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Research Papers

# Multistage Adomian Decomposition Method for Solving NLP Problems Over a Nonlinear Fractional Dynamical System

[+] Author and Article Information
Fırat Evirgen

Department of Mathematics, Balıkesir University, Çağış Campus, 10145 Balıkesir, Turkeyfevirgen@balikesir.edu.tr

Necati Özdemir1

Department of Mathematics, Balıkesir University, Çağış Campus, 10145 Balıkesir, Turkeynozdemir@balikesir.edu.tr

1

Corresponding author.

J. Comput. Nonlinear Dynam 6(2), 021003 (Oct 20, 2010) (6 pages) doi:10.1115/1.4002393 History: Received November 25, 2009; Revised March 15, 2010; Published October 20, 2010; Online October 20, 2010

## Abstract

This paper deals with implementation of the multistage Adomian decomposition method (MADM) to solve a class of nonlinear programming (NLP) problems, which are reformulated with a nonlinear system of fractional differential equations. The multistage strategy is used to investigate the relation between an equilibrium point of the fractional order dynamical system and an optimal solution of the NLP problem. The preference of the method lies in the fact that the multistage strategy gives this relation in an arbitrary longtime interval, while the Adomian decomposition method (ADM) gives the optimal solution just only in the neighborhood of the initial time. The numerical results taken by the fractional order MADM show that these results are compatible with the solution of NLP problem rather than the ADM. Furthermore, in some cases the fractional order MADM can perform more rapid convergency to the optimal solution of optimization problem than the integer order ones.

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## Figures

Figure 1

Comparison of x1 and x2. ◻: 4-term ADM for α=0.9; ◇: 2-term MADM (ΔT=10−5) for α=0.9; ○: 3-term MADM (ΔT=10−5) for α=1.

Figure 2

Comparison of x1 and x2. ◻: 4-term ADM for α=0.95; ◇: 2-term MADM (ΔT=10−5) for α=0.95; ○: 3-term MADM (ΔT=10−5) for α=1.

Figure 3

Comparison of x3 and x4. ◻: 4-term ADM for α=0.95; ◇: 2-term MADM (ΔT=10−5) for α=0.95; ○: 3-term MADM (ΔT=10−5) for α=1.

Figure 4

Comparison of x5. ◻: 4-term ADM for α=0.95; ◇: 2-term MADM (ΔT=10−5) for α=0.95; ○: 3-term MADM (ΔT=10−5) for α=1.

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