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A perturbing nonlinear Schrodinger equation is studied under general complex nonhomogeneities and complex initial conditions for zero boundary conditions. The perturbation method together with the eigenfunction expansion and variational parameters methods are used to introduce an approximate solution for the perturbative nonlinear case for which a power series solution is proved to exist. Using Mathematica, the symbolic solution algorithm is tested through computing the possible approximations under truncation procedures. The method of solution is illustrated through case studies and figures.

The nonlinear Schrodinger (NLS) equation is the principal equation to be analyzed and solved in many fields, see [

Wang et al. [

with initial condition

In this paper, a straight forward solution algorithm is introduced using the transformation from a complex solution to a coupled equations in two real solutions, eliminating one of the solutions to get separate independent and higher-order equations, and finally introducing a perturbative approximate solution to the system.

Consider the nonhomogeneous linear Schrodinger equation:

Eliminating one of the variables in (

Consider the homogeneous nonlinear Schrodinger equation:

The solution of (

At

According to the previous lemma, one can assume the solution of (

As a perturbation solution, one can assume that

Substituting (

Following the solution algorithm described in the previous section for the linear case, the general symbolic algorithm in Figure

In this case,

The absolute value of the zero-order approximation is got from

The absolute value of the first-order approximation can be got using

The absolute value of the second-order approximation can be got using

To examine the proposed solution algorithm, see Figure

Taking

The zero-order approximation of

The zero-order approximation

The zero-order approximation

Taking

Taking

The zero-order approximation of

The zero-order approximation

The zero-order approximation

One can notice the decrease of the solution level and its higher variability.

Taking

Taking

The zero-order approximation of

The zero-order approximation

The zero-order approximation

One can notice that the solution level becomes a little bit higher than that of case study 2.

Taking

One can notice the little increase of the solution level than that of case studies 3 and 4.

Taking

One can notice the small perturbations at small values of

Taking

One can notice the increase of the depth of the perturbations.

Taking

One can notice that the perturbations become smaller than that of case study 8.

Taking the case of

One can notice the little increase in the solution level.

Taking the case of

One can notice the low solution level and high perturbations.

Taking the case of

One can notice that a higher solution level is got compared with case study 11 and less perturbations are got at small values of

Taking the case of

The zero-order approximation of

The zero-order approximation

The zero-order approximation

The first-order approximation of

The first-order approximation of

The first-order approximation of

The first-order approximation

The first-order approximation

The general solution algorithm.

One can notice the oscillations of the solution level compared with case 7.

One can notice that the solution level increases with the increase of

The perturbation technique introduces an approximate solution to the NLS equation with a perturbative nonlinear term for a finite time interval. Using mathematica, the difficult and huge computations problems were fronted to some extent for limited series terms. To get more improved orders, it is expected to face a problem of computation. With respect to the solution level, the effect of the nonhomogeneity is higher than the effect of the initial condition. The initial conditions also cause perturbations for the solution at small values of the space variable. The solution level increases with the increase of