Non-standard initial-boundary-value and interface problems for the linear BBM equation

We derive new integral representations for the solutions of both nonhomogeneous initial–boundary value problems and interface problems associated with the linearized Benjamin–Bona–Mahony equation, posed on the half-line and the full real line, respectively. The boundary and interface constraints, albeit appearing to be nontypical, are naturally and directly indicated by the structure of the BBM PDE. The novel representations are formulated as contour integrals in the complex Fourier plane and are obtained through a systematic implementation of the Fokas unified transform method. The analysis presents notable challenges stemming from the presence of the higher-order mixed-derivative term in the equation and from the generality of our framework. The resulting explicit analytical formulae provide a robust foundation for further investigation into the qualitative behavior of solutions, including their asymptotic properties, spatio-temporal evolution, regularity, and well-posedness. Moreover, the methods and results developed here are expected to inform future studies on nonlinear extensions of the BBM equation and on free-boundary problems in related dispersive systems.