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This paper deals with constructing generalized 'fractional' power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non-linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a new adaption of homotopy analysis method is presented to handle nonlinear problems. The proposed approach is capable of reducing the size of calculations and easily overcome the difficulty arising in calculating complicated integrals. Furthermore, the homotopy polynomials that decompose the nonlinear term of the problem as a series of polynomials are introduced. Then, an algorithm of calculating such polynomials, which makes the solution procedure more straightforward and more effective, is constructed. Numerical examples are examined to highlight the significant features of the developed techniques. The algorithms described in this paper are expected to be further employed to solve nonlinear problems in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]