*(English)*Zbl 1180.65181

Summary: Based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve systems of fractional integro-differential equations. Comparing with the exact solution, the HAM provides us with a simple way to adjust and control the convergence region of the series solution by introducing an auxiliary parameter $\hslash $. Four examples are tested using the proposed technique. It is shown that the solutions obtained by the Adomian decomposition method are only special cases of the HAM solutions.

The present work shows the validity and great potential of the homotopy analysis method for solving linear and nonlinear systems of fractional integro-differential equations. The basic idea described in this article is expected to be further employed to solve other similar nonlinear problems in fractional calculus.

##### MSC:

65R20 | Integral equations (numerical methods) |

45J05 | Integro-ordinary differential equations |

45G15 | Systems of nonlinear integral equations |

26A33 | Fractional derivatives and integrals (real functions) |