A new approach for the analysis of evolution partial differential equations on a finite interval

Abstract We show that, for certain evolution partial differential equations, the solution on a finite interval can be reconstructed as a superposition of restrictions to of solutions to two associated partial differential equations posed on the half‐lines and . Determining the appropriate data for these half‐line problems amounts to solving an inverse problem, which we formulate via the unified transform of Fokas (also known as the Fokas method) and address via a fixed‐point argument in ‐based Sobolev spaces, including fractional ones through interpolation techniques. We illustrate our approach through two canonical examples, the heat equation and the Korteweg‐de Vries (KdV) equation, and provide numerical simulations for the former example. We further demonstrate that the new approach extends to more general evolution partial differential equations, including those with time‐dependent coefficients. A key outcome of this work is that spatial and temporal regularity estimates for problems on a finite interval can be directly derived from the corresponding estimates on the half‐line. These results can, in turn, be used to establish local well‐posedness for related nonlinear problems, as the essential ingredients are the linear estimates within nonlinear frameworks.