Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations

• Published in: Studies in applied mathematics (Cambridge) 2018-07, Vol.141 (1), p.46-88
• Main Authors: Pelloni, Beatrice, Smith, David Andrew
• Format: Article
• Language: English
• Subjects: Boundary value problems; Linear evolution equations; Organic chemistry; Partial differential equations; Representations
• ISSN: 0022-2526; 1467-9590
• DOI: 10.1111/sapm.12212

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution partial differential equations (PDE) with constant coefficients in one space variable. The prototypical example of such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third‐order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as multipoint conditions, and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e., it admits a unique solution. For the second‐order case, we also give criteria that guarantee well posedness.