Frobenius method

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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 second-order ordinary differential equation of the form

z^2u''+p(z)zu'+q(z)u=0

with

u' \equiv {{d u} \over {d z}}   and   u'' \equiv {{d^2 u} \over {d z^2}}

in the vicinity of the regular singular point z=0. We can divide by z^2 to obtain a differential equation of the form

u''+{p(z) \over z}u'+{q(z) \over z^2}u = 0

which will not be solvable with regular power series methods if either p(z)/z or q(z)/z2 are not analytic at z = 0. The Frobenius method enables us 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[edit]

The Method of Frobenius tells us that we can seek a power series solution of the form

u(z)=\sum_{k=0}^\infty A_kz^{k+r}, \qquad (A_0 \neq 0)

Differentiating:

u'(z)=\sum_{k=0}^\infty (k+r)A_kz^{k+r-1}
u''(z)=\sum_{k=0}^\infty (k+r-1)(k+r)A_kz^{k+r-2}

Substituting:

 z^2\sum_{k=0}^\infty (k+r-1)(k+r)A_kz^{k+r-2} + zp(z) \sum_{k=0}^\infty (k+r)A_kz^{k+r-1} + q(z)\sum_{k=0}^\infty A_kz^{k+r}
 = \sum_{k=0}^\infty (k+r-1) (k+r)A_kz^{k+r} + p(z) \sum_{k=0}^\infty (k+r)A_kz^{k+r} + q(z) \sum_{k=0}^\infty A_kz^{k+r}
 = \sum_{k=0}^\infty (k+r-1)(k+r) A_kz^{k+r} + p(z) (k+r) A_kz^{k+r} + q(z) A_kz^{k+r}
 = \sum_{k=0}^\infty \left[(k+r-1)(k+r) + p(z)(k+r) + q(z)\right] A_kz^{k+r}
 = \left[ r(r-1)+p(z)r+q(z) \right] A_0z^r+\sum_{k=1}^\infty \left[ (k+r-1)(k+r)+p(z)(k+r)+q(z) \right] A_kz^{k+r}

The expression

r\left(r-1\right) + p\left(0\right)r + q\left(0\right) = I(r)

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. This detail is important to keep in mind because one can end up with complicated expressions 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. 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

I(k+r)A_k+\sum_{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \over (k-j)!}A_j,

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

I(k+r)A_k+\sum_{j=0}^{k-1} {(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \over (k-j)!} A_j=0
\sum_{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \over (k-j)!}A_j=-I(k+r)A_k
{1\over-I(k+r)}\sum_{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \over (k-j)!}A_j=A_k

The series solution with Ak above,

U_{r}(z)=\sum_{k=0}^{\infty}A_kz^{k+r}

satisfies

z^2U_{r}(z)''+p(z)zU_{r}(z)'+q(z)U_{r}(z)=I(r)z^r

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[edit]

Let us solve

z^2f''-zf'+(1-z)f=0\,

Divide throughout by z2 to give

f''-{1\over z}f'+{1-z \over z^2}f=f''-{1\over z}f'+\left({1\over z^2} - {1 \over z}\right) f = 0

which has the requisite singularity at z = 0.

Use the series solution

\begin{align}
f &= \sum_{k=0}^\infty A_kz^{k+r} \\
f' &= \sum_{k=0}^\infty (k+r)A_kz^{k+r-1} \\
f'' &= \sum_{k=0}^\infty (k+r)(k+r-1)A_kz^{k+r-2}
\end{align}

Now, substituting

\begin{align}
\sum_{k=0}^\infty &(k+r)(k+r-1) A_kz^{k+r-2}-\frac{1}{z} \sum_{k=0}^\infty (k+r)A_kz^{k+r-1} + \left(\frac{1}{z^2} - \frac{1}{z}\right) \sum_{k=0}^\infty A_kz^{k+r} \\
&= \sum_{k=0}^\infty (k+r)(k+r-1) A_kz^{k+r-2} -\frac{1}{z} \sum_{k=0}^\infty (k+r) A_kz^{k+r-1} +\frac{1}{z^2} \sum_{k=0}^\infty A_kz^{k+r} -\frac{1}{z} \sum_{k=0}^\infty A_kz^{k+r} \\
&= \sum_{k=0}^\infty (k+r)(k+r-1)A_kz^{k+r-2}-\sum_{k=0}^\infty (k+r)A_kz^{k+r-2}+\sum_{k=0}^\infty A_kz^{k+r-2}-\sum_{k=0}^\infty A_kz^{k+r-1} \\
&= \sum_{k=0}^\infty (k+r)(k+r-1)A_kz^{k+r-2}-\sum_{k=0}^\infty (k+r) A_kz^{k+r-2} + \sum_{k=0}^\infty A_kz^{k+r-2} - \sum_{k-1=0}^\infty A_{k-1}z^{k+r-2} \\
&= \sum_{k=0}^\infty (k+r)(k+r-1)A_kz^{k+r-2}-\sum_{k=0}^\infty (k+r)A_kz^{k+r-2}+\sum_{k=0}^\infty A_kz^{k+r-2}-\sum_{k=1}^\infty A_{k-1}z^{k+r-2} \\
&= \left \{ \sum_{k=0}^{\infty} \left ((k+r)(k+r-1) - (k+r) + 1\right ) A_kz^{k+r-2} \right \} -\sum_{k=1}^\infty A_{k-1}z^{k+r-2} \\
&= \left \{ \left ( r(r-1) - r +1 \right ) A_0 z^{r-2} + \sum_{k=1}^{\infty} \left ((k+r)(k+r-1) - (k+r) + 1\right ) A_kz^{k+r-2} \right \} - \sum_{k=1}^\infty A_{k-1}z^{k+r-2} \\
&= (r-1)^2  A_0 z^{r-2} + \left \{ \sum_{k=1}^{\infty} (k+r-1)^2 A_kz^{k+r-2} - \sum_{k=1}^\infty A_{k-1}z^{k+r-2} \right \} \\
&= (r-1)^2  A_0 z^{r-2} + \sum_{k=1}^{\infty} \left ( (k+r-1)^2 A_k - A_{k-1} \right ) z^{k+r-2}
\end{align}

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:

(k+1-1)^2 A_k - A_{k-1} =k^2A_k-A_{k-1}=0

hence we have the recurrence relation:

 A_k = \frac{A_{k-1}}{k^2}

Given some initial conditions, we can either solve the recurrence entirely or obtain a solution in power series form.

Since the ratio of coefficients A_k/A_{k-1} is a rational function, the power series can be written as a generalized hypergeometric series.

Z-separate roots[edit]

The previous example involved an indicial polynomial with a repeated root, which has 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 unique.

If the root is repeated, or the roots differ by an integer, then the second solution can be found by the equation:

 y_2 = C y_1 \ln x + \sum_{k=0}^\infty B_kx^{k+r_2}

Where y_1(x) is the first solution (based on the larger root in the case of unequal roots), r_2 is the smaller root, and the constant C and the coefficients B_k are to be determined.

In the particular case where the two roots are coincident the constant C is 1.

See also[edit]

External links[edit]