Equivalent systems for differential equations with polynomially distributed delays
Roland Pulch (Universität Greifswald, Institut für Mathematik und Informatik)
We consider a delay differential equation (DDE) of the form \[y'(t) = f \Big( y(t) , \int_I y(t-\tau) \, g(\tau) \; \mathrm{d} \tau \Big) ,\] which includes a distributed delay in a bounded interval \(I = [\tau_{\min},\tau_{\max}] \subset [0,\infty)\) or an unbounded interval \(I = [0,\infty)\). The mapping \(g : I \rightarrow \mathbb{R}_0^+\) represents a probability density function associated to some probability distribution. In the case of \(I = [0,\infty)\) and using a gamma distribution, it is well known that an equivalent system of ordinary differential equations (ODEs) can be arranged. We investigate the case of a bounded interval \(I\) and a polynomial \(g\). Important instances are a uniform distribution (\(g\) constant) and a beta distribution. We derive an equivalent system, which consists of DDEs including two discrete delays. The properties of this equivalent system are analysed. For comparison, Gaussian quadrature yields a discretisation of the integral in the original DDE. This approach implies a DDE with multiple discrete delays, whose solution approximates the solution of the DDE with distributed delay. We present results of numerical computations, where initial value problems are solved.