IMPLICIT EIGENVALUE/EIGENFUNCTION COMPUTATION

The boundary conditions can cause the eigenvalues to be implicitly defined as is seen by the following example:

y'' = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk1NTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUYsNiVRInlGJy9GNVEldHJ1ZUYnL0Y4USdpdGFsaWNGJ0Y3RitGNw== 0 < x < 1

y(0) = y'(1) + y(1) = 0

Assuming that 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 in the differential equation and using the boundary condition y(0)=0 shows that 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. Plugging into the boundary condition y'(1)+y(1)=0 leads to the equation

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

in order for y to be a nontrivial solution

f:= s -> tan(s) + s;

fsolve(f(s)=0,s=0..Pi);

fsolve(f(s)=0,s=0..10);

plot({tan(s),-s},s=0..10);

plot({tan(s),-s},s=0..20,y=-15..0,thickness=3);

We need to get a solution for tan(s)=-s in each of the intervals [(2n+1)\317\200/2,(2n+3)\317\200/2].

S(0):=Pi/2;

for k from 1 to 50 do S(k):= S(k-1) + Pi; od:

for k from 1 to 50 do

z(k):= fsolve(f(s)=0,s=S(k-1)..S(k));

od;

lambda := n -> -z(n)^2;

phi := (n,x) -> sin(z(n)*x);

plot({phi(1,x),phi(2,x),phi(3,x)},x=0..1,thickness=3);

g:=x -> 2*x + (1/2-2*x)*Heaviside(x-1/2);

plot(g(x),x=0..1,thickness=3);

c:=n -> int(g(x)*phi(n,x),x=0..1)/int(phi(n,x)^2,x=0..1);

G:=(N,x) -> sum(c(n)*phi(n,x),n=1..N);

plot({g(x),G(8,x)},x=0..1,thickness=3);

with(plots):

animate({g(x),G(N,x)},x=0..1,N=1..40,frames=40,thickness=3);