Defining Legendre polynomials in (1,2)

In summary, the Legendre polynomials can be changed to a new variable w by changing the coordinates and choosing appropriate values for A and B. This allows for the calculation of new polynomials in the domain of (1, 2), which can be used to construct an estimation of a random uniform variable using chaos polynomials. The numerical model estimation also confirms the accuracy of this method.
  • #1
confused_engineer
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Hello everyone.

The Legendre polynomials are defined between (-1 and 1) as 1, x, ½*(3x2-1), ½*(5x3-3x)...

My question is how can I switch the domain to (1, 2) and how can I calculate the new polynomials.

I need them to construct an estimation of a random uniform variable by chaos polynomials between 1 and 2

Thanks
 
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  • #2
Change coordinates to obtain polynomials in a new variable w.
Let w = Ax + B and choose A and B so that w= 1 at x = -1 and w =2 at x = 1.
 
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  • #3
Stephen Tashi said:
Change coordinates to obtain polynomials in a new variable w.
Let w = Ax + B and choose A and B so that w= 1 at x = -1 and w =2 at x = 1.
Indeed this checks out with the numerical model estimation.
Thanks.
 

1. What are Legendre polynomials?

Legendre polynomials are a set of orthogonal polynomials that are defined on the interval (-1,1). They are named after the French mathematician Adrien-Marie Legendre and have many applications in mathematics, physics, and engineering.

2. How are Legendre polynomials defined on the interval (1,2)?

Legendre polynomials are typically defined on the interval (-1,1), but they can also be defined on any other interval (a,b) by using a linear transformation. In this case, they are defined on the interval (1,2) by using the transformation x = (2t-3)/2. This allows us to map the interval (1,2) onto the interval (-1,1).

3. What is the role of Legendre polynomials in solving differential equations?

Legendre polynomials are used to solve differential equations that have a separable solution. By expressing the solution as a series of Legendre polynomials, the differential equation can be reduced to a set of algebraic equations, making it easier to solve.

4. How do Legendre polynomials relate to other orthogonal polynomials?

Legendre polynomials are a special case of Jacobi polynomials, which are a generalization of Legendre polynomials. They are also closely related to Chebyshev polynomials, which are defined on the interval (-1,1) and have applications in numerical analysis and approximation theory.

5. What are some real-life applications of Legendre polynomials?

Legendre polynomials have many practical applications in physics, engineering, and statistics. They are used in solving problems related to heat flow, electrostatics, quantum mechanics, and signal processing. They also play a role in spherical harmonics, which are important in geophysics and astrophysics.

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