lin : A Linear ODE Eigenvalue Problem.

This demo illustrates the location of eigenvalues of a linear ODE boundary value problem as bifurcations from the trivial solution family. By means of branch switching an eigenfunction is computed, as is illustrated for the first eigenvalue. This eigenvalue is then continued in two parameters by fixing the $L_2$-norm of the first solution component. The eigenvalue problem is given by the equations

$$\begin{array}{cl} u_1 ' &= u_2 , \\ u_2 ' &= (p_1 \pi)^{2} u_1 , \end{array}$$ (15.4)

with boundary conditions $u_1(0)-p_2=0$ and $u_1(1)=0.$ We add the integral constraint

$$\int_0^{1} u_1(t)^{2} dt - p_3 = 0.$$

Then $p_3$ is simply the $L_2$-norm of the first solution component. In the first two runs $p_2$ is fixed, while $p_1$ and $p_3$ are free. In the third run $p_3$ is fixed, while $p_1$ and $p_2$ are free.

Table 15.4: Commands for running demo lin.

AUTO -COMMAND	ACTION
! mkdir lin	create an empty work directory
cd lin	change directory
demo('lin')	copy the demo files to the work directory
run(c='lin.1')	1st run; compute the trivial solution family and locate eigenvalues
sv('lin')	save output-files as b.lin, s.lin, d.lin
run(c='lin.2',s='lin')	2nd run; compute a few steps along the bifurcating family. Constants changed : IRS, ISW, DSMAX
ap('lin')	append output-files to b.lin, s.lin, d.lin
run(c='lin.3',s='lin')	3rd run; compute a two-parameter curve of eigenvalues. Constants changed : IRS, ISW, ICP(2)
sv('2p')	save the output-files as b.2p, s.2p, d.2p