The noisy convergence phenomena in decomposition method solutions
Journal of Computational and Applied Mathematics
Differential equations with singular coefficients
Applied Mathematics and Computation
Necessary conditions for the appearance of noise terms in decomposition solutions series
Applied Mathematics and Computation
Numerical integration, analytic continuation, and decomposition
Applied Mathematics and Computation
A comparison between Adomian decomposition method and Taylor series method in the series solutions
Applied Mathematics and Computation
A reliable modification of Adomian decomposition method
Applied Mathematics and Computation
A new algorithm for calculating Adomian polynomials for nonlinear operators
Applied Mathematics and Computation
Applied Mathematics and Computation
A mathematical model of Adomian polynomials
Applied Mathematics and Computation
A new algorithm for the decomposition solution of nonlinear differential equations
Computers & Mathematics with Applications
Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation
Computers & Mathematics with Applications
International Journal of Computer Mathematics
Linear and Nonlinear Integral Equations: Methods and Applications
Linear and Nonlinear Integral Equations: Methods and Applications
Mathematical and Computer Modelling: An International Journal
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In this article, we present new algorithms for the nonclassic Adomian polynomials, which are valuable for solving a wide range of nonlinear functional equations by the Adomian decomposition method, and introduce their symbolic implementation in MATHEMATICA. Beginning with Rach's new definition of the Adomian polynomials, we derive the explicit expression for each class of the Adomian polynomials, e.g. A"m=@?"k"="1^mf^(^k^)(u"0)Z"m","k for the Class II, III and IV Adomian polynomials, where the Z"m","k are called the reduced polynomials. These expressions provide a basis for developing improved algorithmic approaches. By introducing the index vectors, the recurrence algorithms for the reduced polynomials are suitably deduced, which naturally lead to new recurrence algorithms for the Class II and Class III Adomian polynomials. MATHEMATICA programs generating these classes of Adomian polynomials are subsequently presented. Computation shows that for computer generation of the Class III Adomian polynomials, the new algorithm reduces the running times compared with the definitional formula. We also consider the number of summands of these classes of Adomian polynomials and obtain the corresponding formulas. Finally, we demonstrate the versatility of the four classes of Adomian polynomials with several examples, which include the nonlinearity of the form f(t,u), explicitly depending on the argument t.