Solve BVP using Eigenfunction Expansion

In summary, the conversation discusses solving a boundary value problem using eigenfunction expansion. The problem involves finding the solution to y''[x]-y[x]=Cos[2x] with the given boundary conditions of y'[0]=y'[Pi]=0. The attempt at a solution involves using Fourier transforms and considering the corresponding Sturm-Liouville problem, but ends with confusion over the use of a negative sign in front of y[x] and the resulting incorrect Fourier expansion.
  • #1
bitty
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Homework Statement


Solve BVP y''[x]-y[x]=Cos[2x] using eigenfunction expansion.
We know y'[0]=y'[Pi]=0

Homework Equations



Fourier it up

The Attempt at a Solution


Is the corresponding Sturm-Liouville problem:
f''[x]+/lambda*f=0?

All the examples we've done have been of form y''[x]+y[x]=g[x] but does a negative sign in front of y[x] change the corresponding S/L problem we use?

It follows that f[x]=Cosnx

Because I proceed to solve for y[x]=Sum[A_n*f[x]] and I find the coefficents and all but when I plot my Fourier expansion for y[x], it is very different from the actual result of y[x]. I'm guessing I wasn't supposed to use f''[x]+/lambda*f=0 to find my eigenfunction expansion, but I'm confused as to why I'm getting the wrong answer
 
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  • #2
Edit: sorry i misunderstood
 

Related to Solve BVP using Eigenfunction Expansion

1. What is a boundary value problem (BVP)?

A boundary value problem is a type of mathematical problem that involves finding a solution to a differential equation or system of differential equations, subject to certain boundary conditions. These conditions specify the values of the solution at the boundaries of the domain in which it is defined.

2. What is an eigenfunction expansion?

An eigenfunction expansion is a method for solving a boundary value problem by representing the solution as a linear combination of eigenfunctions. These eigenfunctions are solutions to the homogeneous version of the differential equation and are determined by the boundary conditions.

3. How does one solve a BVP using eigenfunction expansion?

To solve a BVP using eigenfunction expansion, one must first determine the eigenfunctions of the differential equation and the corresponding eigenvalues. Then, the solution can be expressed as a linear combination of these eigenfunctions, with the coefficients determined by the boundary conditions. This results in a system of linear equations, which can be solved to find the coefficients and thus the solution to the BVP.

4. What are the advantages of using eigenfunction expansion to solve BVPs?

One advantage of using eigenfunction expansion is that it allows for the solution of more complex BVPs by breaking them down into simpler eigenvalue problems. It also provides an analytical solution, which can be useful in understanding the behavior of the system. Additionally, eigenfunction expansion can be applied to a wide range of differential equations and boundary conditions.

5. Are there any limitations to using eigenfunction expansion to solve BVPs?

One limitation of eigenfunction expansion is that it may not always be possible to find the eigenfunctions and eigenvalues analytically. In these cases, numerical methods may be necessary. Additionally, the convergence of the series solution obtained through eigenfunction expansion may be slow, requiring a large number of terms to achieve an accurate solution. This can make the method computationally expensive for certain problems.

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