New ideas for decomposing nonlinearities in differential equations

In this paper we consider the decomposition for the nonlinearity in a differential equation for the solution by decomposition. By analyzing and transforming the Taylor expansion of the nonlinearity about the initial solution component, the decomposition of the nonlinearity is converted to the partitions of the solution sets for a class of Diophantine equations. This conversion simplifies the discussion and presents a new idea for decompositions. We enumerate five types of partitions and their corresponding decomposition polynomials. Each of the last four types contains infinitely many kinds of decomposition polynomials in the form of finite sums. In Types 2, 3 and 4, there is a parameter q and each value of q corresponds to a class of decomposition polynomials. In Type 5, each positive integer sequence {cj} satisfying 1 = c1 ⩽ c2 ⩽ ⋯ and j ⩽ cj for j = 2, 3, … corresponds to a class of decomposition polynomials. Four classes of the Adomian polynomials [R. Rach, A new definition of the Adomian polynomials, Kybernetes 37 (2008) 910–955] are derived as particular cases.