Analytical Approximate Solutions for the Nonlinear Fractional Differential-Difference Equations Arising in Nanotechnology

The aim of this article is by using the fractional complex transform FCT and the optimal homotopy analysis transform method OHATM to find the analytical approximate solutions for nonlinear fractional differential-difference equations FNDDEs arising in physical phenomena such as wave phenomena in fluids, coupled nonlinear optical waveguides and nanotechnology fields. Fractional complex transformation is proposed to convert nonlinear fractional differentialdifference equation to nonlinear differential-difference equation. This optimal approach has general meaning and can be used to get the fast convergent series solution of the different type of nonlinear differential-difference equations. This technique finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. HATM method provides us with a simple way to adjust and control the convergence region of solution series by choosing a proper value for the auxiliary parameter h. So the valid region for h where the series converges is the horizontal segment of each h curve. The results obtained by the HATM show that the approach is easy to implement and computationally very attractive. The numerical solutions show that the proposed method is very efficient and computationally attractive. The results reveal that this method is very effective and powerful to obtain the approximate solutions.