Numerical treatment of oscillary functional differential equations

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 234(2010), doi: 10.1016/j.cam.2010.01.035This preprint is concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear v-methods will also be oscillatory. We conclude with some general theor

Characteristic functions of differential equations with deviating arguments

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