Frobenius method
The Frobenius method , after Ferdinand Georg Frobenius , is a method for solving the ordinary differential equation
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
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.
Theorem of fox
We can bet without restricting the general public . The differential equation is given
where 0 has a pole of maximum first order and 0 has a pole of maximum second order. So you can in the form
with the rows converging in a neighborhood of 0.
The characteristic exponents
are the solutions of the characteristic equation
which results from the comparison of coefficients for in the above differential equation,
and we can arrange them accordingly .
Then the following distinction applies:
- If not an integer, there are two solutions of the form
- If it is an integer, there are two solutions of the form
The radius of convergence corresponds to the minimum of the radius of convergence of the series for and .
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.
A differential equation with meromorphic coefficients for which all singularities (inclusive ) are of the above type is referred to as a Fuchs differential equation .
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
- ↑ L. Fuchs: On the theory of linear differential equations with variable coefficients. In: Journal for pure and applied mathematics. 66 (1866) p. 121.
- ↑ G. Frobenius: About the integration of the linear differential equations by series. In: Journal for pure and applied mathematics. 76 (1873), p. 214.