Homotopy analysis method and its application for solving integro-di erential equations of fractional order

In this paper, based on the homotopy analysis method (HAM), a analytical technique is proposed to solve nonlinear fractional integro-di erential equa- tions (FIDEs). Di erent from all other analytical methods, it provides a simple way to adjust and control the convergence region of series solution by introducing an auxiliary parameter ~. This work illustrates the valid- ity and high accuracy of the homotopy analysis method for the non-linear integro-di erential equations with fractional order. The basic concepts of the HAM can be widely used to solve other strongly non-linear fractional problems. Several numerical examples are given to illustrate the convergence of approximate solutions to analytical solutions. The comparison of the re- sults obtained by the HAM with the exact solutions show that the HAM is very e ective and simple for solving various linear and nonlinear fractional problems.