Natural Solutions to Delay Differential Equations
Mathematica has provided a strong platform for developing a new approach to solving problems containing delay differential equations (DDE). In particular, a method has been developed for finding natural analytic solutions that do not require a priori history to be specified. The "natural" solution will, in some sense, be determined from minimum energy requirements and from the DDE rather than an arbitrary choice of history. There will be some DDE where there can be multiple natural solutions, and additional conditions may be needed to determine an appropriate solution(s). It is also possible to determine natural solutions that satisfy a finite number of starting conditions. All other solutions to the DDE can be approximated by natural solutions corresponding to appropriate choices of starting conditions. An additional advantage of natural analytic solution usage is that solutions can be back calculated to an earlier time than the primary dataset. This could help determine the start of specific features in a dataset that might diagnose process problems or further validate the model. This presentation will illustrate the calculation of natural solutions for DDE with a single time delay for both linear and nonlinear equations, with and without starting conditions being specified. In addition, examples of using this approach for fitting data of processes that include time delays will be demonstrated. In particular, some data from the epidemiology of COVID-19 will be examined.