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In this article we present four analytic recurrence algorithms for the multivariable Adomian polynomials. As special cases, we deduce the four simplified results for the one-variable Adomian polynomials. These algorithms are comprised of simple, orderly and analytic recurrence formulas, which do not require time-intensive operations such as expanding, regrouping, parametrization, and so on. They are straightforward to implement in any symbolic software, and are shown to be very efficient by our verification using MATHEMATICA 7.0. We emphasize that from the summation expressions, An ¼ Pn k¼1Uk n for the multivariable Adomian polynomials and An = Pnk ¼1f ðkÞðu0ÞCk n for the one-variable Adomian polynomials, we obtain the recurrence formulas for the Uk n and the Ck n. These provide a theoretical basis for developing new algorithmic approaches such as for parallel computing. In particular, the recurrence process of one particular algorithm for the one-variable Adomian polynomials does not involve the differentiation operation, but significantly only the arithmetic operations of multiplication and addition are involved; precisely C1n ¼ un ðn P 1Þ and Ck n ¼ 1 n Pnk j¼0 ðj þ 1Þujþ1Ck1 n1j ð2 6 k 6 nÞ. We also discuss several other algorithms previously reported in the literature, including the Adomian–Rach recurrence algorithm [1] and this author’s index recurrence algorithm [23,36].
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