Construction of Analytical Solutions to Fractional Differential Equations Using Homotopy Analysis Method

In this paper, we present an algorithm of the homotopy analysis method (HAM) to obtain symbolic approximate solutions for linear and nonlinear differential equations of fractional order. We show that the HAM is different from all analytical methods; it provides us with a simple way to adjust and control the convergence region of the series solution by introducing the auxiliary parameter ℏ, the auxiliary function �� �� , the initial guess �� ��(�� ) and the auxiliary linear operator . Three examples, the fractional oscillation equation, the fractional Riccati equation and the fractional Lane-Emden equation, are tested using the modified algorithm. The obtained results show that the Adomain decomposition method, Variational iteration method and homotopy perturbation method are special cases of homotopy analysis method. The modified algorithm can be widely implemented to solve both ordinary and partial differential equations of fractional order.

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