Eigenvalue problems for ordinary differential operators
by Aleksas Domarkas
Vilnius University, Faculty of Mathematics and Informatics,
Naugarduko 24, Vilnius, Lithuania
aleksas@ieva.mif.vu.lt
NOTE: In this session we solve eigenvalue problems for second order ordinary differential operators.
1. -y''= lambda*y
Problem
Please input number of examples n (1..4) and execute Section
Solving problem
Eigenvalues:
Eigenfunctions:
Example
Checking the Solution
Checking equation:
Checking boundary conditions:
2. -x^2*y''-y/4 = lambda*y
Solution
3. -x^2*y''-x*y'=lambda*y
Eigenfunctions(not normed):
4. -y''-2*y'=lambda*y
5. -y''-y'/x=lambda*y (Bessel functions)
Checking
Checking boundary condition:
6. Gegenbauer (ultraspherical) polynomials G(n,a,x)
Eigenfunctions we search in polynomials f, .
We assign for example :
Eigenfunctions is Gegenbauer polynomials:
and :
Other relations
1. Generating function for Gegenbauer polynomials is
2.
7. Hermite polynomials H(n,x)
Eigenfunctions we search in polynomials f, + ... .
Eigenfunctions is Hermite polynomials:
1.
:
Checking :
3. If x>=0 then
8. Laguerre polynomials L(n,x)
Eigenfunctions is Laguerre polynomials:
Tes t
9. Legendre polynomials P(n,x)
Eigenfunctions is Legendre polynomials:
1. Generating function for Legendre polynomials is :
10. Chebyshev polynomials T(n,x)
Eigenfunctions is Chebyshev polynomials:
1. Generating function for Chebyshev polynomials is :
11. Chebyshev polynomials of the second kind U(n,x)
Procedure
Eigenfunctions we search in polynomials f,
Eigenfunctions is Chebyshev polynomials of the second kind:
1. Generating function for Chebyshev polynomials of the second kind is :
3.
12. Associated Legendre functions
Eigenfunctions are associated Legendre functions
, , ...
where Legendre polynomials and
We define :
For m=0 :
For m=1
For m=2
For m=3
For m=4
Generating function
Checking for m=1
Checking for m=2
Note
In some references associated Legendre functions are defined by
see in Maple ?LegendreP, in Mathematica LegendreP[n,m,x] .
Examples with associated Legendre functions see in elliptic3d.mws
While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.
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