Frobenius Method
If x_0 is an ordinary point of the ordinary differential equation, expand y in a Taylor series about x_0. Commonly, the expansion point can be taken as x_0=0, resulting in the Maclaurin series
| [画像: y=sum_(n=0)^inftya_nx^n. ] |
(1)
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Plug y back into the ODE and group the coefficients by power. Now, obtain a recurrence relation for the nth term, and write the series expansion in terms of the a_ns. Expansions for the first few derivatives are
If x_0 is a regular singular point of the ordinary differential equation,
| P(x)y^('')+Q(x)y^'+R(x)y=0, |
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solutions may be found by the Frobenius method or by expansion in a Laurent series. In the Frobenius method, assume a solution of the form
so that
Now, plug y back into the ODE and group the coefficients by power to obtain a recursion formula for the a_nth term, and then write the series expansion in terms of the a_ns. Equating the a_0 term to 0 will produce the so-called indicial equation, which will give the allowed values of k in the series expansion.
As an example, consider the Bessel differential equation
Plugging (◇) into (◇) yields
The indicial equation, obtained by setting n=0, is then
| a_0[k(k-1)+k-m^2]=a_0(k^2-m^2)=0. |
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Since a_0 is defined as the first nonzero term, k^2-m^2=0, so k=+/-m. For illustration purposes, ignore k=-m and consider only the case k=m (avoiding the special case m!=1/2), then equation (14) requires that
| a_1(2m+1)=0 |
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(so a_1=0) and
| [a_nn(2m+n)+a_(n-2)]x^(m+n)=0 |
(17)
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for n=2, 3, ..., so
| [画像: a_n=-1/(n(2m+n))a_(n-2) ] |
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for n>1. Plugging back in to (◇), rearranging, and simplifying then gives the series solution that defined the Bessel function of the first kind J_m(x), which is the nonsingular solution to (◇). (Considering the case m=-k proceeds analogously and results in the solution J_(-m)(x)=(-1)^mJ_m(x).)
Fuchs's theorem guarantees that at least one power series solution will be obtained when applying the Frobenius method if the expansion point is an ordinary, or regular, singular point. For a regular singular point, a Laurent series expansion can also be used. Expand y in a Laurent series, letting
| y=c_(-n)x^(-n)+...+c_(-1)x^(-1)+c_0+c_1x+...+c_nx^n+.... |
(19)
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Plug y back into the ODE and group the coefficients by power. Now, obtain a recurrence formula for the c_nth term, and write the Taylor series in terms of the c_ns.
See also
Fuchs's Theorem, Ordinary Differential EquationExplore with Wolfram|Alpha
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References
Arfken, G. "Series Solutions--Frobenius' Method." §8.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 454-467, 1985.Frobenius. "Ueber die Integration der linearen Differentialgleichungen durch Reihen." J. reine angew. Math. 76, 214-235, 1873.Ince, E. L. Ch. 5 in Ordinary Differential Equations. New York: Dover, 1956.Referenced on Wolfram|Alpha
Frobenius MethodCite this as:
Weisstein, Eric W. "Frobenius Method." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FrobeniusMethod.html