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Composing and factoring generalized Green’s operators and ordinary boundary problems. (English) Zbl 1375.34041

Barkatou, Moulay (ed.) et al., Algebraic and algorithmic aspects of differential and integral operators. 5th international meeting, AADIOS 2012, held at the applications of computer algebra conference, ACA 2012, Sofia, Bulgaria, June 25–28, 2012. Selected and invited papers. Berlin: Springer (ISBN 978-3-642-54478-1/pbk). Lecture Notes in Computer Science 8372, 116-134 (2014).
Summary: We consider solution operators of linear ordinary boundary problems with “too many” boundary conditions, which are not always solvable. These generalized Green’s operators are a certain kind of generalized inverses of differential operators. We answer the question when the product of two generalized Green’s operators is again a generalized Green’s operator for the product of the corresponding differential operators and which boundary problem it solves. Moreover, we show that – provided a factorization of the underlying differential operator – a generalized boundary problem can be factored into lower order problems corresponding to a factorization of the respective Green’s operators. We illustrate our results by examples using the Maple package IntDiffOp, where the presented algorithms are implemented.
For the entire collection see [Zbl 1283.68027].

MSC:

34B27 Green’s functions for ordinary differential equations
34-04 Software, source code, etc. for problems pertaining to ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
68W30 Symbolic computation and algebraic computation

Software:

IntDiffOp; Maple
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Full Text: DOI arXiv

References:

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