NISP

NISP Commit Details

Date:2015-03-27 23:59:44 (3 years 5 months ago)
Author:Michael Baudin
Commit:331
Parents: 330
Message:Added S-L equation.
Changes:
M/doc/polychaos/1-orthopoly.tex
M/doc/polychaos/5-particularorthopoly.tex

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doc/polychaos/5-particularorthopoly.tex
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\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}
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}

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Revision: 331