Numerical Methods for Mixed Type Functional Equations

This talk is concerned with the analysis of a linear mixed type (forward-backward) functional equation, that is, a linear functional equation which has a delay and an advance term. We search for a solution of such equation which has a given form at the initial interval $[-1,0]$ and at the final interval $[k-1,k]$. This problem has been studied both analytically and numerically (see[1]). One of the most common approaches for the analysis of this problem is based on its reduction to an initial value problem for a delay differential equation (DDE). Following this approach, we search for an approximate solution in the form of a linear combination of a given set ob basis functions. The basis functions can be extended to the interval $[1,k]$ either numerically (using the finite difference method) or analytically (using recurrence formulae — “method of steps”). Finally, the coefficients of the linear combination are computed by the collocation or least squares method, so that the numerical solution fits the data at the interval $[k-1,k]$. A different approach consists on the transformation of the considered problem into a boundary value problem for a ODE. In this case, standard numerical methods for ODEs can be applied. Numerical results obtained by these methods are presented and compared with the ones, presented in previous works. The advantages and weaknesses of the introduced computational methods are discussed. This is a joint work with P. Lima, P. Lumb and N. Ford.

Reference

1. N. Ford and P. Lumb, Mixed type functional differential equations: a numerical approach (submitted).