Numerical treatment of delay differential equations by Hermite Interpolation

Abstract

Summary. A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the weltknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically.Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.

QMC Methods for the solution of delay differential equations

“…For a constant delay function, we have T −1 (t k−1 ) = t k−1 + τ 0 , and the solution y(t) has k continuous derivatives at t k = t 0 + kau 0 , and y (k+1) (t) in general has a jump discontinuity at t k . If the initial function φ(t) (or its derivatives) have discontinuities, a similar statement holds with t 0 replaced by the discontinuities of the initial function (see [9]). …”

“…However, even if f and φ are smooth, the solution y(t) is only smooth if φ(t) solves the differential equation (8). Otherwise, the solution y(t) will have discontinuous derivatives y (j) (t k ) for j ≥ k at times t k , which are recursively defined by t k = T −1 (t k−1 ) (see [9]). This means that with each additional time interval of length h k = t k − t k−1 , the discontinuities are smoothed out, and all the derivatives up to the k-th are continuous.…”

“…As the value of the derivative of y(t) can be easily calculated for each time t j by simply inserting into the differential equation, Hermite interpolation (as investigated in [9]) is used here since it only requires half the points ordinary Lagrange interpolation would require. Although the interpolating function then also depends on the approximated values of the derivative of y(t), we will suppress this in our notation.…”

“…One should note that in the proof given in [9], it is assumed that the approximation formula z(t) to the retarded values is smooth, and so the set of support points for the interpolation can only be changed between two time steps. Since the QMC integration needs the calculation of a lot of points from a large interval, a significant amount of extrapolation would be involved.…”

Natural Continuous Extensions of Runge-Kutta Methods

“…for every sufficiently smooth matrix-valued function G. It is easily seen that condition (9) implies the following error bounds for the higher derivatives of the NCEs: (11) max \y{k)(t) -uw(t)\= 0(hd-k + x), k = 2,...,d, t0^t^t0 + h and, obviously, u{k)(t) = 0 for k > d + 1.…”