dc.contributor.author Baker, Christopher T. H. * dc.contributor.author Ford, Neville J. * dc.date.accessioned 2019-06-18T11:46:41Z dc.date.available 2019-06-18T11:46:41Z dc.date.issued 2019-04-24 dc.identifier.citation Baker, CTH, & Ford, NJ. (2019 - in press). Characteristic functions of differential equations with deviating arguments, Applied Numerical Mathematics. en dc.identifier.issn 0168-9274 dc.identifier.doi 10.1016/j.apnum.2019.04.010 dc.identifier.uri http://hdl.handle.net/10034/622356 dc.description.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 $$u'(t) = \big\{a u(t) + b u(t+\check{\tau}_1) + c u(t+\check{\tau}_2) \big\} { \red +} \int_{\check{\tau}_3}^{\check{\tau}_4} {{\gamma}_\natural}(s) u(t+s) ds \tag{\hbox{\rd{\star}}} .$$ A wide class of equations ($\rd{\star}$), or of similar type, can be written in the {\lq\lq}canonical{\rq\rq} form $$u'(t) =\DSS \int_{\tau_{\rd \min}}^{\tau_{\rd \max}} u(t+s) d\sigma(s) \quad (t \in \Rset), \hbox{ for a suitable choice of } {\tau_{\rd \min}}, {\tau_{\rd \max}} \tag{\hbox{{\rd \star\star}}}$$ 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 $$\chi(\zeta) ):= \zeta - \DSS \int_{\tau_{\rd \min}}^{\tau_{\rd \max}} \exp(\zeta s) d\sigma(s) \quad (\zeta \in \Cset), \tag{\hbox{{\rd{\star\star\star}}}}$$ %%($\chi(\zeta) \equiv \chi_\sigma (\zeta)$) 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.iso en en dc.publisher Elsevier en dc.relation.url https://www.journals.elsevier.com/applied-numerical-mathematics en dc.relation.url https://www.sciencedirect.com/science/article/pii/S0168927419300972 en dc.rights Attribution-NonCommercial-NoDerivatives 4.0 International * dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/ en dc.subject Differential equations with deviating arguments en dc.subject Equations of mixed-type en dc.subject Characteristic functions en dc.subject Discretisations en dc.subject Oscillatory and non-oscillatory and bounded or unbounded functions en dc.title Characteristic functions of differential equations with deviating arguments en dc.type Article en dc.identifier.eissn 1873-5460 dc.contributor.department University of Manchester; University of Chester en dc.identifier.journal Applied Numerical Mathematics dc.date.accepted 2019-04-18 or.grant.openaccess Yes en rioxxterms.funder unfunded en_US rioxxterms.identifier.project unfunded en_US rioxxterms.version AM en rioxxterms.licenseref.startdate 2020-04-24 refterms.dateFCD 2019-05-09T07:49:29Z refterms.versionFCD AM refterms.dateFOA 2020-04-24T00:00:00Z rioxxterms.publicationdate 2019-04-24
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