Frobenius method

The Frobenius method , after Ferdinand Georg Frobenius , is a method for solving the ordinary differential equation

${\displaystyle u''+p(z)u'+q(z)u=0}$

to find where and as analytic in a neighborhood of is required. The idea is to provide solutions in the form of a generalized power series ${\displaystyle (z-z_{0})p(z)}$ ${\displaystyle (z-z_{0})^{2}q(z)}$ ${\displaystyle z=z_{0}}$

${\displaystyle u(z)=(z-z_{0})^{\alpha }\sum _{n=0}^{\infty }u_{n}(z-z_{0})^{n}}$

and to determine the unknown coefficients by comparing coefficients . The central theorem was first proven by Lazarus Immanuel Fuchs based on the work of Karl Weierstrass and then generalized by Frobenius. ${\displaystyle \alpha ,u_{n}}$

Theorem of fox

We can bet without restricting the general public . The differential equation is given ${\displaystyle z_{0}=0}$

${\displaystyle u''+p(z)u'+q(z)u=0}$

where 0 has a pole of maximum first order and 0 has a pole of maximum second order. So you can in the form ${\displaystyle p(z)}$ ${\displaystyle q(z)}$

${\displaystyle p(z)={\frac {1}{z}}\sum _{n=0}^{\infty }p_{n}z^{n},\qquad q(z)={\frac {1}{z^{2}}}\sum _{n=0}^{\infty }q_{n}z^{n}}$

with the rows converging in a neighborhood of 0.

The characteristic exponents

${\displaystyle \alpha _{1,2}={\frac {1}{2}}\left(1-p_{0}\pm {\sqrt {(p_{0}-1)^{2}-4q_{0}}}\right)}$

are the solutions of the characteristic equation

${\displaystyle \alpha ^{2}+(p_{0}-1)\alpha +q_{0}=0,}$

which results from the comparison of coefficients for in the above differential equation, ${\displaystyle z^{\alpha -2}}$

and we can arrange them accordingly . ${\displaystyle \mathrm {Re} (\alpha _{1})\geq \mathrm {Re} (\alpha _{2})}$

Then the following distinction applies:

• If not an integer, there are two solutions of the form ${\displaystyle \alpha _{1}-\alpha _{2}}$

${\displaystyle u_{j}(z)=z^{\alpha _{j}}\sum _{n=0}^{\infty }u_{j,n}z^{n},\qquad u_{j,0}=1,\quad j=1,2.}$

• If it is an integer, there are two solutions of the form ${\displaystyle \alpha _{1}-\alpha _{2}}$

${\displaystyle u_{1}(z)=z^{\alpha _{1}}\sum _{n=0}^{\infty }u_{1,n}z^{n},\qquad u_{2}(z)=z^{\alpha _{2}}\sum _{n=0}^{\infty }u_{2,n}z^{n}+c\log(z)u_{1}(z),\qquad u_{j,0}=1.}$

The radius of convergence corresponds to the minimum of the radius of convergence of the series for and . ${\displaystyle p(z)}$ ${\displaystyle q(z)}$

The converse also applies: If there are two solutions of the above form, then at 0 has a pole of maximum first order and at 0 has a pole of maximum second order. ${\displaystyle p(z)}$ ${\displaystyle q(z)}$

A differential equation with meromorphic coefficients for which all singularities (inclusive ) are of the above type is referred to as a Fuchs differential equation . ${\displaystyle \infty }$

Generalizations

Fuchs' theorem can be generalized to higher order differential equations and to systems of first order differential equations.

Applications

The following differential equations can be solved using Frobenius' method:

• Bessel's differential equation
• Legendre's differential equation
• Laguerre's differential equation
• hypergeometric differential equation

literature

• Gerald Teschl : Ordinary Differential Equations and Dynamical Systems . American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( free online version ).
• Wolfgang Walter : Ordinary differential equations . 7th edition. Springer, Berlin 2000, ISBN 3-540-67642-2 .

credentials

1. ↑ L. Fuchs: On the theory of linear differential equations with variable coefficients. In: Journal for pure and applied mathematics. 66 (1866) p. 121.
2. ↑ G. Frobenius: About the integration of the linear differential equations by series. In: Journal for pure and applied mathematics. 76 (1873), p. 214.