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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The generalized Green's function for an $ n$th order linear differential operator


Author: John Locker
Journal: Trans. Amer. Math. Soc. 228 (1977), 243-268
MSC: Primary 34B05; Secondary 47E05
MathSciNet review: 0481204
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Abstract: The generalized Green's function $ K(t,s)$ for an nth order linear differential operator L is characterized in terms of the 2nth order differential operators $ L{L^\ast}$ and $ {L^\ast}L$. The development is operator oriented and takes place in the Hilbert space $ {L^2}[a,b]$. Two features of the characterization are a determination of the jumps occurring in the derivatives of orders n, $ n + 1, \ldots ,2n - 1$ at $ t = s$ and a determination of the boundary conditions satisfied by the functions $ K(a, \cdot )$ and $ K(b,\cdot)$. Several examples are given to illustrate the properties of the generalized Green's function.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0481204-0
PII: S 0002-9947(1977)0481204-0
Keywords: Generalized Green's function, differential operator, jump conditions, boundary conditions, generalized inverse, quasi-derivatives, Green's formula
Article copyright: © Copyright 1977 American Mathematical Society