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dc.contributor.authorBaker, Christopher T. H.*
dc.contributor.authorFord, Neville J.*
dc.date.accessioned2019-06-18T11:46:41Z
dc.date.available2019-06-18T11:46:41Z
dc.date.issued2019-04-24
dc.identifier.citationBaker, CTH, & Ford, NJ. (2019 - in press). Characteristic functions of differential equations with deviating arguments, Applied Numerical Mathematics.en
dc.identifier.issn0168-9274
dc.identifier.doi10.1016/j.apnum.2019.04.010
dc.identifier.urihttp://hdl.handle.net/10034/622356
dc.description.abstractThe 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.
dc.language.isoenen
dc.publisherElsevieren
dc.relation.urlhttps://www.journals.elsevier.com/applied-numerical-mathematicsen
dc.relation.urlhttps://www.sciencedirect.com/science/article/pii/S0168927419300972en
dc.rightsAttribution-NonCommercial-NoDerivatives 4.0 International*
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/en
dc.subjectDifferential equations with deviating argumentsen
dc.subjectEquations of mixed-typeen
dc.subjectCharacteristic functionsen
dc.subjectDiscretisationsen
dc.subjectOscillatory and non-oscillatory and bounded or unbounded functionsen
dc.titleCharacteristic functions of differential equations with deviating argumentsen
dc.typeArticleen
dc.identifier.eissn1873-5460
dc.contributor.departmentUniversity of Manchester; University of Chesteren
dc.identifier.journalApplied Numerical Mathematics
dc.date.accepted2019-04-18
or.grant.openaccessYesen
rioxxterms.funderunfundeden_US
rioxxterms.identifier.projectunfundeden_US
rioxxterms.versionAMen
rioxxterms.licenseref.startdate2021-04-24
refterms.dateFCD2019-05-09T07:49:29Z
refterms.versionFCDAM


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