The homotopy perturbation method for discontinued problems arising in nanotechnology

Abstract

a b s t r a c tContinuum hypothesis on nanoscales is invalid, and a differential-difference model is considered as an alternative approach to describing discontinued problems. This paper applies the homotopy perturbation method to a nonlinear differential-difference equation arising in nanotechnology. Comparison of the approximate solution with the exact one reveals that the method is very effective.

Exact Solutions of a Fractional-Type Differential-Difference Equation Related to Discrete MKdV Equation

“…Among those, we can name: Hirota's bilinear method, [10] Casoratian technique, [11] homotopy perturbation method, [12] ADM-Padé technique, [13] etc. But, most of the methods are not easy to handle and require a thorough knowledge of the solution procedure one has in mind.…”

Construction of exact solutions for fractional-type difference-differential equations via symbolic computation

“…As far as we could verify, relatively less work is being performed for the symbolic computation of exact solutions to fractional-type DDEs while there has been a considerable amount of work done in finding exact solutions to polynomial DDEs. In the last decade, due to the increased interest in DDEs, a whole range of analytical solution methods such as Hirota's bilinear method [16], ADM-Padé technique [17], Casoratian technique [18], homotopy perturbation method [19], Exp-function method [20], and so on. were developed by the researchers.…”

Exact and explicit solutions to the discrete nonlinear Schrödinger equation with a saturable nonlinearity

“…In recent years, considerable attention has been given to the problem of exactly solving NDDEs. Some authors [6][7][8][9][10] put forth new modifications of the existing methods to tackle NDDEs. Moreover, Ma and You [11] used Casoratian technique for constructing rational solutions to the Toda lattice equation.…”