pvl : Use of the Routine PVLS.

Consider Bratu's equation

$$\begin{array}{cl} u_1 ' &= u_2 , \\ u_2 ' &= -p_1 e^{u_1} , \\ \end{array}$$ (19.1)

with boundary conditions $u_1(0)=0$, $u_1(1)=0.$ As in demo exp, a solution curve requires one free parameter; here $p_1$.

Note that additional parameters are specified in the user-supplied subroutine PVLS in file pvls.f, namely, $p_2$ (the $L_2$-norm of $u_1$), $p_3$ (the minimum of $u_2$ on the space-interval $[0,1]$ ), $p_4$ (the boundary value $u_2(0)$ ). These additional parameters should be considered as ``solution measures'' for output purposes; they should not be treated as true continuation parameters.

Note also that four free parameters are specified in the AUTO -constants file c.pvl.1, namely, $p_1$, $p_2$, $p_3$, and $p_4$. The first one in this list, $p_1$, is the true continuation parameter. The parameters $p_2$, $p_3$, and $p_4$ are overspecified so that their values will appear in the output. However, it is essential that the true continuation parameter appear first. For example, it would be an error to specify the parameters in the following order : $p_2$, $p_1$, $p_3$, $p_4$.

In general, true continuation parameters must appear first in the parameter-specification in the AUTO constants-file. Overspecified parameters will be printed, and can be defined in PVLS, but they are not part of the intrinsic continuation procedure.

As this demo also illustrates (see the UZR values in c.pvl.1), labeled solutions can also be output at selected values of the overspecified parameters.

Table 19.1: Commands for running demo pvl.

AUTO -COMMAND	ACTION
! mkdir pvl	create an empty work directory
cd pvl	change directory
demo('pvl')	copy the demo files to the work directory
run(c='pvl.1')	compute a solution family
sv('pvl')	save output-files as b.pvl, s.pvl, d.pvl