zbMATH — the first resource for mathematics

Recurrence triangle for Adomian polynomials. (English) Zbl 1190.65031
Summary: A recurrence technique for calculating Adomian polynomials is proposed, the convergence of the series for the Adomian polynomials is discussed, and the dependence of the convergent domain of the solution’s decomposition series \(\sum_{n=0}^\infty u_n\) on the initial component function \(u_{0}\) is illustrated. By introducing the index vectors of the Adomian polynomials the recurrence relations of the index vectors are discovered and the recurrence triangle is given. The method simplifies the computation of the Adomian polynomials. In order to obtain a solution’s decomposition series with larger domain of convergence, we illustrate by examples that the domain of convergence can be changed by choosing a different \(u_{0}\) and a modified iteration.

65D20 Computation of special functions and constants, construction of tables
33E20 Other functions defined by series and integrals
33F05 Numerical approximation and evaluation of special functions
Full Text: DOI
[1] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Dordrecht · Zbl 0802.65122
[2] Adomian, G., Nonlinear stochastic operator equations, (1986), Academic Orlando · Zbl 0614.35013
[3] Adomian, G., Nonlinear stochastic systems theory and applications to physics, (1989), Kluwer Academic Dordrecht · Zbl 0659.93003
[4] Gabet, L., The theoretical foundation of the Adomian method, Comput. math. appl., 27, 41-52, (1994) · Zbl 0805.65056
[5] Cherruault, Y.; Saccomandi, G.; Some, B., New results for convergence of adomian’s method applied to integral equations, Math. comput. modell., 16, 85-93, (1992) · Zbl 0756.65083
[6] Cherruault, Y.; Adomian, G.; Abbaoui, K.; Rach, R., Further remarks on convergence of decomposition method, Int. J. bio-med. comput., 38, 89-93, (1995)
[7] Abbaoui, K.; Cherruault, Y., Convergence of adomian’s method applied to differential equations, Comput. math. appl., 28, 103-109, (1994) · Zbl 0809.65073
[8] Rach, R., A new definition of the Adomian polynomials, Kybernetes, 37, 910-955, (2008) · Zbl 1176.33023
[9] Adomian, G.; Rach, R., Inversion of nonlinear stochastic operators, J. math. anal. appl., 91, 39-46, (1983) · Zbl 0504.60066
[10] Rach, R., A convenient computational form for the Adomian polynomials, J. math. anal. appl., 102, 415-419, (1984) · Zbl 0552.60061
[11] Seng, V.; Abbaoui, K.; Cherruault, Y., Adomian’s polynomials for nonlinear operators, Math. comput. modell., 24, 59-65, (1996) · Zbl 0855.47041
[12] Biazar, J.; Babolian, E.; Kember, G.; Nouri, A.; Islam, R., An alternate algorithm for computing Adomian polynomials in special cases, Appl. math. comput., 138, 523-529, (2003) · Zbl 1027.65076
[13] Zhu, Y.; Chang, Q.; Wu, S., A new algorithm for calculating Adomian polynomials, Appl. math. comput., 169, 402-416, (2005) · Zbl 1087.65528
[14] Wazwaz, A.M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. math. comput., 111, 53-69, (2000) · Zbl 1023.65108
[15] Abdelwahid, F., A mathematical model of Adomian polynomials, Appl. math. comput., 141, 447-453, (2003) · Zbl 1027.65072
[16] Babolian, E.; Javadi, Sh., New method for calculating Adomian polynomials, Appl. math. comput., 153, 253-259, (2004) · Zbl 1055.65068
[17] Gu, H.; Li, Z., A modified Adomian method for system of nonlinear differential equations, Appl. math. comput., 187, 748-755, (2007) · Zbl 1121.65082
[18] Biazar, J.; Shafiof, S.M., A simple algorithm for calculating Adomian polynomials, Int. J. contemp. math. sci., 2, 975-982, (2007) · Zbl 1145.65014
[19] Azreg-Aïnou, M., A developed new algorithm for evaluating Adomian polynomials, Comput. model. eng. sci., 42, 1-18, (2009) · Zbl 1357.65067
[20] Chen, W.; Lu, Z., An algorithm for Adomian decomposition method, Appl. math. comput., 159, 221-235, (2004) · Zbl 1062.65059
[21] Choi, H.W.; Shin, J.G., Symbolic implementation of the algorithm for calculating Adomian polynomials, Appl. math. comput., 146, 257-271, (2003) · Zbl 1033.65036
[22] Pourdarvish, A., A reliable symbolic implementation of algorithm for calculating Adomian polynomials, Appl. math. comput., 172, 545-550, (2006) · Zbl 1088.65021
[23] Rach, R., On the Adomian (decomposition) method and comparisons with picard’s method, J. math. anal. appl., 128, 480-483, (1987) · Zbl 0645.60067
[24] Bellomo, N.; Sarafyan, D., On adomian’s decomposition method and some comparisons with picard’s iterative scheme, J. math. anal. appl., 123, 389-400, (1987) · Zbl 0624.60079
[25] Wazwaz, A.M.; El-Sayed, S.M., A new modification of the Adomian decomposition method for linear and nonlinear operators, Appl. math. comput., 122, 393-405, (2001) · Zbl 1027.35008
[26] El-Wakil, S.A.; Abdou, M.A., New applications of variational iteration method using Adomian polynomials, Nonlinear dyn., 52, 41-49, (2008) · Zbl 1170.76356
[27] Podlubny, I., Fractional differential equations, (1999), Academic San Diego · Zbl 0918.34010
[28] Wazwaz, A.M., A reliable modification of Adomian decomposition method, Appl. math. comput., 102, 77-87, (1999) · Zbl 0928.65083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.