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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 residu- al error in optimal HAM. All optimal convergence-control parameters can be found concurrently. This method can deal with optimal HAM which has fi- nite convergence-control parameters. Two nonlinear fractional-order differen- tial equations are given to illustrate the proposed algorithm. The comparison reveals that optimal HAM combined with PSO is effective and reliable. Mean- while, 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.

AN OPTIMAL HOMOTOPY ANALYSIS METHOD BASED ON PARTICLE SWARM OPTIMIZATION: APPLICATION TO FRACTIONAL-ORDER DIFFERENTIAL EQUATION