We use the two-dimensional heat equation as an illustrative example to show that the unified transform is capable of constructing analytical solutions for linear evolution partial differential equations (PDEs) in two spatial dimensions involving non-separable boundary conditions. Such non-separable boundary value problems apparently cannot be solved by the usual transforms. We note that the unified transform always yields integral expressions which, in contrast to the expressions obtained by the usual transforms, have the advantage that are uniformly convergent at the boundary. Thus, even for the cases of separable boundary value problems where the usual transforms can be implemented, the unified transform provides alternative solution expressions which have advantages for both numerical and asymptotic considerations. The former advantage is illustrated by providing the numerical evaluation of a typical boundary value problem, by extending the approach of Flyer and Fokas (2008Proc. R. Soc.464, 1823–1849). This work is the two-dimensional continuation of the heat equation with oblique Robin boundary conditions which was analysed in Mantzavinos and Fokas (2013Eur. J. Appl. Math.24(6), 857–886.

We use the heat equation as an illustrative example to show that the unified method introduced by one of the authors can be employed for constructing analytical solutions for linear evolution partial differential equations in one spatial dimension involving non-separable boundary conditions as well as non-local constraints. Furthermore, we show that for the particular case in which the boundary conditions become separable, the unified method provides an easier way for constructing the relevant classical spectral representations avoiding the classical spectral analysis approach. We note that the unified method always yields integral expressions which, in contrast to the series or integral expressions obtained by the standard transform methods, are uniformly convergent at the boundary. Thus, even for the cases that the standard transform methods can be implemented, the unified method provides alternative solution expressions which have advantages for both numerical and asymptotic considerations. The former advantage is illustrated by providing the numerical evaluation of typical boundary value problems.