Our prototypical examples are the heat, linear Schr\"odinger, Airy, linearized Korteweg de Vries, and biharmonic Schr\"odinger equations. In each problem, one of the listed equations is posed on one spatial half line, and another on the other half line, with appropriate interface conditions. These problems are solved by means of novel extensions of Fokas's unified transform method and the explicit solution formulae are evaluated using Filon quadrature.