MATLAB Mathematics - SERC - Index of

5 Differential Equations
Note The Robertson problem appears as an example in the prolog to
LSODI [4].
In hb1ode, the problem is solved with initial conditions y 1 ( 0) = 1 , y 2 ( 0) = 0 ,
y 3 ( 0) = 0 to steady state. These differential equations satisfy a linear
conservation law that is used to reformulate the problem as the DAE
y′ 1 = – 0.04y 1 + 10 4 y 2 y 3
y′ 2 = 0.04y 1 – 10 4 y 2 y 3 3 10 7 2
– ⋅ y 2
0 = y 1 + y 2 + y 3 – 1
These equations do not have a solution for y( 0)
with components that do not
sum to 1. The problem has the form of My′ = fty ( , ) with
M
=
1 0 0
0 1 0
0 0 0
M is singular, but hb1dae does not inform the solver of this. The solver must
recognize that the problem is a DAE, not an ODE. Similarly, although
consistent initial conditions are obvious, the example uses an inconsistent
value y 3 ( 0) = 10 – 3 to illustrate computation of consistent initial conditions.
To run this example, click on the example name, or type hb1dae at the
command line. Note that hb1dae:
• Imposes a much smaller absolute error tolerance on y 2 than on the other
components. This is because y 2 is much smaller than the other components
and its major change takes place in a relatively short time.
• Specifies additional points at which the solution is computed to more clearly
show the behavior of y 2 .
• Multiplies y by 10 4 2 to make y 2 visible when plotting it with the rest of the
solution.
• Uses a logarithmic scale to plot the solution on the long time interval.
5-36