Is there any way one could apply Gram-Schmidt Orthonormalization in finding solutions to differential equations Iteratively?
Do you mean expressing the solution in an orthonormal base ? If yes, this is the starting point of the theory of orthogonal polynomials in L2. For instance, in order to solve the ODE for prolate spheroidal functions, you decompose the corresponding solns into Legendre polynomials, which are solns of a related (but simpler) ODE.
Dear @Soh, Gram-Schmidt Orthogonalization Process is a process that is used to find a set of orthogonal polynomials on a closed interval [a,b] with respect to a weight function w(x); thereafter, these polynomials can be used in the solution of an ODE or the least-squares approximation or any other application. In solving ODEs, the solution is written in terms of the orthogonal polynomials, substituted in the ODE, and thereafter using the orthogonality property to find the coefficients of the solution as explained by dear @David.
Thank you all for the question and the comments. I used Schmidt Orthogonalization procedure. I can suggest to consult my paper appear at JPB vol. 35, p.3365-3376 (2002); Pramana J. Phys. vol.63, p.1065-1072 (2004);
In addition please search if details are available in the papers:
* Astrophysics & Space Science, 283, 415 (2003).
* Term values of Na-like iron group highly stripped ions using coupled-cluster theory, Hasi Ray, J. Phys. B Lett. 35, L299 (2002).
* Studies on E2-transition in Na-like iron group ion:CoXVII, Hasi Ray, Astronomy and Astrophysics 391, 1173 (2002).
* Studies on E2-transitions in Na-like highly stripped ion:FeXVI, Hasi Ray, Astrophysical Journal 579, 914 (2002).
I have used Schmidt Orthogonalization produre to form the basis-orbitals for atomic wavefunction of a many-electron system.
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