# NISP Commit Details

Date: 2015-03-27 23:59:44 (3 years 8 months ago) Michael Baudin 331 330 Added S-L equation.

## File differences

doc/polychaos/1-orthopoly.tex
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 which leads to \ref{eq-weightsjacobi3}.␍␊ \end{proof}␍␊ ␍␊ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%␍␊ ␍␊ \subsection{Sturm-Liouville equations}␍␊ \label{sec-Sturm}␍␊ ␍␊ \begin{definition}␍␊ (\emph{Sturm-Liouville equation})␍␊ \label{def-Sturm}␍␊ Let $p$, $q$, and $w$ be functions on $\RR$, with $w(x)>0$, for ␍␊ any $x\in \RR$. ␍␊ The differential equation ␍␊ \begin{eqnarray*}␍␊ \frac{d}{dx} \left( p(x)\frac{dy}{dx}\right) + q(x)y = -\lambda w(x) y,␍␊ \end{eqnarray*}␍␊ where $y$ is a function of $x$ is called a Sturm-Liouville ␍␊ equation. ␍␊ \end{definition}␍␊ ␍␊ Let $L$ be the differential operator defined by :␍␊ \begin{eqnarray*}␍␊ L = \frac{1}{w(x)} \left(- \frac{d}{dx} \left( p(x)\frac{d}{dx}\right) + q(x)\right).␍␊ \end{eqnarray*}␍␊ The operator $L$ is called Sturm-Liouville operator. ␍␊ The S-L differential equation is the eigenvalue equation~:␍␊ \begin{eqnarray*}␍␊ Ly = \lambda y.␍␊ \end{eqnarray*}␍␊ ␍␊ Provided suitable boundary conditions are chosen, the ␍␊ S-L operator $L$ is symmetric (or self-adjoint or Hermitian, ␍␊ if complex functions are allowed), i.e. :␍␊ \begin{eqnarray*}␍␊ (Lf,g)=(f,Lg),␍␊ \end{eqnarray*}␍␊ for any functions $f$ and $g$ in the weighted Hilbert space. ␍␊ ␍␊ In practice, we must prescribe some appropriate boundary conditions to the ␍␊ Sturm-Liouville (S-L) equation. ␍␊ If a value $\lambda$ exists, then it is an eigenvalue of the ␍␊ S-L equation. ␍␊ Under some hypotheses, the eigenvalues $\lambda_1,\lambda_2,...$ of the ␍␊ S-L problem are real. ␍␊ For each eigenvalue $\lambda_n$, the solution $y_n(x)$ is the ␍␊ corresponding eigenvector function. ␍␊ The normalized eigenfunctions form an orthonormal basis in the ␍␊ weighted Hilbert space associated with the scalar product :␍␊ \begin{eqnarray*}␍␊ \int_\RR y_n(x) y_m(x) w(x) dx = \delta_{nm},␍␊ \end{eqnarray*}␍␊ for $n,m\geq 1$.␍␊ ␍␊ As we are going to see, all orthogonal polynomials are associated ␍␊ with a S-L equation. ␍␊ ␍␊ % http://www.iitg.ernet.in/physics/fac/charu/courses/ph402/SturmLiouville.pdf␍␊ ␍␊ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%␍␊ ␍␊ \subsection{Notes and references}␍␊
doc/polychaos/5-particularorthopoly.tex
 299 299 300 300 301 301 302 303 304 305 306 307 308 309 310 311 312 302 313 303 314 304 315 ... ... 551 562 552 563 553 564 565 566 567 568 569 570 571 572 573 554 574 555 575 556 576 ... ... 834 854 835 855 836 856 857 858 859 860 861 862 863 864 865 837 866 838 867 839 868 ... ... 1171 1200 1172 1201 1173 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1174 1213 1175 1214 1176 1215
 \end{eqnarray*}␍␊ where we have used the Gaussian integral proved in the section \ref{sec-gaussinteg}.␍␊ \end{proof}␍␊ ␍␊ Hermite polynomials are solutions of the following differential equation :␍␊ \begin{eqnarray*}␍␊ y'' - xy' = -n y.␍␊ \end{eqnarray*}␍␊ This leads to the following S-L equation :␍␊ \begin{eqnarray*}␍␊ \left(e^{-\frac{x^2}{2}}y'\right)' =- n e^{-\frac{x^2}{2}} y.␍␊ \end{eqnarray*}␍␊ The eigenvalue is $\lambda=n$, for $n=0,1,...$.␍␊ ␍␊ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%␍␊ ␍␊ \subsection{Legendre polynomials}␍␊ and concludes the proof.␍␊ \end{proof}␍␊ ␍␊ Legendre polynomials are solutions of the following differential equation :␍␊ \begin{eqnarray*}␍␊ (1-x^2) y'' -2xy' = - n(n+1) y.␍␊ \end{eqnarray*}␍␊ The associated S-L equation is :␍␊ \begin{eqnarray*}␍␊ \left((1-x^2) y'\right)' = -n(n+1) y.␍␊ \end{eqnarray*}␍␊ ␍␊ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%␍␊ ␍␊ \subsection{Laguerre polynomials}␍␊ which concludes the second part of the proof.␍␊ \end{proof}␍␊ ␍␊ Laguerre polynomials are solutions of the following differential equation :␍␊ \begin{eqnarray*}␍␊ x y'' + (1-x)y' = - n y .␍␊ \end{eqnarray*}␍␊ The associated S-L equation is :␍␊ \begin{eqnarray*}␍␊ \left(xe^{-x} y'\right)' = -ne^{-x} y.␍␊ \end{eqnarray*}␍␊ ␍␊ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%␍␊ ␍␊ \subsection{Chebyshev polynomials of the first kind}␍␊ for $k=1,...,n$, which concludes the proof.␍␊ \end{proof}␍␊ ␍␊ Chebyshev polynomials are solutions of the following differential equation :␍␊ \begin{eqnarray*}␍␊ (1-x^2) y'' -xy' = - n^2 y,␍␊ \end{eqnarray*}␍␊ for $x\in[-1,1]$.␍␊ The associated S-L equation is :␍␊ \begin{eqnarray*}␍␊ \left(\frac{1}{\sqrt{1-x^2}} y'\right)' = -\frac{n^2}{\sqrt{1-x^2}}y.␍␊ \end{eqnarray*}␍␊ ␍␊ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%␍␊ ␍␊ \subsection{Accuracy of evaluation of the polynomial}␍␊

Revision: 331