Nonlinear fractional differential equations reflect the true nature of physical and biological models with non-locality memory effects. This paper considers nonlinear unknown analytical solutions. The Adomian decomposition power series methods are adopted to approximate two approaches illustrated compared by means four numerical examples.

Analysis of the nonlinear Schrödinger vortex reconnection is given in terms of coordinate-time power series. The lowest order terms in these series correspond to a solution of the linear Schrödinger equation and provide several interesting properties of the reconnection process, in particular the nonsingular character of reconnections, the anti-parallel configuration of vortex filaments and a s...

A time-fractional heat equation arising in a quiescent medium is established, and its approximate analytical solution obtained by the fractional power series method. The results show that method performs extremely well terms of efficiency simplicity.

on the basis of a reproducing kernel space, an iterative algorithm for solving the one-dimensional linear and nonlinear schrödinger equations is presented. the analytical solution is shown in a series form in the reproducing kernel space and the approximate solution is constructed by truncating the series. the convergence of the approximate solution to the analytical solution is also proved. th...