On Barenblatt's pseudoparabolic equation with forcing on the half‐line via the Fokas method

Abstract

A novel technique is presented for explicitly solving inhomogeneous initial‐boundary‐value problems (IBVPs) (Dirichlet, Neumann and Robin) on the half‐line, for a well‐known pseudo‐parabolic partial differential equation. This so‐called Barenblatt's equation arises in a plethora of important applications, ranging from heat‐mass transfer, solid‐fluid‐gas dynamics and materials science, to mechanical, chemical and petroleum engineering, as well as electron physics, radiation and diffusive processes. Our approach is based on the extension of the Fokas method, so that it can be applied to problems with mixed derivatives. First, we derive formally effective solution representations and then justify a posteriori their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each type of IBVP, the novel formulae are utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain and the problem's well‐posedness. Furthermore, importantly, our solutions’ numerical advantages are demonstrated and highlighted by way of a concrete and illustrative example. Our rigorous approach can be extended to IBVPs for other significant models.