Introduction

The RPSM was produced as an efficient method for definite worths of coefficients of the power series solution for fuzzy differential equations (Arqub, 2013). The RPSM is constituted with an repeated algorithm. This method is effective and easily to obtain power series solution for forcibly linear and nonlinear equations lacking linearization, perturbation, or discretization. Unlike the other series method, the RPSM does not want to match the coefficients of the comparable conditions and a repeated connection is'nt needed. Present method calculates the coefficients of the power series by a bond of algebraic equations of some variables. Besides, the RPSM does not need any transforming while changing from the low-order to the higher-order; thus the present method can be worked straight to the given example by selecting an suitable initial estimate approximation. This method have tested to be powerful, effective, and can easily handle a broad class of linear and nonlinear problems (Arqub, El-Ajou, Bataineh, & Hashim, 2013;El-Ajou, Arqub, Al Zhour, & Momani, 2013; Arqub, El-Ajou, Al Zhour, & Momani, 2014; Arqub, El-Ajou, & Momani, 2015; El-Ajou, Arqub, & Momani, 2015; Ich, Körpınar, Al-Qurashi, & Baleanu, 2016; Tchier, Ich, Körpınar, & Baleanu, 2016; Mishra & Sen, 2016; Mishra, Agarwal, & Sen, 2016).

The homotopy analysis transform method (Hatm) is a compounding of the homotopy analysis method (HAM) and Laplace transform method (Khan, Gondala, Hussain, & Vanani, 2012; Gondal, Arife, Khan, & Hussain, 2011; Kumar, Singh, & Kumar, 2014; Kumar, Kumar, & Baleanu, 2016). The profit of this method is its potentiality of combination two powerful methods for finding exact and approximate analytical solutions for nonlinear equations. HATM solves nonlinear problems without using Adomian's polynomials and He's polynomials is a net profit of this method over the Adomian's decomposition method (ADM) and the homotopy perturbation transform method (HPTM).

The purpose of this work is to utilize RPSM and HATM to find the numerical solutions for the linear Schrodinger Equation 1 (Wazwaz, 2008).

and the nonlinear Schrodinger Equation 2 (Wazwaz, 2008).

where:

is a constant and is a complex function. Equation 2 considers the time development of a free molecule. It is applied variational iteration method to handle approximate solutions of these equations by (Wazwaz, 2008). It is studied sine-cosine function method for nonlinear Schrodinger equation by (Jawad, Kumar, & Biswas, 2013).

The Schrodinger equations apply in several area of physics, containing nonlinear optics, plasma physics, superconductivity and quantum mechanics. The linear Schrodinger Equation 1 is commonly used by applying the spectral transform among other methods (Wazwaz, 2002).The nonlinear Schrodinger's Equation 2 acts a critical function in several fields of physical, biological, and engineering sciences. It seems in several applied fields, containing liquid dynamics, nonlinear optics, plasma physics, and proteinchemistry.

The outline of the remainder of this paper is as follows. In the next section, we explained Hatm. In Sections 3, we applied RPSM and Hatm for linear and nonlinear Schrodinger's Equations. In Sections 4, is formed graphics and is drew tables for reliable of obtained solutions in Figure 1-4. Finally, some concluding remarks are given.

Material and method

In N(u(x)) = r(x) equation, N a general ordinary or partial nonlinear differential operator containing every two cases. The linear case is L+R, where L is the greatest order linear operator and R is the resting of the linear operator. Thus, Equation 3,

where:

Nu is the nonlinear cases. By using Laplace transform on both sides of .

From the property of Laplace transform, we have Equation 4,

We describe the nonlinear element:where is real functions of x, t and z on condition that . Now we make a homotopy, Equation 5:

in Equation 5, ; Laplace transform,; an assisting parameter,; embedding parameter,; an assisting function,; a unknown function and u₀(x, t); an initial condition of u(x, t).

Then, for z = 0 and z = 1, it gives,,, respectively. Thus, Equation 6:

where, Equation 7,

Then we have Equation 8,

The present equation can be concluded from the 0th- order deformation.

Describe the vector .

Differentiating Equation 6n times with respect to the embedding parameter z and then setting z = 0 and finally dividing them by n!, we have the socalled nth-order deformation Equation 9:

Using inverse Laplace transform, we obtain Equation 10,

where:, and Equation 11,

Results and discussion

Numerical applications for Schrodinger's equations

Example 1: We first study the linear Schrodinger Equation 12,

It isfound the exact solution for (Equation 12) as Equation 13:

by (Jawad et al., 2013).

For applications of Hatm; applying the Laplace transform on both sides of Equation 12, we have

We describe a nonlinear operator as and thus Equation 14:

The nth-order deformation equation is given by . Applying the inverse Laplace transform, we rewrite Equation 15:

Solving Equation 14, for n = 1, 2, 3, ..., we get Equation 16:

Hence, the 5th-order HATM solution (for h = -1) is given by Equation 17;

For applications of RPSM;

We consider (Equation 12) equation and his initial condition. We apply the RPSM to find out series solution for this equation subject to given initial conditions by replacing its power series expansion with its truncated residual function. From this equation a repetition formula for the calculation of coefficients is supplied, while coefficients in PS expansion can be calculated repeatedly from the truncated residual function (El-Ajou et al., 2013; 2015).

Theorize that the solution yields the expanse form, Equation 18:

Next, we let u_kto denote k. truncated series of u, Equation 19:

where:

u₀ = f₀(x) = u(x, 0) = f(x), Equation 19 can be written as Equation 20:

At first, to find the value of coefficients in series expansion of Equation 20, we define residual function Res; for Equation 12 as Res = u_t+ iu_xx and the k-th residual function, Res_k as follows Equation 21:

(Arqub, 2013; Arqub et al., 2013; El-Ajou et al., 2013) show that Res = 0 and for each and .

Then, when t = 0 for each . To determine f₁(x), we write k = 1 in Equation 21, Equation 22:

where:

u₁ = f(x) + tf₁(x) for u₀ = f₀(x) = f(x) = u(x, 0) = e^3ix.

From Equation 22 we deduce that Res₁ = 0 (t = 0) and thus, Equation 23:

Therefore, the -st RPS approximate solutions are Equation 24:

Similarly, to find out the form of the second unknown coefficient f₂(x), we write u₂ = f(x) + tf₁(x) + t²f₂(x) in Res₂. (t = 0) and thus, Equation 25:

Therefore, the 2-st RPS approximate solutions are Equation 26:

Similarly, we write u₄ = f(x) + tf₁(x) + t²f₂(x) + t³f₃(x) + t⁴f₄(x) in Res4. (t = 0) nd thus, acoording Equation 27:

Therefore, the 4-st RPS approximate solutions are.

Example 2: We now study the cubic nonlinear Schrodinger Equation 28,

The exact solution for (Equation 28) is (Wazwaz, 2008), acoording Equation 29.

For applications of Hatm; applying the Laplace transform on both sides of Equation 28, we have.

We describe a nonlinear operator as and thus.

The nth-order deformation equation is given by .

Applying the inverse Laplace transform, we have Equation 30:

Solving Equation 30, for n = 1, 2, 3, ..., we get u₀(x, t) = e^ix, u₁(x, t) = 3ie^ixht .

Hence, the 5th-order Hatm solution (for h = -1) is given by Equation 31:

For applications of RPSM;

We the cubic (Equation 28) nonlinear Schrodinger equation and his initial condition.

If we apply the RPSM, we can write, Equation 32:

Therefore, the 4-st RPS approximate solutions are Equation 33:

Final considerations

In this section, we formed graphics and drew tables for reliable to above obtained solutions.

Table 1-4 clarify the convergence of the approximate solutions to the exact solution. In these tables, comparison among approximate solutions with known results is made. These results obtained by using RPSM and Hatm.

Conclusion

In this work we have demonstrated efficiency of the Homotopy analysis transform method (Hatm) and Residual power series method (RPSM) for finding series solutions of linear and nonlinear Schrödinger equations. These methods are applied successfully and series solutions are compared known exact solutions. Graphical and numerical consequences are introduced to illustrate the solutions. The solution finded by using RPSM is more convenient as compared to Hatm solution.Thus, it is concluded that the RPSM becomes powerful and efficient in finding numerical solutions than Hatm by assuming h = -1. We conclude that consequences emphasizes the powers of these methods in handling a wide variety of nonlinear problems.

References

Arqub, O. A. (2013). Series solution of fuzzy differential equations under strongly generalized differentiability. Journal of Advanced Research in Applied Mathematics, 5(1), 31-52. doi: 10.5373/jaram.1447.051912

Arqub, O. A., El-Ajou, A., & Momani, S. (2015). Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. Journal of Computational Physics, 293, 385-399. doi: 10.1016/j.jcp.2014.09.034

Arqub, O. A., El-Ajou, A., Al Zhour, Z., & Momani, S. (2014). Multiple solutions of nonlinear boundary value problems of fractional order: a new analytic iterative technique. Entropy, 16(1), 471-493. doi: 10.3390/e16010471

Arqub, O. A., El-Ajou, A., Bataineh, A., & Hashim, I. (2013). A representation of the exact solution of generalized Lane Emden equations using a new analytical method. Abstract and Applied Analysis, 2013, 1-10. doi: 10.1155/2013/378593

El-Ajou, A., Arqub, O. A., & Momani, S. (2015). Approximate analytical solution of the nonlinearfractional KdV-Burgers equation: a new iterative algorithm. Journal of Computational Physics, 293, 81-95. doi: 10.1016/j.jcp.2014.08.004

El-Ajou, A., Arqub, O. A., Al Zhour, Z., & Momani, S. (2013). New results on fractional power series: theories and applications. Entropy, 15(12), 5035-5323. doi: 10.3390/e15125305

Gondal, M. A., Arife, A. S., Khan, M., & Hussain, I. (2011). An efficient numerical method for solving linear and nonlinear partial differential equations by combining homotopy analysis and transform method. World Applied Sciences Journal, 14(12), 1786-1791.

Ich, M., Körpınar, Z. S., Al-Qurashi, M. M., & Baleanu, D. (2016). A new method for approximate solutions of some nonlinear equations: Residual power series method. Advances in Mechanical Engineering, 8(4), 1-7. doi: 10.1177/1687814016644580

Jawad, A. J. M., Kumar, S., & Biswas, A. (2013). Solition solutions of a few nonlinear wave equations in engineering sciences. Scientia Iranica, 21(3), 861-869.

Khan, M., Gondala, M. A., Hussain, I., & Vanani, S. K. (2012). A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on a semiinfinite domain. Mathematical and Computer Modelling, 55(3-4), 1143-1150. doi: 10.1016/j.mcm.2011.09.038

Kumar, D., Singh, J., & Kumar, S. (2014). Numerical computation of Klein-Gordonequations arising in quantum field theory by using homotopy analysistransfrom method. Alexandria Engineering Journal, 53(2), 469-474. doi: 10.1016/j.aej.2014.02.001

Kumar, S., Kumar, A., & Baleanu, D. (2016). Two analytical methods for time-fractional nonlinear coupled Boussinesq-Burger's equations arise in propagation of shallow water waves. Nonlinear Dynamics, 85(2), 699-715.

Mishra, L. N., & Sen, M. (2016). On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order. Applied Mathematics and Computation, 285, 174-183. doi: 10.1016/j.amc.2016.03.002

Mishra, L. N., Agarwal, R. P., & Sen, M. (2016). Solvability and asymptotic behavior for some nonlinear quadratic integral equation involving Erdelyi-Kober fractional integrals on the unbounded interval. Progress in Fractional Differentiation and Applications, 2(3), 153-168. doi: 10.18576/pfda/020301

Tchier, F., Ich, M., Körpınar, Z. S., & Baleanu, D. (2016). Solutions of the time fractional Reaction diffusion equations with residual power series method. Advances in Mechanical Engineering, 8(10), 1-10. doi: 10.1177/1687814016670867

Wazwaz, A. M. (2002). Partial differential equations: methods and applications. The Netherlands, NL: Balkema Publishers.

Wazwaz, A. M. (2008). A study on linear and nonlinear Schrodingerequations by thevariational iteration method. Chaos, Solitons and Fractals, 37(4), 1136-1142. doi: 10.1016/j.chaos.2006.10.009

Author notes

zelihakorpinar@gmail.com