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An efficient algorithm for the multivariable Adomian polynomials. (English) Zbl 1204.65022
The Adomian decomposition methods are efficient techniques for solving nonlinear functional equations. This problem appears in coupled nonlinear differential equations \(Lu+Ru+Nu=g(t)\), where \(L+R\) is the linear part and \(N\) is a nonlinear operator. The method consists in the decomposition of \(Nu=f(u)\) in the series of Adomian polynomials \(A_n=\frac{1}{n!}\frac{d^n}{d\lambda_n}[f(\sum^\infty_{n=0}u_n\lambda^n)]_{\lambda=0}\) depending of certain initial solutions \(u_i\). The author gives an algorithm for rapid generation of the multivariable polynomials in question and tests it with a MATHEMATICA program.

MSC:
65D20 Computation of special functions and constants, construction of tables
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A34 Nonlinear ordinary differential equations and systems, general theory
65L05 Numerical methods for initial value problems
12Y05 Computational aspects of field theory and polynomials (MSC2010)
Software:
Mathematica
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References:
[1] Adomian, G., Stochastic systems, (1983), Academic New York · Zbl 0504.60067
[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] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Dordrecht · Zbl 0802.65122
[5] Wazwaz, A.M., Partial differential equations and solitary waves theory, (2009), Higher Education Press Beijing · Zbl 1175.35001
[6] Rach, R.; Adomian, G.; Meyers, R.E., A modified decomposition, Comput. math. appl., 23, 17-23, (1992) · Zbl 0756.35013
[7] Wazwaz, A.M., A reliable modification of Adomian decomposition method, Appl. math. comput., 102, 77-87, (1999) · Zbl 0928.65083
[8] 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
[9] Fang, J.Q.; Yao, W.G., Inverse operator method for solutions of nonlinear dynamical equations and some typical applications, Acta phys. sin., 42, 1375-1384, (1993) · Zbl 0837.65072
[10] Somali, S.; Gokmen, G., Adomian decomposition method for nonlinear sturm – liouville problems, Surv. math. appl., 2, 11-20, (2007) · Zbl 1132.34333
[11] Soliman, A.A.; Abdou, M.A., The decomposition method for solving the coupled modified KdV equations, Math. comput. model., 47, 1035-1041, (2008) · Zbl 1144.65318
[12] Kaya, D.; El-Sayed, S.M., Adomian’s decomposition method applied to systems of nonlinear algebraic equations, Appl. math. comput., 154, 487-493, (2004) · Zbl 1058.65056
[13] Hosseini, M.M., Adomian decomposition method for solution of nonlinear differential algebraic equations, Appl. math. comput., 181, 1737-1744, (2006) · Zbl 1106.65071
[14] Biazar, J., Solution of systems of integral-differential equations by Adomian decomposition method, Appl. math. comput., 168, 1232-1238, (2005) · Zbl 1082.65594
[15] Ray, S.S.; Bera, R.K., An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. math. comput., 167, 561-571, (2005) · Zbl 1082.65562
[16] Momani, S., An explicit and numerical solutions of the fractional KdV equation, Math. comput. simul., 70, 110-118, (2005) · Zbl 1119.65394
[17] Daftardar-Gejji, V.; Bhalekar, S., Solving multi-term linear and nonlinear diffusion-wave equations of fractional order by Adomian decomposition method, Appl. math. comput., 202, 113-120, (2008) · Zbl 1147.65106
[18] Cherruault, Y., Convergence of adomian’s method, Kybernetes, 18, 31-38, (1989) · Zbl 0697.65051
[19] Gabet, L., The theoretical foundation of the Adomian method, Comput. math. appl., 27, 41-52, (1994) · Zbl 0805.65056
[20] Abbaoui, K.; Cherruault, Y., Convergence of adomian’s method applied to differential equations, Comput. math. appl., 28, 103-109, (1994) · Zbl 0809.65073
[21] Abbaoui, K.; Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. math. appl., 29, 103-108, (1995) · Zbl 0832.47051
[22] Rach, R., A new definition of the Adomian polynomials, Kybernetes, 37, 910-955, (2008) · Zbl 1176.33023
[23] Adomian, G.; Rach, R., Inversion of nonlinear stochastic operators, J. math. anal. appl., 91, 39-46, (1983) · Zbl 0504.60066
[24] Rach, R., A convenient computational form for the Adomian polynomials, J. math. anal. appl., 102, 415-419, (1984) · Zbl 0552.60061
[25] Adomian, G.; Rach, R., Generalization of Adomian polynomials to functions of sever al variables, Comput. math. appl., 24, 11-24, (1992) · Zbl 0765.34005
[26] Abbaoui, K.; Cherruault, Y.; Seng, V., Practical formulae for the calculus of multivariable Adomian polynomials, Math. comput. model., 22, 89-93, (1995) · Zbl 0830.65010
[27] Wazwaz, A.M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. math. comput., 111, 53-69, (2000) · Zbl 1023.65108
[28] Abdelwahid, F., A mathematical model of Adomian polynomials, Appl. math. comput., 141, 447-453, (2003) · Zbl 1027.65072
[29] 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
[30] Babolian, E.; Javadi, Sh., New method for calculating Adomian polynomials, Appl. math. comput., 153, 253-259, (2004) · Zbl 1055.65068
[31] Zhu, Y.; Chang, Q.; Wu, S., A new algorithm for calculating Adomian polynomials, Appl. math. comput., 169, 402-416, (2005) · Zbl 1087.65528
[32] Biazar, J.; Shafiof, S.M., A simple algorithm for calculating Adomian polynomials, Int. J. contemp. math. sci., 2, 975-982, (2007) · Zbl 1145.65014
[33] Gu, H.; Li, Z., A modified Adomian method for system of nonlinear differential equations, Appl. math. comput., 187, 748-755, (2007) · Zbl 1121.65082
[34] Azreg-Aı¨nou, M., A developed new algorithm for evaluating Adomian polynomials, Comput. model. eng. sci., 42, 1-18, (2009) · Zbl 1357.65067
[35] Duan, J.S., Recurrence triangle for Adomian polynomials, Appl. math. comput., 216, 1235-1241, (2010) · Zbl 1190.65031
[36] 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
[37] Chen, W.; Lu, Z., An algorithm for Adomian decomposition method, Appl. math. comput., 159, 221-235, (2004) · Zbl 1062.65059
[38] Pourdarvish, A., A reliable symbolic implementation of algorithm for calculating Adomian polynomials, Appl. math. comput., 172, 545-550, (2006) · Zbl 1088.65021
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