Tuesday, February 2, 2010

Frobenius Series Solution

Consider the second order linear differential equation

(1) [Graphics:Images/FrobeniusSeriesMod_gr_1.gif].

Rewrite this equation in the form [Graphics:Images/FrobeniusSeriesMod_gr_2.gif], then use the substitutions [Graphics:Images/FrobeniusSeriesMod_gr_3.gif] and [Graphics:Images/FrobeniusSeriesMod_gr_4.gif] and rewrite the differential equation (1) in the form

(2) [Graphics:Images/FrobeniusSeriesMod_gr_5.gif].

Definition (Analytic):

The functions [Graphics:Images/FrobeniusSeriesMod_gr_6.gif] and [Graphics:Images/FrobeniusSeriesMod_gr_7.gif] are analytic at [Graphics:Images/FrobeniusSeriesMod_gr_8.gif] if they have Taylor series expansions with radius of convergence [Graphics:Images/FrobeniusSeriesMod_gr_9.gif] and [Graphics:Images/FrobeniusSeriesMod_gr_10.gif], respectively. That is

[Graphics:Images/FrobeniusSeriesMod_gr_11.gif] which converges for [Graphics:Images/FrobeniusSeriesMod_gr_12.gif]
and
[Graphics:Images/FrobeniusSeriesMod_gr_13.gif] which converges for [Graphics:Images/FrobeniusSeriesMod_gr_14.gif]

Definition (Ordinary Point):

If the functions [Graphics:Images/FrobeniusSeriesMod_gr_15.gif] and [Graphics:Images/FrobeniusSeriesMod_gr_16.gif] are analytic at [Graphics:Images/FrobeniusSeriesMod_gr_17.gif], then the point [Graphics:Images/FrobeniusSeriesMod_gr_18.gif] is called an ordinary point of the differential equation

[Graphics:Images/FrobeniusSeriesMod_gr_19.gif].

Otherwise, the point [Graphics:Images/FrobeniusSeriesMod_gr_20.gif] is called a singular point.

Definition (Regular Singular Point):

Assume that [Graphics:Images/FrobeniusSeriesMod_gr_21.gif] is a singular point of (1) and that [Graphics:Images/FrobeniusSeriesMod_gr_22.gif] and [Graphics:Images/FrobeniusSeriesMod_gr_23.gif] are analytic at [Graphics:Images/FrobeniusSeriesMod_gr_24.gif].

They will have Maclaurin series expansions with radius of convergence [Graphics:Images/FrobeniusSeriesMod_gr_25.gif] and [Graphics:Images/FrobeniusSeriesMod_gr_26.gif], respectively. That is

[Graphics:Images/FrobeniusSeriesMod_gr_27.gif] which converges for [Graphics:Images/FrobeniusSeriesMod_gr_28.gif]
and
[Graphics:Images/FrobeniusSeriesMod_gr_29.gif] which converges for [Graphics:Images/FrobeniusSeriesMod_gr_30.gif]

Then the point [Graphics:Images/FrobeniusSeriesMod_gr_31.gif] is called a regular singular point of the differential equation (1).

Method of Frobenius:

This method is attributed to the german mathemematican Ferdinand Georg Frobenius (1849-1917 ). Assume that [Graphics:Images/FrobeniusSeriesMod_gr_32.gif] is regular singular point of the differential equation

[Graphics:Images/FrobeniusSeriesMod_gr_33.gif].


A Frobenius series (generalized Laurent series) of the form

[Graphics:Images/FrobeniusSeriesMod_gr_34.gif]

can be used to solve the differential equation. The parameter [Graphics:Images/FrobeniusSeriesMod_gr_35.gif] must be chosen so that when the series is substituted into the D.E. the coefficient of the smallest power of [Graphics:Images/FrobeniusSeriesMod_gr_36.gif] is zero. This is called the indicial equation. Next, a recursive equation for the coefficients is obtained by setting the coefficient of [Graphics:Images/FrobeniusSeriesMod_gr_37.gif] equal to zero. Caveat: There are some instances when only one Frobenius solution can be constructed.

Definition (Indicial Equation):

The parameter [Graphics:Images/FrobeniusSeriesMod_gr_38.gif] in the Frobenius series is a root of the indicial equation

[Graphics:Images/FrobeniusSeriesMod_gr_39.gif].

Assuming that the singular point is [Graphics:Images/FrobeniusSeriesMod_gr_40.gif], we can calculate [Graphics:Images/FrobeniusSeriesMod_gr_41.gif] as follows:

[Graphics:Images/FrobeniusSeriesMod_gr_42.gif]
and
[Graphics:Images/FrobeniusSeriesMod_gr_43.gif]

The Recursive Formulas:

For each root [Graphics:Images/FrobeniusSeriesMod_gr_65.gif] of the indicial equation, recursive formulas are used to calculate the unknown coefficients [Graphics:Images/FrobeniusSeriesMod_gr_66.gif]. This is custom work because a numerical value for [Graphics:Images/FrobeniusSeriesMod_gr_67.gif] is easier use.

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Introductory Methods of Numerical Analysis