MATLAB Mathematics - SERC - Index of
MATLAB Mathematics - SERC - Index of MATLAB Mathematics - SERC - Index of
5 Differential Equations Function ddeset ddeget Description Create/alter the DDE options structure. Extract properties from options structure created with ddeset. DDE Initial Value Problem Examples These examples illustrate the kind of problems you can solve using dde23. Click the example name to see the code in an editor. Type the example name at the command line to run it. Note The Differential Equations Examples browser enables you to view the code for the DDE examples, and also run them. Click on the link to invoke the browser, or type odeexamples('dde')at the command line. Example ddex1 ddex2 Description Straightforward example Cardiovascular model with discontinuities Additional examples are provided by “Tutorial on Solving DDEs with DDE23,” available at http://www.mathworks.com/dde_tutorial. Introduction to Initial Value DDE Problems The DDE solver can solve systems of ordinary differential equations y′ () t = ftyt ( , ()yt , ( – τ 1 ),…, yt ( – τ k )) where t is the independent variable, y is the dependent variable, and y′ represents dy ⁄ dt . The delays (lags) τ 1 , …, τ k are positive constants. 5-50
Initial Value Problems for DDEs Using a History to Specify the Solution of Interest In an initial value problem, we seek the solution on an interval [ t 0 , t f ]. with t 0 < t f . The DDE shows that y′ () t depends on values of the solution at times prior to t. In particular, y′ ( t 0 ) depends on yt ( 0 – τ 1 ),…, yt ( 0 – τ k ). Because of this, a solution on [ t 0 , t f ] depends on its values for t ≤ t 0 , i.e., its history St (). Propagation of Discontinuities Generally, the solution yt () of an IVP for a system of DDEs has a jump in its first derivative at the initial point t 0 because the first derivative of the history function does not satisfy the DDE there. – + S′ ( t 0 ) ≠ y′ ( t 0 ) = ft ( 0 , yt ( 0 ), St ( 0 – τ 1 ), …, St ( 0 – τ k )) A discontinuity in any derivative propagates into the future at spacings of τ 1 , τ 2 , …, τ k . 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” on page 5-57 for more information. DDE Solver This section describes: • The DDE solver, dde23 • DDE solver basic syntax The DDE Solver The function dde23 solves initial value problems for delay differential equations (DDEs) with constant delays. It integrates a system of first-order differential equations y′ () t = ftyt ( , ()yt , ( – τ 1 ),…, yt ( – τ k )) on the interval [ t 0 , t f ], with t 0 < t f and given history yt () = St () for t ≤ t 0 . 5-51
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5 Differential Equations<br />
Function<br />
ddeset<br />
ddeget<br />
Description<br />
Create/alter the DDE options structure.<br />
Extract properties from options structure created with ddeset.<br />
DDE Initial Value Problem Examples<br />
These examples illustrate the kind <strong>of</strong> problems you can solve using dde23.<br />
Click the example name to see the code in an editor. Type the example name at<br />
the command line to run it.<br />
Note The Differential Equations Examples browser enables you to view the<br />
code for the DDE examples, and also run them. Click on the link to invoke the<br />
browser, or type odeexamples('dde')at the command line.<br />
Example<br />
ddex1<br />
ddex2<br />
Description<br />
Straightforward example<br />
Cardiovascular model with discontinuities<br />
Additional examples are provided by “Tutorial on Solving DDEs with DDE23,”<br />
available at http://www.mathworks.com/dde_tutorial.<br />
Introduction to Initial Value DDE Problems<br />
The DDE solver can solve systems <strong>of</strong> ordinary differential equations<br />
y′ () t = ftyt ( , ()yt , ( – τ 1 ),…,<br />
yt ( – τ k ))<br />
where t is the independent variable, y is the dependent variable, and y′<br />
represents dy ⁄ dt . The delays (lags) τ 1 , …,<br />
τ k are positive constants.<br />
5-50
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