How to solve eigenvalue problems with mixed boundary condition?

In summary: That looks similar to the one-dimensional time-independent Schrodinger equation; looking at resources for numerical solution of that might be useful.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
  • #1
wdlang
307
0
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
 
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  • #2
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.
 
  • #3
CompuChip said:
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 [itex]A \cos (kx) + B \sin (kx)[/itex], 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.
 
  • #4
CompuChip said:
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
 
  • #5
wdlang said:
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 Schrodinger equation; looking at resources for numerical solution of that might be useful.
 
  • #6
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.
 

1. What is an eigenvalue problem with mixed boundary condition?

An eigenvalue problem is a mathematical problem that involves finding the eigenvalues (or characteristic values) of a given linear transformation or matrix. A mixed boundary condition in this context means that the boundary conditions for the problem involve a combination of different types, such as both Dirichlet and Neumann conditions.

2. How do I set up an eigenvalue problem with mixed boundary condition?

To set up an eigenvalue problem with mixed boundary condition, you will need to specify the differential equation or matrix that represents the problem, as well as the boundary conditions for each type of boundary. This can be done using techniques such as the separation of variables method or the finite element method.

3. What are some techniques for solving eigenvalue problems with mixed boundary condition?

Some common techniques for solving eigenvalue problems with mixed boundary condition include the shooting method, the Rayleigh quotient method, and the variational method. Each of these methods involves different mathematical approaches to finding the eigenvalues and corresponding eigenvectors of the problem.

4. How can I check if my solution to an eigenvalue problem with mixed boundary condition is correct?

One way to check the correctness of your solution is to plug it back into the original problem and see if it satisfies both the differential equation and all of the boundary conditions. You can also compare your results to known solutions or use numerical methods to verify your solution.

5. Are there any software programs or tools available for solving eigenvalue problems with mixed boundary condition?

Yes, there are many software programs and tools available for solving eigenvalue problems with mixed boundary condition. Some popular options include MATLAB, Mathematica, and COMSOL Multiphysics. These programs often have built-in functions and algorithms specifically designed for solving eigenvalue problems.

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