5.10 Some Applications of Legendre Polynomials

5.10 Some Applications of Legendre Polynomials

• In this section, we look at the details of the problems solved using Legendre polynomials.
• The problem was stated in the previous section.

• The eigenfunctions of Eq were found in Section 5.9.
• We have to solve Eq.

• The three-dimensional heat equation is satisfied by the temperature on a spherical shell.

• As found in the previous section.
• There is an animated version of the solution on the CD.

• The second problem is not as familiar.

• Problems in spherical coordinates are where these functions occur frequently.

• We can put together some solutions.

• We have obtained some information, so let's summarize it.

• Product solutions of the problem were found in Section 5.8, Part B.

• It's possible to use odd-order Legendre polynomials.

• Before attempting a solution, think about the situation.

• Nodal surfaces are surfaces where the function is 0 in Part C of this section.

• There is a potential problem in the exterior of a sphere.

• The tip is not straight.

• The variables are changed to simplify the problem.