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In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form
in the vicinity of the regular singular point .
One can divide by to obtain a differential equation of the form
History: Frobenius' Actual Contributions
Frobenius' contribution[1] was not so much in all the possible forms of the series solutions involved (see below). These forms had all been established earlier,[2] by Fuchs.[3][4] The indicial polynomial (see below) and its role had also been established by Fuchs.[2]
A first contribution by Frobenius to the theory was to show that - as regards a first, linearly independent solution, which then has the form of an analytical power series multiplied by an arbitrary power r of the independent variable (see below) - the coefficients of the generalized power series obey a recurrence relation, so that they can always be straightforwardly calculated.
A second contribution by Frobenius was to show that, in cases in which the roots of the indicial equation differ by an integer, the general form of the second linearly independent solution (see below) can be obtained by a procedure which is based on differentiation[5] with respect to the parameter r, mentioned above.
A large part of Frobenius' 1873 publication[1] was devoted to proofs of convergence of all the series involved in the solutions, as well as establishing the radii of convergence of these series.
Explanation of Frobenius Method: first, linearly independent solution
The method of Frobenius is to seek a power series solution of the form
Differentiating:
Substituting the above differentiation into our original ODE:
The expression
Using this, the general expression of the coefficient of zk + r is
These coefficients must be zero, since they should be solutions of the differential equation, so
The series solution with Ak above,
If we choose one of the roots to the indicial polynomial for r in Ur(z), we gain a solution to the differential equation. If the difference between the roots is not an integer, we get another, linearly independent solution in the other root.
Example
Let us solve
Divide throughout by z2 to give
Use the series solution
Now, substituting
From (r − 1)2 = 0 we get a double root of 1. Using this root, we set the coefficient of zk + r − 2 to be zero (for it to be a solution), which gives us:
Given some initial conditions, we can either solve the recurrence entirely or obtain a solution in power series form.
Since the ratio of coefficients is a rational function, the power series can be written as a generalized hypergeometric series.
"The exceptional cases": roots separated by an integer
The previous example involved an indicial polynomial with a repeated root, which gives only one solution to the given differential equation. In general, the Frobenius method gives two independent solutions provided that the indicial equation's roots are not separated by an integer (including zero).
If the root is repeated or the roots differ by an integer, then the second solution can be found using:
Example: consider the following differential equation (Kummer's equation with a = 1 and b = 2):
Tandem Recurrence Relations for Series Coefficients in the Exceptional Cases
In cases in which roots of the indicial polynomial differ by an integer (including zero), the coefficients of all series involved in second linearly independent solutions can be calculated straightforwardly from tandem recurrence relations.[5] These tandem relations can be constructed by further developing Frobenius' original invention of differentiating with respect to the parameter r, and using this approach to actually calculate the series coefficients in all cases.[5]
See also
External links
- Weisstein, Eric W. "Frobenius Method". MathWorld.
- Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. (Draft version available online at https://www.mat.univie.ac.at/~gerald/ftp/book-ode/). Chapter 4 contains the full method including proofs.
References
- ^ a b Frobenius, Ferdinand Georg (1968) [Originally in Journal für die reine und angewandte Mathematik 76, 214-235 (1873)]. "Uber die Integration der linearen Differentialgleichungen durch Reihen". Gesammelte Abhandlungen (in German). Berlin: Springer-Verlag. pp. 84–105.
- ^ a b Gray, Jeremy (1986). Linear Differential Equations and Group Theory from Riemann to Poincare. Boston: Birkhauser. ISBN 0-8176-3318-9.
- ^ Fuchs, Lazarus Immanuel (1865). "Zur Theorie der linearen Differentialgleichungen mit veranderlichen Coefficienten". Gesammelte Mathematische Werke von L. Fuchs (in German). University Of Michigan Library.
- ^ Fuchs, Lazarus Immanuel (1866). "Zur Theorie der linearen Differentialgleichungen mit veranderlichen Coefficienten". Journal für die reine und angewandte Mathematik. 66: 159–204.
- ^ a b c van der Toorn, Ramses (27 December 2022). "Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points". Axioms. 12 (1): 32. doi:10.3390/axioms12010032. ISSN 2075-1680.