| 2024 |
Normatov B, Smith DA, 'The Airy equation with nonlocal conditions', STUDIES IN APPLIED MATHEMATICS, 152, 543-567 (2024) [C1]
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| 2024 |
Pelloni B, Smith DA, 'Revivals, or the Talbot effect, for the Airy equation', STUDIES IN APPLIED MATHEMATICS, 153 (2024) [C1]
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| 2022 |
Fokas AS, Pelloni B, Smith DA, 'Time-periodic linear boundary value problems on a finite interval', Quarterly of Applied Mathematics, 80 481-506 (2022)
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| 2022 |
Aitzhan S, Bhandari S, Smith DA, 'Fokas Diagonalization of Piecewise Constant Coefficient Linear Differential Operators on Finite Intervals and Networks', Acta Applicandae Mathematicae, 177 (2022)
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| 2022 |
Smith DA, Toh WY, 'Linear evolution equations on the half-line with dynamic boundary conditions', EUROPEAN JOURNAL OF APPLIED MATHEMATICS, 33, 505-537 (2022)
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| 2021 |
Boulton L, Olver PJ, Pelloni B, Smith DA, 'New revival phenomena for linear integro-differential equations', STUDIES IN APPLIED MATHEMATICS, 147, 1209-1239 (2021)
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| 2020 |
Olver PJ, Sheils NE, Smith DA, 'REVIVALS AND FRACTALISATION IN THE LINEAR FREE SPACE SCHRODINGER EQUATION', QUARTERLY OF APPLIED MATHEMATICS, 78, 161-192 (2020)
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| 2018 |
Miller PD, Smith DA, 'The diffusion equation with nonlocal data', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 466, 1119-1143 (2018)
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| 2018 |
Pelloni B, Smith DA, 'Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations', Studies in Applied Mathematics, 141 46-88 (2018)
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| 2018 |
Kesici E, Pelloni B, Pryer T, Smith D, 'A numerical implementation of the unified Fokas transform for evolution problems on a finite interval', EUROPEAN JOURNAL OF APPLIED MATHEMATICS, 29, 543-567 (2018)
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| 2016 |
Fokas AS, Smith DA, 'Evolution PDEs and augmented eigenfunctions. Finite interval', Advances in Differential Equations, 21, 735-766 (2016) [C1]
The so-called unified or Fokas method expresses the solution of an initial-boundary value problem (IBVP) for an evolution PDE in the finite interval in terms of an inte... [more]
The so-called unified or Fokas method expresses the solution of an initial-boundary value problem (IBVP) for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple IBVPs, which will be referred to as problems of type I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated IBVPs, which will be referred to as problems of type II, there does not exist a classical transform pair and the solution cannot be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalized eigenfunctions described in the sixties by Gel'fand and his co-authors.
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| 2016 |
Deconinck B, Sheils NE, Smith DA, 'The Linear KdV Equation with an Interface', COMMUNICATIONS IN MATHEMATICAL PHYSICS, 347, 489-509 (2016)
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| 2016 |
Pelloni B, Smith DA, 'Evolution PDEs and augmented eigenfunctions. Half-line', Journal of Spectral Theory, 6 185-213 (2016)
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| 2015 |
Sheils NE, Smith DA, 'Heat equation on a network using the Fokas method', JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 48 (2015)
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| 2013 |
Pelloni B, Smith DA, 'Spectral theory of some non-selfadjoint linear differential operators', Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences, 469 (2013)
We give a characterization of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, an... [more]
We give a characterization of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator S with the properties of the solution of a corresponding boundary value problem for the partial differential equation ¿tq ± iSq=0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular, whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator. Copyright © The Royal Society 2013.
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| 2012 |
Smith DA, 'Well-posed two-point initial-boundary value problems with arbitrary boundary conditions', MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 152, 473-496 (2012)
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