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Characteristic functions of differential equations with deviating arguments
Baker, Christopher T. H. Ford, Neville J.
Baker, Christopher T. H.
Ford, Neville J.
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2019-04-24
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Abstract
The material here is motivated by the discussion of solutions of linear homogeneous and autonomous differential equations with deviating arguments. If $a, b, c$ and $\{\check{\tau}_\ell\}$ are real and ${\gamma}_\natural$ is real-valued and continuous, an example with these parameters is
\begin{equation} u'(t) = \big\{a u(t) + b
u(t+\check{\tau}_1) + c u(t+\check{\tau}_2) \big\} { \red +}
\int_{\check{\tau}_3}^{\check{\tau}_4} {{\gamma}_\natural}(s) u(t+s)
ds \tag{\hbox{$\rd{\star}$}} .
\end{equation}
A wide class of equations ($\rd{\star}$), or of similar type, can be written in the {\lq\lq}canonical{\rq\rq} form
\begin{equation} u'(t) =\DSS
\int_{\tau_{\rd \min}}^{\tau_{\rd \max}} u(t+s) d\sigma(s) \quad (t \in \Rset),
\hbox{ for a suitable choice of } {\tau_{\rd \min}}, {\tau_{\rd \max}}
\tag{\hbox{${\rd \star\star}$}} \end{equation}
where $\sigma$ is of bounded variation and the integral is a Riemann-Stieltjes integral. For equations written in the form (${\rd{\star\star}}$), there is a
corresponding characteristic function \begin{equation} \chi(\zeta) ):= \zeta -
\DSS
\int_{\tau_{\rd \min}}^{\tau_{\rd \max}} \exp(\zeta s) d\sigma(s) \quad (\zeta \in
\Cset), \tag{\hbox{${\rd{\star\star\star}}$}}
\end{equation}
%%($ \chi(\zeta) \equiv \chi_\sigma (\zeta)$)
whose zeros (if one considers appropriate subsets of equations (${\rd \star\star}$) -- the literature provides additional information on the subsets to which we refer) play a r\^ole in the study of oscillatory or non-oscillatory solutions, or of bounded or unbounded solutions. We show that the related discussion of the zeros of $\chi$ is facilitated by observing and exploiting some simple and fundamental properties of characteristic functions.
Citation
Baker, C. T. H., & Ford, N. J. (2020). Characteristic functions of differential equations with deviating arguments. Applied Numerical Mathematics, 149, 17-29. https://doi.org/10.1016/j.apnum.2019.04.010
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Elsevier
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Applied Numerical Mathematics
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Article
Language
en
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ISSN
0168-9274
EISSN
1873-5460