Mathematics

Introduction to Initial Value DDE Problems

The DDE solver can solve systems of ordinary differential equations

where is the independent variable, is the dependent variable, and represents . The delays (lags) are positive constants.

Using a History to Specify the Solution of Interest

In an initial value problem, we seek the solution on an interval . with . The DDE shows that depends on values of the solution at times prior to . In particular, depends on

. Because of this, a solution on depends on its values for , i.e., its history .

Propagation of Discontinuities

Generally, the solution of an IVP for a system of DDEs has a jump in its first derivative at the initial point because the first derivative of the history function does not satisfy the DDE there

A discontinuity in any derivative propagates into the future at spacings of .

For reliable and efficient integration of DDEs, a solver must track discontinuities in low order derivatives and deal with them. For DDEs with constant lags, the solution gets smoother as the integration progresses, so after a while the solver can stop tracking a discontinuity. See Discontinuities for more information.

DDE Function Summary

DDE Solver