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| Format: | Preprint |
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2004
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| Online Access: | https://arxiv.org/abs/math/0409464 |
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| _version_ | 1866913467875721216 |
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| author | Bueler, Ed |
| author_facet | Bueler, Ed |
| contents | We present a Chebyshev collocation method for linear ODE and DDE problems. We first give a posteriori estimates for the accuracy of the approximate solution of a scalar ODE initial value problem. Examples of the success of the estimate are given. For linear, periodic DDEs with integer delays we define and discuss the monodromy operator U as our main goal is reliable estimation of the stability of such DDEs. We prove a theorem which gives a posteriori estimates for eigenvalues of U, our main result. This result is based on a generalization to operators on Hilbert spaces of the Bauer-Fike theorem for (matrix) eigenvalue perturbation problems. We generalize these results to systems of DDEs. A delayed, damped Mathieu equation example is given. The computation of good bounds on ODE fundamental solutions is an important technical issue; an a posteriori method for such bounds is given. Certain technical issues are also addressed, namely the evaluation of polynomials and the estimation of L^\infty norms of analytic functions. Generalization to the non-integer delays case is also considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0409464 |
| institution | arXiv |
| publishDate | 2004 |
| record_format | arxiv |
| spellingShingle | Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates Bueler, Ed Numerical Analysis Spectral Theory 65Q05, 65F, 15A42 We present a Chebyshev collocation method for linear ODE and DDE problems. We first give a posteriori estimates for the accuracy of the approximate solution of a scalar ODE initial value problem. Examples of the success of the estimate are given. For linear, periodic DDEs with integer delays we define and discuss the monodromy operator U as our main goal is reliable estimation of the stability of such DDEs. We prove a theorem which gives a posteriori estimates for eigenvalues of U, our main result. This result is based on a generalization to operators on Hilbert spaces of the Bauer-Fike theorem for (matrix) eigenvalue perturbation problems. We generalize these results to systems of DDEs. A delayed, damped Mathieu equation example is given. The computation of good bounds on ODE fundamental solutions is an important technical issue; an a posteriori method for such bounds is given. Certain technical issues are also addressed, namely the evaluation of polynomials and the estimation of L^\infty norms of analytic functions. Generalization to the non-integer delays case is also considered. |
| title | Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates |
| topic | Numerical Analysis Spectral Theory 65Q05, 65F, 15A42 |
| url | https://arxiv.org/abs/math/0409464 |