zbMATH — the first resource for mathematics

On the order of convergence of Adomian method. (English) Zbl 1044.65043
The authors consider an approximate solution of the equation \(y-N(y)=f\) where \(N\) is a nonlinear operator from a Hilbert space onto itself. They state (without proof) that the Adomian decomposition method for the above equation is equivalent to solving the equation \(S=N(y_0+S)\) by iteration, \(S_{n+1}=N(y_0+S_n)\), and obtain the order of convergence of \((S_n)\) to \(S\) under certain smoothness assumptions on \(N\). Here \(y_0\) is supposed to be \(f\) and \(S=y-y_0\).

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
Full Text: DOI
[1] Adomian, G.; Adomian, G.E., A global method for solution of complex systems, Math. model., 5, 521-568, (1984) · Zbl 0556.93005
[2] Adomian, G.; Rach, R., On the solution of algebraic equations by the decomposition method, Math. anal. appl., 105, 1, 141-166, (1985) · Zbl 0552.60060
[3] Adomian, G.; Rach, R.; Sarafyan, D., On the system of equations containing radicals, J. math. anal. appl., 111, 2, 423-426, (1985) · Zbl 0579.60060
[4] Cherruault, Y., Convergence of Adomian’s method, Kyberneters, 8, 2, 31-38, (1988) · Zbl 0697.65051
[5] Adomian, G.; Sarafyan, D., Numerical solution of differential equations in the deterministic limit of stochastic theory, Appl. math. comput., 8, 111-119, (1981) · Zbl 0466.65046
[6] Adomian, G.; Rach, R., Non-linear stochastic differential delay equations, J. math. anal. appl., 91, 1, (1983) · Zbl 0504.60067
[7] Bellman, R.; Adomian, G., Partial differential equations, (1985), Reidel Dordrecht
[8] Adomian, G., Convergent series solution of non-linear equations, J. comput. appl. math., 11, 225-230, (1964) · Zbl 0549.65034
[9] Adomian, G., Non-linear stochastic dynamical systems in physical problems, J. math. anal. appl., 111, 1, (1985) · Zbl 0582.60067
[10] Abbaoui, K.; Cherruault, Y., Convergence of Adomian’s method applied to non-linear equations, Math. compact. model., 20, 9, 69-73, (1994) · Zbl 0822.65027
[11] Chambadal, L., Dictionnaire de mathematiques, (1981), Hachette Paris
[12] Saaty, T.L.; Bram, J., Nonlinear mathematics, (1964), McGraw-Hill New York · Zbl 0198.00102
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.