Frobenius method

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

to obtain a differential equation of the form

which will not be solvable with regular power series methods if either p(z)/z or q(z)/z2 is not analytic at z = 0.

The Frobenius method enables one to create a power series solution to such a differential equation, provided that p(z) and q(z) are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist (and are finite).

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.

The method of Frobenius is to seek a power series solution of the form

Substituting the above differentiation into our original ODE:

is 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.

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 zk + r is

These coefficients must be zero, since they should be solutions of the differential equation, so

If we choose one of the roots to the indicial polynomial for r in Ur(z), we gain a solution to the differential equation.

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.

is a rational function, the power series can be written as a generalized hypergeometric series.

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).

is the smaller root, and the constant C and the coefficients

the recurrence relation places no restriction on the coefficient for the term

If it is set to zero then with this differential equation all the other coefficients will be zero and we obtain the solution 1/z.

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.