![Singular point with indicial roots one half and negative one](https://upload.wikimedia.org/wikipedia/commons/9/9f/Singular_point_with_indicial_roots_one_half_and_negative_one.png)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the method of Frobenius, named after
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous ...
, is a way to find an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
solution for a second-order
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
of the form
with
and
.
in the vicinity of the
regular singular point
In mathematics, in the theory of ordinary differential equations in the complex plane \Complex, the points of \Complex are classified into ''ordinary points'', at which the equation's coefficients are analytic functions, and ''singular points'', at ...
.
One can divide by
to obtain a differential equation of the form
which will not be solvable with regular
power series methods if either or are not
analytic at . 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).
Explanation
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
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 ''r''th 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.
Example
Let us solve
Divide throughout by ''z''
2 to give
which 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
is a
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
, the power series can be written as a
generalized hypergeometric series
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...
.
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:
where
is the first solution (based on the larger root in the case of unequal roots),
is the smaller root, and the constant and the coefficients
are to be determined. Once
is chosen (for example by setting it to 1) then and the
are determined up to but not including
, which can be set arbitrarily. This then determines the rest of the
In some cases the constant must be zero. For example, consider the following differential equation (
Kummer's equation
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
with and ):
The roots of the indicial equation are −1 and 0. Two independent solutions are
and
so we see that the logarithm does not appear in any solution. The solution
has a power series starting with the power zero. In a power series starting with
the recurrence relation places no restriction on the coefficient for the term
which can be set arbitrarily. If it is set to zero then with this differential equation all the other coefficients will be zero and we obtain the solution .
See also
*
Fuchs' theorem
In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form
y'' + p(x)y' + q(x)y = g(x)
has a solution expressible by a generalised Frobenius series when p(x), q(x) and g(x) are analytic ...
*
Regular singular point
In mathematics, in the theory of ordinary differential equations in the complex plane \Complex, the points of \Complex are classified into ''ordinary points'', at which the equation's coefficients are analytic functions, and ''singular points'', at ...
*
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
External links
*
* {{cite book
, surname = Teschl
, given = Gerald
, authorlink=Gerald Teschl
, title = Ordinary Differential Equations and Dynamical Systems
, publisher=
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, place =
Providence
Providence often refers to:
* Providentia, the divine personification of foresight in ancient Roman religion
* Divine providence, divinely ordained events and outcomes in Christianity
* Providence, Rhode Island, the capital of Rhode Island in the ...
, year = 2012
, isbn= 978-0-8218-8328-0
, url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ (Draft version available online at https://www.mat.univie.ac.at/~gerald/ftp/book-ode/). Chapter 4 contains the full method including proofs.
Ordinary differential equations