http://www.bio-physics.at/wiki/index.php?title=Special_Functions&feed=atom&action=history Special Functions - Revision history 2022-07-07T02:03:24Z Revision history for this page on the wiki MediaWiki 1.19.3 http://www.bio-physics.at/wiki/index.php?title=Special_Functions&diff=1600&oldid=prev Andreas Piehler: Created page with "By solving Partial Differential Equations by Separation of Variables, one reduces the PDE in several ODEs. Very common ODEs that originate from PDEs are special functi..." 2013-03-26T09:01:06Z <p>Created page with &quot;By solving <a href="/wiki/index.php?title=Partial_Differential_Equations" title="Partial Differential Equations">Partial Differential Equations</a> by <a href="/wiki/index.php?title=Separation_of_Variables" title="Separation of Variables">Separation of Variables</a>, one reduces the PDE in several ODEs. Very common ODEs that originate from PDEs are special functi...&quot;</p> <p><b>New page</b></p><div>By solving [[Partial Differential Equations]] by [[Separation of Variables]], one reduces the PDE in several ODEs. Very common ODEs that originate from PDEs are special functions like<br /> *[[Bessel's Differential Equation]]<br /> *[[Legendre's Differential Equation]]<br /> *Laguerre Differential Equation<br /> *Chebyshev Differential Equation <br /> <br /> This ODEs that they belong to a greater class of Oridinary Differential Equations, the so called Sturm-Liouville Differential Equations, thus the discussion of special functions will lead us to the Sturm-Liouville theory. <br /> <br /> Let us consider a general ODE of the form<br /> \begin{align}<br /> y''+r(x)y'+q(x)y+ \lambda \rho(x) y=0 \label{gode}<br /> \end{align}<br /> In analogy to the [[Integrierender Faktor|Integrating Factor]] method to solve ODEs, we multiply \ref{gode} by the Integrating Factor $u=e^{\int r(x) dx}$. <br /> \begin{align}<br /> u(x)y''+u(x) r(x)y'+u(x) q(x)y+ u(x)\lambda \rho(x) y=0 <br /> \end{align}<br /> Then we can write the same equation as<br /> \begin{align}<br /> (u(x)y')'+\underbrace{u(x) q(x)}_{\tilde{q}(x)}y+ \lambda \underbrace{u(x)\rho(x)}_{\tilde{\rho}(x)} y=0 <br /> \end{align}<br /> with $(u(x)y')'=u(x)y''+u(x) r(x)y'$ since $u'=r \, u$<br /> &lt;div style=&quot;background:#FEF5CA;border:1px solid #797979;border-radius:10px;padding:5px 15px 5px 15px;&quot;&gt;<br /> This is the '''Sturm-Liouville equation'''<br /> \begin{align}<br /> \left [ u(x)\frac{dy}{dx} \right]'+\tilde{q}(x)y+ \lambda \tilde{\rho}(x) y=0\\<br /> \end{align}<br /> &lt;/div&gt;<br /> Sturm-Liouville equations have the special property, that their eigenfunctions are orthogonal. This key-property allows to expand arbitrary functions from initial conditions into an [[Orthogonality|orthogonal series]]. In the following a few well known equations are represented in Sturm-Liouville form<br /> *Simple harmonic equation<br /> \begin{align}<br /> (y')'+\omega^2y=0\\<br /> \end{align}<br /> *Legendre's equation<br /> \begin{align}<br /> \left [ (1-x^2)\frac{dy}{dx} \right]'+\ell(\ell+1) y=0\\<br /> \end{align}<br /> *Bessel's equation<br /> \begin{align}<br /> \left [ \xi\frac{dy}{dx} \right]'+ \left( a^2 \xi - \frac{n^2}{\xi} \right) y=0\\<br /> \end{align}</div> Andreas Piehler