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Series solutions of non-linear Riccati differential equations with fractional order. (English) Zbl 1197.34006

Summary: Based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve non-linear Riccati differential equation with fractional order. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter $\hslash /2\pi$. Besides, it is proved that well-known Adomian’s decomposition method is a special case of the homotopy analysis method when $\hslash /2\pi -1$. This work illustrates the validity and great potential of the homotopy analysis method for the non-linear differential equations with fractional order. The basic ideas of this approach can be widely employed to solve other strongly non-linear problems in fractional calculus.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

##### MSC:
 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) 34A08 Fractional differential equations 26A33 Fractional derivatives and integrals (real functions)