Approximate Analytical Solution of Two Coupled Time Fractional Nonlinear Schrödinger Equations

Abstract

In this article, we consider the well known nonlinear Schrödinger equation (NLS) with fractional time derivative and derive its approximate analytical solution using the homotopy analysis method (HAM). We also applied HAM to two coupled time fractional NLS and constructed its approximate solution. The question of convergence of the obtained solution is discussed. The obtained approximate analytical periodic wave solution, solitary wave solution and the effect of time fractional order α are shown graphically.

Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives

“…More recently, much attention has been paid to the solutions of FDEs using various methods, such as the Adomian decomposition method (2005) [8], the first integral method (2014) [9], the Lie group theory method (2012,2015) [10,11], the homotopy analysis method (2016) [12], the inverse differential operational method (2016) [13][14][15], the F-expansion method (2017) [16], M-Wright transforms (2017) [17], exponential differential operators (2017,2018) [18,19], and so on. In reality, the finding of exact solutions of the FDEs is hard work and remains a problem.…”

Applications of Homogenous Balance Principles Combined with Fractional Calculus Approach and Separate Variable Method on Investigating Exact Solutions to Multidimensional Fractional Nonlinear PDEs

“…In the past few years, there were excessive studies on the (1 + 1)-dimensional fractional nonlinear PDEs(fnPDEs), and a lot of excellent tools were used for solving them, which include Adomian decomposition method [27,28], homotopy analysis method (HAM) [29,30], fractional variational method [31,32], Lie symmetry method [33][34][35][36][37][38][39][40], invariant subspace method [40][41][42][43][44], and homogenous balance principle (HBP) [45,46]. Recently, a few of the above methods were also applied to solve several (2 + 1)-dimensional fnPDEs [47][48][49][50].…”

Modified least squares homotopy perturbation method for solving fractional partial differential equations

“…Zayed et al [12]) where the method employed to solve fractional partial differential equations by turning them into nonlinear ordinary differential equations of integer orders. In [13] T. Bakkyaraj and R. Sahadevan applied homotopy analysis method to obtain the approximate analytical solution of two coupled time fractional nonlinear Schrödinger equations. Zigen Ouyang [14] obtained some conditions for the existence of the solutions of a class of nonlinear fractional order partial differential equations with delay.…”