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Volume 6, Number 1, 2016, Pages 103-118                                                                DOI:10.11948/2016009
An optimal homotopy analysis method based on particle swarm optimization: application to fractional-order differential equation
Li-Guo Yuan,Zeeshan Alam
Keywords:Optimal homotopy analysis method; particle swarm optimization; fractional-order differential equation; Caputo derivative; residual error.
      This paper describes a new problem-solving mentality of finding optimal parameters in optimal homotopy analysis method (optimal HAM). We use particle swarm optimization (PSO) to minimize the exact square residual error in optimal HAM. All optimal convergence-control parameters can be found concurrently. This method can deal with optimal HAM which has finite convergence-control parameters. Two nonlinear fractional-order differential equations are given to illustrate the proposed algorithm. The comparison reveals that optimal HAM combined with PSO is effective and reliable. Meanwhile, we give a sufficient condition for convergence of the optimal HAM for solving fractional-order equation, and try to put forward a new calculation method for the residual error.
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