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 formwith and .
z=0
One can divide by
z2
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.
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 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.
The method of Frobenius is to seek a power series solution of the form
Differentiating:
Substituting the above differentiation into our original ODE:
The expressionis known as the indicial polynomial, which is quadratic in r. The general definition of the indicial polynomial is the coefficient of the lowest power of z in the infinite series. In this case it happens to be that this is the rth coefficient but, it is possible for the lowest possible exponent to be r − 2, r − 1 or, something else depending on the given differential equation. This detail is important to keep in mind. In the process of synchronizing all the series of the differential equation to start at the same index value (which in the above expression is k = 1), one can end up with complicated expressions. However, in solving for the indicial roots attention is focused only on the coefficient of the lowest power of z.
Using this, the general expression of the coefficient of is
These coefficients must be zero, since they should be solutions of the differential equation, so
The series solution with above,satisfies
If we choose one of the roots to the indicial polynomial for r in, 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.
Let us solve
Divide throughout by z2 to givewhich has the requisite singularity at z = 0.
Use the series solution
Now, substituting
From we get a double root of 1. Using this root, we set the coefficient of to be zero (for it to be a solution), which gives us:hence we have the recurrence relation:
Given some initial conditions, we can either solve the recurrence entirely or obtain a solution in power series form.
Since the ratio of coefficients
Ak/Ak-1
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:where
y1(x)
r2
Bk
B0
Bk
B | |
r1-r2 |
Bk.
Example: consider the following differential equation (Kummer's equation with and):The roots of the indicial equation are −1 and 0. Two independent solutions are
1/z
ez/z,
(ez-1)/z
z-1
z0,
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. 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.
. Teschl . Gerald . Gerald Teschl . Ordinary Differential Equations and Dynamical Systems . . . 2012 . 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.