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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 $\hbar/2 \pi$. Besides, it is proved that well-known Adomian’s decomposition method is a special case of the homotopy analysis method when $\hbar/2 \pi - 1$. 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.

34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
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