### Abstract

We derive the solution representation for a large class of nonlocal boundary

value problems for linear evolution PDEs with constant coecients

in one space variable. The prototypical 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 dierent 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. that it admits a unique solution. For the second

order case, we also give criteria that guarantee well-posedness.

value problems for linear evolution PDEs with constant coecients

in one space variable. The prototypical 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 dierent 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. that it admits a unique solution. For the second

order case, we also give criteria that guarantee well-posedness.

Original language | English |
---|---|

Pages (from-to) | 46-88 |

Journal | Studies in Applied Mathematics |

Volume | 141 |

Issue number | 1 |

Early online date | 25 Mar 2018 |

DOIs | |

Publication status | Published - Jul 2018 |

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## Profiles

## Beatrice Pelloni

- School of Mathematical & Computer Sciences - Professor
- School of Mathematical & Computer Sciences, Mathematics - Professor

Person: Academic (Research & Teaching)

## Cite this

Pelloni, B., & Smith, D. A. (2018). Nonlocal and multipoint boundary value problems for linear evolution equations.

*Studies in Applied Mathematics*,*141*(1), 46-88. https://doi.org/10.1111/sapm.12212