Eigenfunction expansion in Legendre polynomials

In summary, the conversation discusses how to use eigenfunction expansion in Legendre polynomials to find the bounded solution of a differential equation on a specific interval. The main focus is on determining the eigenfunctions and using them to solve the equation. The post also includes an attachment with further information and asks for clarification on the variable u(x).
  • #1
hi10
1
0

Homework Statement



How to use eigenfunction expansion in Legendre polynomials to find the bounded solution of
(1-x^2)f'' - 2xf' + f = 6 - x - 15x^2 on -1<= x <= 1

Homework Equations



eigenfunction expansion

The Attempt at a Solution



[r(x)y']' + [ q(x) + λ p(x) ] = f(x)
In this case, r = 1-x^2 , q = 1 , p = 0 , f = 6 - x -15 x^2 , r(-1) = r (1) = 0

Thanks for any help!
 

Attachments

  • q.bmp
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  • #2
First what are the eigenfunctions you want to use? In other words, what are the solutions to Legendre's equation.
 
  • #3
I have the same problem.

the attachment from hi10 is what i was thinking.

Does anyone know what u(x) is in hi10's post for determining what an is?
 

1. What is the purpose of using eigenfunction expansion in Legendre polynomials?

Eigenfunction expansion in Legendre polynomials is a mathematical technique used to express a function as a linear combination of eigenfunctions, which are solutions to a specific differential equation. This method is often used in physics and engineering to solve problems involving boundary value equations and Fourier series.

2. How do Legendre polynomials differ from other types of polynomials?

Legendre polynomials are a special type of orthogonal polynomial, meaning they are mutually perpendicular when evaluated over a certain interval. They are commonly used in eigenfunction expansions because of their unique properties, such as being orthogonal and complete over the interval [-1, 1].

3. What are the applications of eigenfunction expansion in Legendre polynomials?

Eigenfunction expansion in Legendre polynomials has many practical applications in fields such as physics, engineering, and mathematics. It is commonly used to solve problems involving heat transfer, quantum mechanics, and vibrations. It is also used in image and signal processing to compress data and reduce noise.

4. How do you determine the coefficients in an eigenfunction expansion using Legendre polynomials?

The coefficients in an eigenfunction expansion using Legendre polynomials can be determined by using a weighted inner product. This involves multiplying both sides of the expansion by the corresponding eigenfunction and integrating over the interval of interest. This process results in a system of equations that can be solved to find the coefficients.

5. Can eigenfunction expansion using Legendre polynomials be generalized to other types of polynomials?

Yes, eigenfunction expansion can be generalized to other types of orthogonal polynomials, such as Chebyshev polynomials and Hermite polynomials. However, each type of polynomial has its own unique properties and applications, so the specific method of expansion may vary depending on the polynomial being used.

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