“Frobenius Method”的概念、定义、习题解题方法及翻译-数学辞典

单词

Frobenius Method

释义

Frobenius Method

If is an ordinary point of the Ordinary Differential Equation, expand in a Taylor Series about , letting

(1)

Plug

back into the ODE and group the Coefficients by Power. Now, obtain a Recurrence Relation for the

thterm, and write the Taylor Series in terms of the

s. Expansions for the first few derivatives are

			(2)
			(3)

			(4)

is a regular singular point of the Ordinary Differential Equation,

(5)

solutions may be found by the Frobenius method or by expansion in a Laurent Series. In the Frobenius method, assumea solution of the form

(6)

so that

			(7)
			(8)
			(9)

Now, plug

back into the ODE and group the Coefficients by Power to obtain a recursionFormula for the

th term, and then write the Taylor Series in terms of the

s. Equating the

term to0 will produce the so-called Indicial Equation, which will give the allowed values of

in the Taylor Series.

Fuchs's Theorem guarantees that at least one Power series solution will be obtained when applying the Frobeniusmethod if the expansion point is an ordinary, or regular, Singular Point. For a regularSingular Point, a Laurent Series expansion can also be used. Expand in aLaurent Series, letting

(10)

Plug

back into the ODE and group the Coefficients by Power. Now, obtain a recurrenceFormula for the

th term, and write the Taylor Expansion in terms of the

See also Fuchs's Theorem, Ordinary Differential Equation

References

Arfken, G. ``Series Solutions--Frobenius' Method.'' §8.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 454-467, 1985.

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