How to solve eigenvalue problems with mixed boundary condition?

[wdlang]

suppose function f is define on the interval [0,1]

it satisfies the eigenvalue equation f'' + E f=0, and it satisfies the boundary conditions

f'(0)+ f(0)=0, f(1)=0.

How to solve this eigenvalue problem numerically?

the mixed boundary condition at x=0 really makes it difficult

 

[CompuChip]

Have you found the general solution? It is f(x) = a sin(kx) + b cos(kx) for a known constant k and constants a and b to be determined.

The two equations f'(0) + f(0) = 0 and f(1) = 0 will give you two equations in the two unknowns a and b.

However, I would double check the question if I were you, because as you posted it a = b = 0 is the only solution, leading to f(x) = 0.

 

[pasmith]

Have you found the general solution? It is f(x) = a sin(kx) + b cos(kx) for a known constant k and constants a and b to be determined.

The two equations f'(0) + f(0) = 0 and f(1) = 0 will give you two equations in the two unknowns a and b.

However, I would double check the question if I were you, because as you posted it a = b = 0 is the only solution, leading to f(x) = 0.

The point of eigenvalue problems is that E - which determines your k - is unknown; the object is to find those values of E for which non-zero solutions f are possible.

Here, we have the general solution $A \cos (kx) + B \sin (kx)$, where k is also unknown. Substituting this into the boundary conditions gives two equations for the three unknowns; we have to add the condition that at least one of A and B is non-zero to determine the permissible values of k.

 

[wdlang]

Have you found the general solution? It is f(x) = a sin(kx) + b cos(kx) for a known constant k and constants a and b to be determined.

The two equations f'(0) + f(0) = 0 and f(1) = 0 will give you two equations in the two unknowns a and b.

However, I would double check the question if I were you, because as you posted it a = b = 0 is the only solution, leading to f(x) = 0.

actually i am more interested in the numerical solution

because my eigenvalue equation will be modified in future as

f'' + V(x) f + E f =0,

where V(x) is an arbitrary real function.

so the problem is to device a numerical scheme to do it

 

[pasmith]

actually i am more interested in the numerical solution

because my eigenvalue equation will be modified in future as

f'' + V(x) f + E f =0,

where V(x) is an arbitrary real function.

so the problem is to device a numerical scheme to do it

That looks similar to the one-dimensional time-independent Schrödinger equation; looking at resources for numerical solution of that might be useful.

 

[the_wolfman]

The simplest way is to build a shooting code.

First let's note that if

g(x) is an solution of your equation then ag(x) is also a solution.

This allows us to pick f(0) = 1 which also gives us f'(0) =-1. We will use this for all the following calculations.

Next treat x as a time coordinate and using standard techniques for advancing in time we advance the ode in x from 0 to 1. You do a range of assumed values for E and note the value f(1) for each E.

The values of E where f(1) is close to 0 are approximate eigenvalues.

This is the basic idea. Typically people use root finding algorithms and interpolation to improve accuracy and performance.