Identification of the initial function for discretized delay differential equations

In the continuous problem (finding an initial function that gives rise to a solution of a DDE) studied in 2004 by Baker and Parmuzin, the function is obtained by minimizing a functional Sαβ,γ(ϕ). Here, we use a stepsize h to introduce a discrete version of the problem, along with h-dependent discrete functionals (S˜αβ,γh(ϕ˜)) that simulate Sαβ,γ(ϕ). Conditions for a minimum of S˜αβ,γh(ϕ˜) are explored through an analysis of its first variation P˜αβ,γh(ϕ˜), and an iterative technique for obtaining the minimum is written down. In order to explore the properties of this iteration, it is convenient to relate it to an iterative algorithm for the solution of a discretized integral equation (a summation equation), for which the properties of the “kernel” can be obtained. A rôle for adjoint equations and fundamental solutions in the discrete case is established. The final part of the paper consists of a report of numerical experiments that demonstrate the performance of the algorithm.