Use this URL to cite or link to this record in EThOS: | https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.507233 |
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Title: | Very singular solutions of odd-order PDEs, with linear and nonlinear dispersion | ||||||
Author: | Fernandes, Ray Stephen |
ISNI:
0000 0004 2675 402X
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Awarding Body: | University of Bath | ||||||
Current Institution: | University of Bath | ||||||
Date of Award: | 2008 | ||||||
Availability of Full Text: |
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Abstract: | |||||||
Asymptotic properties of solutions of the linear dispersion equation ut = uxxx in R × R+, and its (2k + 1)th-order generalisations are studied. General Hermitian spectral theory and asymptotic behaviour of its kernel, for the rescaled operator B = D3 + 1 3 yDy + 1 3 I, is developed, where a complete set of bi-orthonormal pair of eigenfunctions, {ψβ}, {ψ∗β }, are found. The results apply to the construction of VSS (very singular solutions) of the semilinear equation with absorption ut = uxxx − |u|p−1u in R × R+, where p > 1, which serves as a basic model for various applications, including the classic KdV area. Finally, the nonlinear dispersion equations such as ut = (|u|nu)xxx in R × R+, and ut = (|u|nu)xxx − |u|p−1u in R × R+, where n > 0, are studied and their “nonlinear eigenfunctions” are constructed. The basic tools include numerical methods and “homotopy-deformation” approaches, where the limits n → 0 and n → +∞ turn out to be fruitful. Local existence and uniqueness is proved and some bounds on the highly oscillatory tail are found. These odd-order models were not treated in existing mathematical literature, from the proposed point of view.
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Supervisor: | Galaktionov, Victor | Sponsor: | Not available | ||||
Qualification Name: | Thesis (Ph.D.) | Qualification Level: | Doctoral | ||||
EThOS ID: | uk.bl.ethos.507233 | DOI: | Not available | ||||
Keywords: | nonlinear dispersion ; PDEs ; odd-order ; VSS | ||||||
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