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[parent] Frobenius method (Topic)

Let us consider the linear homogeneous differential equation

$\displaystyle \sum_{\nu=0}^n k_\nu(x) y^{(n-\nu)}(x) = 0$
of order $ n$. If the coefficient functions $ k_\nu(x)$ are continuous and the coefficient $ k_0(x)$ of the highest order derivative does not vanish on a certain interval (resp. a domain in $ \mathbb{C}$), then all solutions $ y(x)$ are continuous on this interval (resp. domain). If all coefficients have the continuous derivatives up to a certain order, the same concerns the solutions.

If, instead, $ k_0(x)$ vanishes in a point $ x_0$, this point is in general a singular point. After dividing the differential equation by $ k_0(x)$ and then getting the form

$\displaystyle y^{(n)}(x)+\sum_{\nu=1}^n c_\nu(x)y^{(n-\nu)}(x) = 0,$
some new coefficients $ c_\nu(x)$ are discontinuous in the singular point. However, if the discontinuity is restricted so, that the products
$\displaystyle (x-x_0)c_1(x),\quad (x-x_0)^2c_2(x),\quad \ldots,\quad (x-x_0)^nc_n(x)$
are continuous, and even analytic in $ x_0$, the point $ x_0$ is a regular singular point of the differential equation.

We introduce the so-called Frobenius method for finding solution functions in a neighbourhood of the regular singular point $ x_0$, confining us to the case of a second order differential equation. When we use the quotient forms

$\displaystyle (x-x_0)c_1(x) := \frac{p(x)}{r(x)},\quad (x-x_0)^2c_2(x) := \frac{q(x)}{r(x)},$
where $ r(x)$, $ p(x)$ and $ q(x)$ are analytic in a neighbourhood of $ x_0$ and $ r(x) \neq 0$, our differential equation reads
$\displaystyle (x-x_0)^2r(x)y''(x)+(x-x_0)p(x)y'(x)+q(x)y(x) = 0.$ (1)

Since a simple change $ x\!-\!x_0\mapsto x$ of variable brings to the case that the singular point is the origin, we may suppose such a starting situation. Thus we can study the equation
$\displaystyle x^2r(x)y''(x)+xp(x)y'(x)+q(x)y(x) = 0,$ (2)

where the coefficients have the converging power series expansions
$\displaystyle r(x) = \sum_{n=0}^\infty r_nx^n,\quad p(x) = \sum_{n=0}^\infty p_nx^n,\quad q(x) = \sum_{n=0}^\infty q_nx^n$ (3)

and
$\displaystyle r_0 \neq 0.$
In the Frobenius method one examines whether the equation (2) allows a series solution of the form
$\displaystyle y(x) = x^s\sum_{n=0}^\infty a_nx^n = a_0x^s+a_1x^{s+1}+a_2x^{s+2}+\ldots,$ (4)

where $ s$ is a constant and $ a_0 \neq 0$.

Substituting (3) and (4) to the differential equation (2) converts the left hand side to

  $\displaystyle [r_0s(s\!-\!1)\!+\!p_0s\!+\!q_0]a_0x^s+$    
  $\displaystyle [[r_0(s\!+\!1)s\!+\!p_0(s\!+\!1)\!+\!q_0]a_1\!+\![r_1s(s\!-\!1)\!+\!p_1s\!+\!q_1]a_0]x^{s+1}+$    
  $\displaystyle [[r_0(s\!+\!2)(s\!+\!1)\!+\!p_0(s\!+\!2)\!+\!q_0]a_2\!+\![r_1(s\!... ...1(s\!+\!1)\!+\!q_1]a_1\!+\![r_2s(s\!-\!1)\!+\!p_2s\!+\!q_2]a_0]x^{s+2}\!+\ldots$    

Our equation seems clearer when using the notations $ f_\nu(s) := r_\nu{s}(s\!-\!1)+p_\nu{s}+q_nu$:
$\displaystyle f_0(s)a_0x^s+[f_0(s\!+\!1)a_1+f_1(s)a_0]x^{s+1}+[f_0(s\!+\!2)a_2+f_1(s\!+\!1)a_1+f_2(s)a_0]x^{s+2}+\ldots = 0$ (5)

Thus the condition of satisfying the differential equation by (4) is the infinite system of equations
\begin{align*}\begin{cases}f_0(s)a_0 = 0\\ f_0(s\!+\!1)a_1+f_1(s)a_0 = 0\\ f_0(s... ...a_1+f_2(s)a_0 = 0\\ \qquad\cdots\qquad\cdots\qquad\cdots \end{cases}\end{align*} (6)

In the first place, since $ a_0 \neq 0$, the indicial equation
$\displaystyle f_0(s) \equiv r_0s^2+(p_0-r_0)s+q_0 = 0$ (7)

must be satisfied. Because $ r_0 \neq 0$, this quadratic equation determines for $ s$ two values, which in special case may coincide.

The first of the equations (6) leaves $ a_0\,(\neq 0)$ arbitrary. The next linear equations in $ a_n$ allow to solve successively the constants $ a_1,\,a_2,\,\ldots$ provided that the first coefficients $ f_0(s\!+\!1)$, $ f_0(s\!+\!2),$$ \ldots$ do not vanish; this is evidently the case when the roots of the indicial equation don't differ by an integer (e.g. when the roots are complex conjugates or when $ s$ is the root having greater real part). In any case, one obtains at least for one of the roots of the indicial equation the definite values of the coefficients $ a_n$ in the series (4). It is not hard to show that then this series converges in a neighbourhood of the origin.

For obtaining the complete solution of the differential equation (2) it suffices to have only one solution $ y_1(x)$ of the form (4), because another solution $ y_2(x)$, linearly independent on $ y_1(x)$, is gotten via mere integrations; then it is possible in the cases $ s_1\!-\!s_2 \in\mathbb{Z}$ that $ y_2(x)$ has no expansion of the form (4).

Bibliography

1
PENTTI LAASONEN: Matemaattisia erikoisfunktioita. Handout No. 261. Teknillisen Korkeakoulun Ylioppilaskunta; Otaniemi, Finland (1969).



"Frobenius method" is owned by pahio.
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See Also: Fuchsian singularity, Bessel's equation

Other names:  method of Frobenius
Also defines:  indicial equation
Keywords:  Frobenius' method, series solution

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Cross-references: linearly independent, converges, real part, complex conjugates, integer, linear equations, quadratic equation, infinite, series, power series, equation, origin, variable, neighbourhood, regular singular point, analytic, products, discontinuous, differential equation, singular point, point, derivatives, solutions, interval, vanish, continuous, functions, coefficient, homogeneous differential equation
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This is version 13 of Frobenius method, born on 2008-01-07, modified 2008-09-06.
Object id is 10179, canonical name is FrobeniusMethod.
Accessed 1706 times total.

Classification:
AMS MSC: 34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions)
 15A06 (Linear and multilinear algebra; matrix theory :: Linear equations)
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