Piecewise finite series solution of nonlinear initial value differential problem. (English) Zbl 1166.65350

Summary: We apply a piecewise finite series as a hybrid analytical-numerical technique for solving some systems of nonlinear ordinary differential equations (ODEs). The finite series is generated by using the Adomian decomposition method, which is an analytical method that gives the solution based on a power series and has been successfully used in a wide range of problems in applied mathematics.
We study the influence of the step size and the truncation order of the piecewise finite series Adomian (PFSA) method on the accuracy of the solutions when applied to nonlinear ODEs. Numerical comparisons between the PFSA method with different time steps and truncation orders against Runge-Kutta type methods are presented. Based on the numerical results we propose a low value truncation order approach with small time step size. The numerical results show that the PFSA method is accurate and easy to implement with the proposed approach.


65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
Full Text: DOI


[1] Abdulaziz, O., Further accuracy tests on Adomian decomposition method for chaotic systems, Chaos soliton fract., 36, 5, 405-1411, (2008)
[2] Biazar, J.; Montazeri, R., A computational method for solution of the prey and predator problem, Appl. math. comput., 163, 841-847, (2005) · Zbl 1060.65612
[3] Biazar, J., Solution of the epidemic model by Adomian decomposition method, Appl. math. comput., 173, 2, 1101-1106, (2006) · Zbl 1087.92051
[4] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Boston · Zbl 0802.65122
[5] Hashim, I.; Noorani, M.S.M.; Ahmad, R.; Bakar, S.A.; Ismail, E.S.; Zakaria, A.M., Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos soliton fract., 28, 5, 1149-1158, (2006) · Zbl 1096.65066
[6] Adomian, G., A review of the decomposition method in applied mathematics, Math. anal. appl., 135, 2, 501-544, (1988) · Zbl 0671.34053
[7] Repaci, A., Nonlinear dynamical systems: on the accuracy of adomian’s decomposition method, Appl. math. lett., 3, 4, 35-39, (1990) · Zbl 0719.93041
[8] Ruan, J.; Lu, Z., A modified algorithm for Adomian decomposition method with applications to lotka – volterra systems, Math. comput. modell., 46, 9-10, 1214-1224, (2007) · Zbl 1133.65046
[9] Chen, B.; Company, R.; Jódar, L.; Roselló, M.D., Constructing accurate polynomial approximations for nonlinear differential initial value problems, Appl. math. comput., 193, 2, 523-534, (2007) · Zbl 1193.65111
[10] Chen, B.; Solis, F., Discretizations of nonlinear differential equations using explicit finite order methods, J. comput. appl. math., 90, 2, 171-183, (1998) · Zbl 0940.65074
[11] Mickens, R.E., Discretizations of nonlinear differential equations using explicit nonstandard methods, J. comput. appl. math., 110, 1, 181-185, (1999) · Zbl 0940.65079
[12] Mickens, R.E., A nonstandard finite-difference scheme for the lotka – volterra system, Appl. numer. math., 45, 3, 309-314, (2003) · Zbl 1025.65047
[13] Mickens, R.E., Application of nonstandard finite difference schemes, (2000), World Scientific Publishing Co. Pte. Ltd. · Zbl 1237.76105
[14] Abbaoui, K.; Cherruault, Y., Convergence of adomian’s method applied to differential equations, Comput. math. appl., 28, 5, 103-109, (1994) · Zbl 0809.65073
[15] Wazwaz, A.M., The decomposition method applied to systems of partial differential equations and to the reaction – diffusion Brusselator model, Appl. math. comput., 110, 2-3, 251-264, (2000) · Zbl 1023.65109
[16] Kincaid, D.; Cheney, W., Numerical analysis, (2002), Brooks/Cole Publishing Company Pacific Grove, CA
[17] Olek, S., An accurate solution to the multispecies lotka – volterra equations, SIAM rev., 36, 3, 480-488, (1994) · Zbl 0802.92018
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.