Novel exact solutions for PDEs with mixed boundary conditions

We develop methods for the solution of inhomogeneous Robin-type boundary value problems (BVPs) that arise for certain linear parabolic partial differential equations (PDEs) on a half-line, as well as a second-order generalization. We are able to obtain nonstandard solutions to equations arising in a range of areas, including mathematical finance, stochastic analysis, hyperbolic geometry, and mathematical physics. Our approach uses the odd and even Hilbert transforms. The solutions we obtain and the method itself seem to be new.