Derivatives of Legendre Series Expansions

[Vepomuk]

I am working on an advanced fundamental engineering theory. For that I need to solve a system of differential equations in R² by expanding my variables as Legendre series expansion.

Thus: u(x,y)=$$\sum$$$$\sum$$A_mnP_m(x)P_n(y)

The equations contain of each variable derivatives up to the fourth order. How would I go about this?

-The most straight forward way as I see it is to write out the series up to the point where I truncate and take derivatives, however that seems to be laborious, even if I use a program like Mathematica or Maple. Besides, it is "un-elegant"

-Then again, I could also write out the series up to truncation but leave the Polynomials as dⁿ/dxⁿ(P_m(x)). There are equations, amongst others by Brychkov, that give solutions to the derivatives of the polynomials directly. To my knowledge there are no such equations for the coefficients A_mn. Still, this feels not right.

I know my way with series expansions, but not this complicated with derivatives involved and in 2-D. Would anyone enlighten me with a push in the right direction?

 

[jvicens]

Hello, thank you for sharing your work and question with us. Solving a system of differential equations in R2 using Legendre series expansion is definitely a complex task. Here are some steps that you can follow to approach this problem:

1. First, make sure that you have a clear understanding of the problem and the equations involved. It might be helpful to write out the equations in full and break them down into smaller parts to make them easier to work with.

2. Familiarize yourself with the properties of Legendre polynomials and their series expansion. This will help you understand the equations and how to manipulate them.

3. As you mentioned, using a program like Mathematica or Maple can definitely make the process easier. These programs have built-in functions for calculating derivatives of Legendre polynomials, which can save you time and effort.

4. Consider using a technique called separation of variables to break down the system of differential equations into smaller, more manageable equations. This involves assuming that the solution can be written as a product of two functions, one dependent on x and the other on y.

5. Once you have separated the equations, you can use the series expansion for each variable and substitute it into the equations. This will give you a system of equations in terms of the coefficients Amn.

6. From here, you can use methods such as substitution or elimination to solve for the coefficients. Alternatively, you can use numerical methods or approximation techniques to solve for the coefficients.

7. Finally, once you have obtained the coefficients, you can substitute them back into the series expansion to obtain the solution for u(x,y).

I hope this helps guide you in the right direction. It may also be helpful to consult with other experts in the field for further guidance and insights. Best of luck with your work!