User:EGM6341.s11.team1.Chiu/PEA1 F09 Mtg44

Mtg 44: Tue, 08 Dec 09

[[media: Egm6321.f09.mtg44.djvu| page44-1]]

Rodrigues' formula for $$ \color{blue} P_{n}(x) \ $$ (Legendre)


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ P_{n}(x)= \frac{(-1)^{n}}{2^{n}n!} \frac{\mathrm{d}^{n} }{\mathrm{d} x^{n}} (1-x^{2})^{n} \ $$


 * }

Other "orthogonal" functions (in particular polynomial) Laguerre, Hermite, Jacobi $$ \to \ $$ Legendre, Chebyshev

Application: Interpretation, Integration, Solution of PDEs. Unified general theory of orthogonal functions. Note: Also includes Bessel functions. Definition: (after some transformation)

End Rodrigues' formula for $$ \color{blue} P_{n}(x) \ $$

Fundamental Equation of Hypergeometric Type (FEHT)


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ a_{2}(x)y''+a_{1}(x)y'+a_{o}y=0 \ $$


 * }

[[media: Egm6321.f09.mtg44.djvu| page44-2]]


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ a_{0} \in \mathbb{P}_{0} \ $$ constant


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ a_{1}(x) \in \mathbb{P}_{1} \ $$ linear


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ a_{2}(x) \in \mathbb{P}_{2} \ $$ quadratic


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

or: $$ a_{i}(x) \in \mathbb{P}_{i}, \ i=0,1,2 \ $$


 * }

End Fundamental Equation of Hypergeometric Type

Example Legendre equation


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ a_{2}(x)=1-x^{2} \ $$ $$ a_{1}(x)=-2x \ $$ $$ a_{0}=n(n+1) \ $$


 * }

End Example

A solution of FEHT is a function of HT (i.e., FHT)

Theorem 1:

All derivative of a FHT, which is a solution of a FEHT, is again a FHT.

End Theorem 1

Definition


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ v_{1}(x):=y^{(1)}(x) \equiv y'(x) \ $$ $$ v_{n}(x):=y^{(n)}(x) \ $$ $$ y \ $$ is a FHT, i.e., solution of a FEHT


 * }

[[media: Egm6321.f09.mtg44.djvu| page44-3]]


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ a_{2}^{(n)}v''_{n}+a_{1}^{(n)}v'_{n}+a_{0}^{(n)}v_{n}=0 \ $$


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ a_{2}^ \overset{ \color{blue} \overset{ \textbf{not} \ nth \ derivative}{\downarrow}}{(n)} \equiv a_{2} \in \mathbb{P}_{2} \ $$


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ a_{1}^{(n)}=na \overset{ \color{blue} \overset{ prime = 1st \ derivative }{\downarrow}}'_{2}+a_{1} \in \mathbb{P}_{1} \ $$


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ a_{0}^{(n)}=c_{2}a''_{2}+c_{1}a'_{1}+c_{0}a_{0} \in \mathbb{P}_{0} \ $$


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ c_{0}=1, \ c_{1}=n, \ c_{2}= \frac{1}{2} n(n-1) \ $$


 * }

End Theorem 1

Theorem 2: "Self-adjoint" form


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ (a_{2} \rho y')'+a_{0} \rho y=0 \ $$ at $$ (a_{2} \rho)'=a_{1} \rho \ $$


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ (a_{2}^{(n)} \rho_{n} v_{n}')'+a_{0}^{n} \rho_{n} v_{n}=0 \ $$ at $$ (a_{2}^{(n)} \rho_{n})'=a_{1}^{(n)} \rho_{n} \ $$


 * }

[[media: Egm6321.f09.mtg44.djvu| page44-4]]


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ ( \rho_{n+1}v_{n+1})'+a_{0}^{(n)}( \rho_{n}v_{n})=0 \ $$


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ y(x)= \frac{1}{A_{n} \rho(x)} \left [ (a_{2}(x))^{n} \rho(x) v_{n}(x) \right ]^ \overset{ \color{blue} \overset{nth \ derivative}{\downarrow}}{(n)} \ $$


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ A_{n}:=(-1)^{n} \prod_{k=0}^{n-1}a_{0}^{(k)} \in \mathbb{P}_{0} \ $$ constant


 * }

End Theorem 2

Theorem 3:


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

If $$ v_{n}(x)=y_{n}^{(n)}(x) \equiv \frac{\mathrm{d}^{n} }{\mathrm{d} x^{n}}y_{n}(x) \ $$ at $$ y_{n} \in \mathbb{P}_{n} \Rightarrow v_{n}(x)= \ $$constant


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

By Theorem 2: $$ y_{n}(x)= \frac{b_{0}}{A_{n} \rho(x)} \left [ (a_{2}(x))^{n} \rho(x) \right ]^\overset{ \color{blue} \overset{nth \ derivative}{\downarrow}} {(n)} \ $$


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ y_{n} \in \mathbb{P}_{n} \Rightarrow y_{n}(x)= \sum_{j=0}^{n} b_{j}x^{j} \ $$


 * }

End Theorem 3

Canonical forms of classical orthogonal polynomial

[[media: Egm6321.f09.mtg44.djvu| page44-5]]


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ \rho(x)=(1-x)^{\alpha}(1+x)^{\beta} \ $$ for $$ a_{2}(x)=1-x^{2} \ $$ Jacobi (Legendre, Chebyshev)


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ \rho(x)=x^{\alpha}e^{-x} \ $$ for $$ a_{2}(x)=x \ $$ laguerre


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

$$ \rho(x)=e^{-x^{2}} \ $$ for $$ a_{2}(x)=1 \ $$ Hermite


 * }


 * {| style="width:100%" border="0"


 * style="width:95%" |
 * style="width:95%" |

Legendre polynomial: $$ \alpha=\beta=0 \ $$


 * }

End Canonical forms of classical orthogonal polynomial