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Elzaki Transform Homotopy Analysis Techniques for Solving 

Fractional (2+1)-D and (3+1)-D Nonlinear Schrodinger Equations 

 

Sandeep Sharma1, Inderdeep Singh2 

1,2Sant Baba Bhag Singh University, Jalandhar-144030, Punjab, India 

Email: 1sandeepsharma200@gmail.com ,2inderdeeps.ma.12@gmail.com 

 

Article History: 

Received: 08-06-2024 

Revised: 09-07-2024 

Accepted: 29-07-2024 

Abstract:  

In this research, New homotopy analysis method for solving the fractional (2+1) D and 

(3+1) D non-linear Schrödinger equations by Elzaki. To solve these equations , the Elzaki 

transform is applied jointly to the Homotopy analysis method (HAM). This has proved 

efficient in tackling fractional calculus and nonlinear dynamics since correct solutions are 

offered and they converge at a faster rate. The accuracy of the proposed technique has been 

corroborated by analyzing various examples for which the latter were used for solving high-

dimensional non-linear Schrodinger equation, which indicates that the technique is quite 

resilient as well as efficient; thus, making it an effective tool in theoretical physics and other 

applied sciences. 

Keywords: Elzaki Transform, Homotopy Analysis Method, (2+1)-D- and (3+1)-D 

nonlinear fractional Schrodinger equations. 

 

1. Introduction 

This research paper is devoted to finding the semi-analytical solutions of (2+1)-D and (3+1)-D 

nonlinear fractional Schrodinger equations of the form: 

𝑖𝑤𝑡
𝛼(Ω) + 𝑎∆2𝑤(Ω) + 𝛼(Ω)𝑤(Ω) − 𝛽|𝑤|2𝑤(Ω) = 0,                            (1) 

 

with initial condition 𝑤(Ω, 0) = 𝑤0(Ω) and 𝑖2 = −1. Here, Ω is either (𝑥, 𝑦) or (𝑥, 𝑦, 𝑧), 𝑎, 𝛽 are 

constants and 𝛼 is a function of variables 𝑥, 𝑦 and 𝑧.    

Elzaki integral transform has been introduced in [1] for solving differential equations. In [2], various 

applications of Elzaki transform had been used for solving several mathematical models in the PDE 

(Partial Differential Equations) form. In [3], the authors have presented a comparison study between 

Laplace and Elzaki transforms. For solving various differential equations, a new transform called 

Sumudu transform-based technique had been utilized in [4]. A brief discussion about integral 

transform for solving differential equations had been represented in [5]. Homotopy analysis 

approaches had been applied for solving generalized Benjamin-Bona-Mahony equation and fifth-order 

KdV eqns in [6-7]. Homotopy analysis addressed nonlinear issues in science and engineering [8]. A 

comparison study has been presented in [9] for solving various differential equations. For this purpose, 

Homotopy analysis method and Homotoy perturbation method have been used. In [10], fractional Kdv-

Burgers-Kuramoto eqn had been solved with the help of Homotopy analysis approach. Homotopy 

analysis [11] finds semi-analytical solutions for nonlinear fractional differential equations. Linear 

along with nonlinear fractional diffusion- wave eqns had been solved by using Homotopy analysis 

approach in [12]. In [13], homotopy analysis solves linear and nonlinear Schrodinger equations. To 



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solve heat radiation equations, Homotopy analysis was devised [14]. 2D Schrodinger eqns were solved 

utilizing a compact boundary value approach [18]. In [19], decomposition solves cubic Schrodinger 

equations.       

This research paper is constituted as follows: Section 2 consists of the basic definitions of fractional 

calculus and the basic properties of the Elzaki transform. “Homotopy analysis approach has been 

discussed in Section 3. The suggested scheme on the basis of the combination of Homotopy analysis 

as well as the Elzaki transform approach for solving (2+1)-D and (3+1)-D nonlinear fractional 

Schrodinger equations in Section 4. Test experiments have been performed to solve nonlinear (2+1)-

D and (3+1)-D fractional” Schrodinger equations in Section 5. The conclusion has been discussed in 

Section 6. 

2. Basic Of Fractional Calculus And Elzaki Transform 

This section covers fractional calculus basics. 

Definition 2.1. A real function ℎ(𝑡) ∈ 𝐶𝜇, 𝑡 > 0, 𝜇 ∈ ℛ if ∃ 𝑞 ∈ ℛ; (𝑞 > 𝜇), s.t  ℎ(𝑡) = 𝑡𝑞𝑚(𝑡),  

where 𝑚(𝑡) ∈ 𝐶[0,∞) &  ℎ(𝑡) ∈ 𝐶𝜇
𝑛  if ℎ(𝑛) ∈ 𝐶𝜇, 𝑛 ∈ 𝑁. 

Definition  2.2. The Caputo fractional derivative of ℎ(𝜏) is written as: 

𝜕𝛼

𝜕𝜏𝛼
ℎ(𝜏) = 𝐽(𝑛−𝛼) 𝜕𝑛

𝜕𝜏𝑛
ℎ(𝜏) =

1

Γ(𝑛 − 𝛼)
∫(𝜏 − Ω)𝑛−𝛼−1

𝜏

0

ℎ𝑛(Ω)𝑑Ω, 

where ℎ ∈ 𝐶−1
𝑛 , 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ ℕ, 𝜏 > 0. Here, 

𝜕𝛼

𝜕𝜏𝛼 is Caputo derivative operator & Γ as Gamma 

function. 

Definition 2.3.The function 𝑔1(𝑡) Elzaki transform has been expressed as: 

𝐸{𝑔1(𝑡)} = 𝑣 ∫ 𝑔1(𝑡). 𝑒
−

𝑡
𝑣𝑑𝑡

∞

0

,                  𝑡 > 0 

 

Definition 2.4. For 2 parameters 𝑎 & 𝑏, the Mittag-Leffler function is defined as:  

𝐸𝑎,𝑏(𝜏) = ∑
𝜏𝑛

Γ(𝑎𝑛 + 𝑏)
, 𝑎, 𝑏 > 0

∞

𝑛=0

 

SOME BASIC PROPERTIES 

• The Caputo fractional derivative Elzaki transform is: 

𝐸 {
𝜕𝛼

𝜕𝜏𝛼
ℎ(𝜏)} =

𝐸{ℎ(𝜏)}

𝑣𝛼
− ∑ 𝑣𝑘−𝛼+2ℎ𝑘(0),   

𝑛−1

𝑘=0

 𝑛 − 1 < 𝑘 ≤ 𝑛                     

• Below are the Elzaki transformations of certain partial derivatives: 

 

a) 𝐸 [
𝜕

𝜕𝑡
𝑓(𝑥, 𝑡)] =

𝐸[𝑓(𝑥,𝑡)]

𝑣
− 𝑣. 𝑓(𝑥, 0), 



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b) 𝐸 [
𝜕2

𝜕𝑡2
𝑓(𝑥, 𝑡)] =

1

𝑣2
𝐸[𝑓(𝑥, 𝑡)] − 𝑓(𝑥, 0) − 𝑣.

𝜕𝑓

𝜕𝑡
(𝑥, 0), 

c) 𝐸 [
𝜕

𝜕𝑥
𝑓(𝑥, 𝑡)] =

𝑑

𝑑𝑥
𝐸[𝑓(𝑥, 𝑡)], 

d) 𝐸 [
𝜕2

𝜕𝑥2
𝑓(𝑥, 𝑡)] =

𝑑2

𝑑𝑥2
𝐸[𝑓(𝑥, 𝑡)]. 

 

• The Elzaki transform of certain functions is provided in the list: 

𝐸(1) = 𝑣2,          𝐸(𝑡) = 𝑣3, 𝐸(𝑡𝑛) = 𝑛! 𝑣𝑛+2, 

𝐸(𝑒𝑎𝑡) =
𝑣2

1 − 𝑎𝑣
,   𝐸(sin 𝑎𝑡) =

𝑎𝑣3

1 + 𝑎2𝑣2
 

3. Homotopy Analysis Method [15-17] 

Take into account the subsequent nonlinear differential equation 

  𝑁[𝑤(Ω, 𝑡)] = 0                                                                               (2)       

Where 𝑤(Ω, 𝑡) as an unknown function, 𝑁 as a nonlinear operator, and Ω may be {𝑥, 𝑦} or {𝑥, 𝑦, 𝑧}. 

The variables 𝑥, 𝑦, 𝑧, and 𝑡 as the temporal and spatial independent variables, correspondingly. 

Utilizing the classical Homotopy method “(invented by Liao) 

(1 − 𝑝)𝐿[ 𝜑(Ω, 𝑡; 𝑝) − 𝑤0(Ω, 𝑡)] = 𝑝ℎ𝑁[𝜑(Ω, 𝑡; 𝑝)]                                        (3)                            

Where ℎ  is a nonzero auxiliary parameter,  𝑝 ∈ [0,1] is an embedding parameter, 𝐿 is an auxiliary 

linear operator, 𝜑( Ω, 𝑡; 𝑝) as an unknown function and  𝑤0(Ω, 𝑡)  is as 𝑤(Ω, 𝑡)  initial guess  .  If  𝑝 =

0 &  𝑝 = 1, it holds  

𝜑( Ω, 𝑡; 0) = 𝑤0(Ω, 𝑡), 

and 

𝜑( Ω, 𝑡; 1) = 𝑤(Ω, 𝑡)                            

 Therefore as 𝑝 rises from 0-1, solution  𝜑( Ω, 𝑡; 𝑝) which has been differs from the initial guess 

𝑤0(Ω, 𝑡) to solution 𝑤(Ω, 𝑡). Expanding  𝜑( Ω, 𝑡; 𝑝) n Taylor series regarding  𝑝 , then we have  

𝜑( Ω, 𝑡; 𝑝) = 𝑤0(Ω, 𝑡) + ∑ 𝑤𝑚  (Ω, 𝑡)𝑝𝑚

∞

𝑚=1

                                         (4) 

Where,   

𝑤𝑚(Ω, 𝑡) =
1

𝑚!
 
𝜕𝑚𝜑(Ω, 𝑡; 𝑝)

𝜕𝑝𝑚
|
𝑝=0

   

 

If the auxiliary linear operator, auxiliary parameter ℎ, initial guess, and auxiliary function which have 

been  appropriately selected, then the series (4) converges at 𝑝 = 1 and we get  

𝑤(Ω, 𝑡) = 𝑤0(Ω, 𝑡) + ∑ 𝑤𝑚 

∞

𝑚=1

(Ω, 𝑡),                                                   (5) 



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This should be a valid solution to the original nonlinear eqn. The governing eqn could be derived from 

the 0-order deformation eqn (3) based on definition (5).  

Define the vector  

𝑤𝑛⃗⃗⃗⃗  ⃗ = {𝑤0(Ω, 𝑡), 𝑤1(Ω, 𝑡), 𝑤2(Ω, 𝑡) …… . . 𝑤𝑛(Ω, 𝑡)} 

Differentiating the zero- order deformation eqn (3), 𝑚 −times regarding embedding parameter 𝑝. After 

that putting 𝑝 = 0 and then dividing it with 𝑚!, then the 𝑚th-order deformation eqn is: 

𝐿 [𝑤𝑚(Ω, 𝑡) − 𝜒𝑚 𝑤𝑚−1 (Ω, 𝑡)] = ℎ 𝑅𝑚[𝑤𝑚−1 (Ω, 𝑡)] 

where 

                 𝑅𝑚(𝑤𝑚−1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗ =
1

𝑚−1!
 
𝜕𝑚−1 𝑁[𝜑(Ω,𝑡;𝑝)

𝜕𝑝𝑚−1     |
𝑝=0

 

and 

𝜒𝑚 = {
0,    𝑚 ≤ 1
1,     𝑚 > 1

. 

4. Elzaki Transform Homotopy Analysis Method 

Rewrite” Equation (1) as: 

𝑤𝑡
𝛼(Ω) = 𝑖{𝑎∆2𝑤(Ω) + 𝛼(Ω)𝑤(Ω) − 𝛽𝑤2�̅�}. 

Taking Elzaki transform both sides, we obtain 

𝐸{𝑤𝑡
𝛼(Ω)} = 𝑖𝐸{𝑎∆2𝑤(Ω) + 𝛼(Ω)𝑤(Ω) − 𝛽𝑤2�̅�}. 

Using applications of Elzaki transform as well as an initial condition, we obtain 

𝐸{𝑤(Ω, 𝑡)} = 𝑣2𝑤0(Ω) + 𝑣𝛼𝑖𝐸{𝑎∆2𝑤(Ω) + 𝜓(Ω)𝑤(Ω) − 𝛽𝑤2�̅�}. 

Taking the nonlinear part as: 

𝑅[𝜑(Ω, 𝑡; 𝑝)] = 𝐸(𝜑) − 𝑣2𝑤0(Ω) − 𝑣𝛼𝑖𝐸{𝑎∆2𝜑(Ω) + 𝜓(Ω)𝜑(Ω) − 𝛽𝜑2�̅�}. 

We formulate the zero-order deformation eqn under the given assumption. 𝐻(𝑥, 𝑦, 𝑧, 𝑡) = 1, we have 

(1 − 𝑝)𝐸{𝜑(Ω, 𝑡) − 𝑤0(Ω, 𝑡)} = 𝑝ℎ𝑅[𝜑(Ω, 𝑡; 𝑝)]. 

When 𝑝 = 0 & 𝑝 = 1, we get,  

{
𝜑(Ω, 𝑡; 0) = 𝑤0(Ω, 0)

𝜑(Ω, 𝑡; 1) = 𝑤(Ω, 𝑡).
 

Hence, we obtain the eqn of deformation of order m.  

𝐸{𝑤𝑚(Ω, 𝑡) − 𝜒𝑚𝑤𝑚−1(Ω, 𝑡)} = ℎ𝑅𝑚(𝑤𝑚−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗(Ω, 𝑡)).                                  

Inverse Elzaki transforms both sides, we obtain, 

𝑤𝑚(Ω, 𝑡) − 𝜒𝑚𝑤𝑚−1(Ω, 𝑡) = 𝐸−1{ℎ𝑅𝑚(𝑤𝑚−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗(Ω, 𝑡))}.                                 

From the above Eqon, we get 



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𝑤1(Ω, 𝑡) = −𝐸−1{𝑅1(𝑤0⃗⃗⃗⃗  ⃗(Ω, 𝑡))}, 

𝑤2(Ω, 𝑡) = 𝑤1(Ω, 𝑡) − 𝐸−1{𝑅2(𝑤1⃗⃗ ⃗⃗  (Ω, 𝑡))}, 

𝑤3(Ω, 𝑡) = 𝑤2(Ω, 𝑡) − 𝐸−1{𝑅3(𝑤2⃗⃗⃗⃗  ⃗(Ω, 𝑡))}, 

⋮ 

Therefore, the solution is:  

𝑤(Ω, 𝑡) = 𝑤0 + 𝑤1 + 𝑤2 + ⋯ 

 

5. Test examples: 

In “this Section, we will perform some test examples to find semi-analytical solutions of nonlinear 

fractional (2+1)-D and (3+1)-D nonlinear fractional” Schrodinger equations. 

Example 1: Consider the (3+1)-D fractional nonlinear Schrodinger eqn of form 

𝑖𝑤𝑡
𝛼 + 𝑤𝑥𝑥 + 𝑤𝑦𝑦 + 𝑤𝑧𝑧 + 4|𝑤|2𝑤 = 0,                                   (6) 

with initial condition 𝑤(𝑥, 𝑦, 𝑧, 0) = 𝑒𝑖(𝑥+𝑦+𝑧). The problem (𝛼 = 1)𝑡ℎ𝑒 exact solution is:  

𝑤(𝑥, 𝑦, 𝑧, 𝑡) = 𝑒𝑖(𝑥+𝑦+𝑧+𝑡) 

Rewrite the given problem as: 

𝑖𝑤𝑡
𝛼 = −(𝑤𝑥𝑥 + 𝑤𝑦𝑦 + 𝑤𝑧𝑧 + 4𝑤2�̅�), 

It implies  

𝑤𝑡
𝛼 = 𝑖(𝑤𝑥𝑥 + 𝑤𝑦𝑦 + 𝑤𝑧𝑧 + 4𝑤2�̅�)                                              (7) 

Taking Elzaki transform to both the sides of Eqn (7), we get, 

𝐸[𝑤𝑡
𝛼] = 𝐸[𝑖(𝑤𝑥𝑥 + 𝑤𝑦𝑦 + 𝑤𝑧𝑧 + 4𝑤2�̅�)] 

This implies  

𝐸[𝑤(𝑥, 𝑦, 𝑧, 𝑡)] = ∑ 𝑣𝑖+2

𝑛−1

𝑖=0

𝑤(𝑖)(𝑥, 𝑦, 𝑧, 0) + 𝑣𝛼𝑖𝐸[𝑤𝑥𝑥 + 𝑤𝑦𝑦 + 𝑤𝑧𝑧 + 4𝑤2�̅�] 

After applying initial conditions, we get 

𝐸[𝑤(𝑥, 𝑦, 𝑧, 𝑡)] = 𝑣2. 𝑒𝑖(𝑥+𝑦+𝑧) + 𝑣𝛼𝑖𝐸[𝑤𝑥𝑥 + 𝑤𝑦𝑦 + 𝑤𝑧𝑧 + 4𝑤2�̅�] 

Or 

𝐸[𝑤(𝑥, 𝑦, 𝑧, 𝑡)] − 𝑣2. 𝑒𝑖(𝑥+𝑦+𝑧) − 𝑣𝛼𝑖𝐸[𝑤𝑥𝑥 + 𝑤𝑦𝑦 + 𝑤𝑧𝑧 + 4𝑤2�̅�] = 0 

The nonlinear component is defined as: 

𝑅[𝜑(𝑥, 𝑦, 𝑧, 𝑡; 𝑝)] = 𝐸[𝜑] − 𝑣2. 𝑒𝑖(𝑥+𝑦+𝑧) − 𝑣𝛼𝑖𝐸[𝜑𝑥𝑥 + 𝜑𝑦𝑦 + 𝜑𝑧𝑧 + 4𝜑2�̅�]          (8) 

We formulate the zero-order deformation eqn under the given assumption 𝐻(𝑥, 𝑦, 𝑧, 𝑡) = 1, we “have 



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(1 − 𝑝)𝐸{𝜑(𝑥, 𝑦, 𝑧, 𝑡) − 𝑤0(𝑥, 𝑦, 𝑧, 𝑡)} = 𝑝ℎ𝑅[𝜑(𝑥, 𝑦, 𝑧, 𝑡; 𝑝)] 

When 𝑝 = 0 & 𝑝 = 1, we have  

{
𝜑(𝑥, 𝑦, 𝑧, 𝑡; 0) = 𝑤0(𝑥, 𝑦, 𝑧, 0)

𝜑(𝑥, 𝑦, 𝑧, 𝑡; 1) = 𝑤(𝑥, 𝑦, 𝑧, 𝑡)
 

So, the mth-order deformation eqn  

𝐸{𝑤𝑚(𝑥, 𝑦, 𝑧, 𝑡) − 𝜒𝑚𝑤𝑚−1(𝑥, 𝑦, 𝑧, 𝑡)} = ℎ𝑅𝑚(𝑤𝑚−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗(𝑥, 𝑦, 𝑧, 𝑡))                     (9) 

By inverse Elzaki transform both sides, we obtain 

𝑤𝑚(𝑥, 𝑦, 𝑧, 𝑡) − 𝜒𝑚𝑤𝑚−1(𝑥, 𝑦, 𝑧, 𝑡) = 𝐸−1{ℎ𝑅𝑚(𝑤𝑚−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗(𝑥, 𝑦, 𝑧, 𝑡))}                 (10) 

From Equation (10) (Taking ℎ = −1), we obtain 

𝑤1(𝑥, 𝑦, 𝑧, 𝑡) = −𝐸−1{𝑅1(𝑤0⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑧, 𝑡))}, 

𝑤2(𝑥, 𝑦, 𝑧, 𝑡) = 𝑤1(𝑥, 𝑦, 𝑧, 𝑡) − 𝐸−1{𝑅2(𝑤1⃗⃗ ⃗⃗  (𝑥, 𝑦, 𝑧, 𝑡))}, 

𝑤3(𝑥, 𝑦, 𝑧, 𝑡) = 𝑤2(𝑥, 𝑦, 𝑧, 𝑡) − 𝐸−1{𝑅3(𝑤2⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑧, 𝑡))}, 

⋮ 

Where,” 

𝑅1(𝑤0⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑧, 𝑡)) = 𝐸[𝑤0] − 𝑣2. 𝑒𝑖(𝑥+𝑦+𝑧) − 𝑣𝛼𝑖𝐸[(𝑤0)𝑥𝑥 + (𝑤0)𝑦𝑦 + (𝑤0)𝑧𝑧 + 4𝑤0
2𝑤0̅̅̅̅ ], 

𝑅2(𝑤1⃗⃗ ⃗⃗  (𝑥, 𝑦, 𝑧, 𝑡)) = 𝐸[𝑤1] − 𝑣𝛼𝑖𝐸[(𝑤1)𝑥𝑥 + (𝑤1)𝑦𝑦 + (𝑤1)𝑧𝑧 + 4𝑤0
2𝑤1̅̅̅̅ + 8𝑤0𝑤0̅̅̅̅ 𝑤1], 

𝑅3(𝑤2⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑧, 𝑡)) = 𝐸[𝑤2] 

−𝑣𝛼𝑖𝐸[(𝑤2)𝑥𝑥 + (𝑤2)𝑦𝑦 + (𝑤2)𝑧𝑧 + 4𝑤0
2𝑤2̅̅̅̅ + 8𝑤0𝑤1̅̅̅̅ 𝑤1 + 8𝑤0𝑤0̅̅̅̅ 𝑤2 + 4𝑤0̅̅̅̅ 𝑤1

2], 

⋮ 

After simplifications, we obtain 

𝑅1(𝑤0⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑧, 𝑡)) = −𝑖. 𝑣𝛼+2𝑒𝑖(𝑥+𝑦+𝑧), 

𝑅2(𝑤1⃗⃗ ⃗⃗  (𝑥, 𝑦, 𝑧, 𝑡)) = 𝑒𝑖(𝑥+𝑦+𝑧)(𝑖𝑣𝛼+2 + 𝑣𝛼+3), 

𝑅3(𝑤2⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑧, 𝑡)) = 𝑒𝑖(𝑥+𝑦+𝑧){−𝑣𝛼+3 − 𝑖𝑣𝛼+4}, 

⋮ 

Therefore,  

𝑤1(𝑥, 𝑦, 𝑧, 𝑡) = 𝑖
𝑡𝛼

(𝛼)!
𝑒𝑖(𝑥+𝑦+𝑧), 

𝑤2(𝑥, 𝑦, 𝑧, 𝑡) = 𝑖2
𝑡𝛼+1

(𝛼 + 1)!
𝑒𝑖(𝑥+𝑦+𝑧), 



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𝑤3(𝑥, 𝑦, 𝑧, 𝑡) = 𝑖3
𝑡𝛼+2

(𝛼 + 2)!
𝑒𝑖(𝑥+𝑦+𝑧), 

⋮ 

For 𝛼 = 1, the solution is: 

𝑤(𝑥, 𝑦, 𝑧, 𝑡) = 𝑤0 + 𝑤1 + 𝑤2 + 𝑤3 + ⋯ 

Or 

𝑤(𝑥, 𝑦, 𝑧, 𝑡) = 𝑒𝑖(𝑥+𝑦+𝑧) {1 + (𝑖𝑡) +
(𝑖𝑡)2

2!
+ ⋯} = 𝑒𝑖(𝑥+𝑦+𝑧+𝑡) 

 

Figure 1: Physical behavior of solutions of real part for 𝑧 = 2 and 𝑡 = 0.5 

 

Figure 2: Physical behavior of solutions of imaginary part for 𝑧 = 2 and 𝑡 = 0.5 

-5 -4 -3 -2 -1 0 1 2 3 4 5 -5

0

5
-1

-0.5

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0.5

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y

w (Real Part)

x

w
 (

R
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a
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P

a
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)

-5 -4 -3 -2 -1 0 1 2 3 4 5 -5

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-1

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1

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w(Imaginary part)

x

w
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 p
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)



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Figure 3: Physical behavior of solutions of real part for 𝑧 = 10 and 𝑡 = 2 

 

Figure 4: Physical behavior of solutions of imaginary part for 𝑧 = 10 and 𝑡 = 2 

Figures 1 & 2 show the real & the imaginary part solutions' physical behavior of Example 1 at 𝑧 = 2,

𝑡 = 0.5 respectively.  Figures 3 & 4 show the real & the imaginary part solution's physical behavior 

of Example 1 at 𝑧 = 10, 𝑡 = 2 respectively.   

Example 2: Consider the (2+1)-D fractional nonlinear Schrodinger eqn of the form 

𝑖𝑤𝑡
𝛼 = −

1

4
𝑤𝑥𝑥 −

1

4
𝑤𝑦𝑦 − 𝑤sin2𝑥 sin2𝑦 + |𝑤|2𝑤,                                   (11) 

with initial “condition 𝑤(𝑥, 𝑦, 𝑧, 0) = sin 𝑥 sin 𝑦. The exact solution to the problem (𝛼 = 1) is:  

𝑤(𝑥, 𝑦, 𝑡) = 𝑒−𝑖𝑡/2 sin 𝑥 sin 𝑦 

Rewrite the given problem as: 

𝑤𝑡
𝛼 = 𝑖 (

1

4
𝑤𝑥𝑥 +

1

4
𝑤𝑦𝑦 + 𝑤sin2𝑥 sin2𝑦 − 𝑤2�̅�)                               (12) 

Taking Elzaki transform to both sides of Eqn (12), we obtain 

𝐸[𝑤𝑡
𝛼] = 𝐸 [𝑖 (

1

4
𝑤𝑥𝑥 +

1

4
𝑤𝑦𝑦 + 𝑤sin2𝑥 sin2𝑦 − 𝑤2�̅�)] 

This implies  

-2 -1 0 1 2 -2
0

2-1

-0.5

0

0.5

1

y

w (Real part)

x

w(
Re

al
 p

ar
t)

-2 -1 0 1 2 -2
0

2
-1

-0.5

0

0.5

1

y
x

w( Imaginary part)

w
 (I

m
ag

in
ar

y 
pa

rt)



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𝐸[𝑤(𝑥, 𝑦, 𝑡)] = ∑ 𝑣𝑖+2

𝑛−1

𝑖=0

𝑤(𝑖)(𝑥, 𝑦, 0) + 𝑣𝛼𝑖𝐸 [
1

4
𝑤𝑥𝑥 +

1

4
𝑤𝑦𝑦 + 𝑤sin2𝑥 sin2𝑦 − 𝑤2�̅�] 

After applying initial conditions, we obtain 

𝐸[𝑤(𝑥, 𝑦, 𝑡)] = 𝑣2. sin 𝑥 sin 𝑦 + 𝑣𝛼𝑖𝐸 [
1

4
𝑤𝑥𝑥 +

1

4
𝑤𝑦𝑦 + 𝑤sin2𝑥 sin2𝑦 − 𝑤2�̅�] 

Or 

𝐸[𝑤(𝑥, 𝑦, 𝑡)] − 𝑣2. sin 𝑥 sin 𝑦 − 𝑣𝛼𝑖𝐸 [
1

4
𝑤𝑥𝑥 +

1

4
𝑤𝑦𝑦 + 𝑤sin2𝑥 sin2𝑦 − 𝑤2�̅�] = 0 

The nonlinear component is: 

𝑅[𝜑(𝑥, 𝑦, 𝑡; 𝑝)] = 𝐸[𝜑] − 𝑣2. sin 𝑥 sin 𝑦 − 𝑣𝛼𝑖𝐸 [
1

4
𝜑𝑥𝑥 +

1

4
𝜑𝑦𝑦 + 𝜑sin2𝑥 sin2𝑦 − 𝜑2�̅�] (13) 

We build the zero-order deformation eqn with the assumption 𝐻(𝑥, 𝑦, 𝑡) = 1,  

(1 − 𝑝)𝐸{𝜑(𝑥, 𝑦, 𝑡) − 𝑤0(𝑥, 𝑦, 𝑡)} = 𝑝ℎ𝑅[𝜑(𝑥, 𝑦, 𝑡; 𝑝)] 

When 𝑝 = 0 &   𝑝 = 1, we get 

{
𝜑(𝑥, 𝑦, 𝑡; 0) = 𝑤0(𝑥, 𝑦, 0)

𝜑(𝑥, 𝑦, 𝑡; 1) = 𝑤(𝑥, 𝑦, 𝑡)
 

Therefore, the mth-order deformation eqn  

𝐸{𝑤𝑚(𝑥, 𝑦, 𝑡) − 𝜒𝑚𝑤𝑚−1(𝑥, 𝑦, 𝑡)} = ℎ𝑅𝑚(𝑤𝑚−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗(𝑥, 𝑦, 𝑡))                     (14) 

Inverse Elzaki transforms both sides and gives 

𝑤𝑚(𝑥, 𝑦, 𝑡) − 𝜒𝑚𝑤𝑚−1(𝑥, 𝑦, 𝑡) = 𝐸−1{ℎ𝑅𝑚(𝑤𝑚−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗(𝑥, 𝑦, 𝑡))}                 (15) 

From Equation (15) (Taking ℎ = −1), we obtain 

𝑤1(𝑥, 𝑦, 𝑡) = −𝐸−1{𝑅1(𝑤0⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑡))}, 

𝑤2(𝑥, 𝑦, 𝑡) = 𝑤1(𝑥, 𝑦, 𝑡) − 𝐸−1{𝑅2(𝑤1⃗⃗ ⃗⃗  (𝑥, 𝑦, 𝑡))}, 

𝑤3(𝑥, 𝑦, 𝑡) = 𝑤2(𝑥, 𝑦, 𝑡) − 𝐸−1{𝑅3(𝑤2⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑡))}, 

⋮ 

where 

𝑅1(𝑤0⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑡)) = 𝐸[𝑤0] − 𝑣2. sin 𝑥 sin 𝑦 

−𝑣𝛼𝑖𝐸 [
1

4
(𝑤0)𝑥𝑥 +

1

4
(𝑤0)𝑦𝑦 + 𝑤0sin

2𝑥 sin2𝑦 − 𝑤0
2𝑤0̅̅̅̅ ], 

𝑅2(𝑤1⃗⃗ ⃗⃗  (𝑥, 𝑦, 𝑡)) = 𝐸[𝑤1] − 𝑣𝛼𝑖𝐸 [
1

4
(𝑤1)𝑥𝑥 +

1

4
(𝑤1)𝑦𝑦 + 𝑤1sin

2𝑥 sin2𝑦 − 𝑤0
2𝑤1̅̅̅̅ − 2𝑤0𝑤0̅̅̅̅ 𝑤1], 

𝑅3(𝑤2⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑡)) = 𝐸[𝑤2] 



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−𝑣𝛼𝑖𝐸 [
1

4
(𝑤2)𝑥𝑥 +

1

4
(𝑤2)𝑦𝑦 + 𝑤2sin

2𝑥 sin2𝑦 − 𝑤0
2𝑤2̅̅̅̅ − 2𝑤0𝑤1̅̅̅̅ 𝑤1 − 2𝑤0𝑤0̅̅̅̅ 𝑤2 − 𝑤0̅̅̅̅ 𝑤1

2], 

⋮ 

After simplifications, we obtain 

𝑅1(𝑤0⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑡)) = 𝑖. 𝑣𝛼+2 sin 𝑥 sin 𝑦, 

𝑅2(𝑤1⃗⃗ ⃗⃗  (𝑥, 𝑦, 𝑡)) = sin 𝑥 sin 𝑦 (−
𝑖

2
𝑣𝛼+2 +

1

4
𝑣𝛼+3), 

𝑅3(𝑤2⃗⃗⃗⃗  ⃗(𝑥, 𝑦, 𝑡)) = sin 𝑥 sin 𝑦 {−
1

4
𝑣𝛼+3 −

𝑖

8
𝑣𝛼+4}, 

⋮ 

Therefore,  

𝑤1(𝑥, 𝑦, 𝑧, 𝑡) = −𝑖
(𝑡 2⁄ )

𝛼

𝛼!
sin 𝑥 sin 𝑦, 

𝑤2(𝑥, 𝑦, 𝑧, 𝑡) = 𝑖2
(𝑡 2⁄ )

𝛼+1

(𝛼 + 1)!
sin 𝑥 sin 𝑦, 

𝑤3(𝑥, 𝑦, 𝑧, 𝑡) = −𝑖3
(𝑡 2⁄ )

𝛼+2

(𝛼 + 2)!
sin 𝑥 sin 𝑦, 

⋮ 

For 𝛼 = 1, the solution” is: 

𝑤(𝑥, 𝑦, 𝑧, 𝑡) = 𝑤0 + 𝑤1 + 𝑤2 + 𝑤3 + ⋯ 

Or 

𝑤(𝑥, 𝑦, 𝑧, 𝑡) = sin 𝑥 sin 𝑦 {1 + (
𝑖𝑡

2
) +

(
𝑖𝑡
2)

2

2!
+ ⋯} = 𝑒−𝑖𝑡/2 sin 𝑥 sin 𝑦 

which is totally equal to the exact solution. 



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Figure 5: Physical behavior of real part solution at  𝑡 = 0.5 

 

Figure 6: Physical behavior of imaginary part solution at  𝑡 = 0.5 

 

Figure 7: Physical behavior of the solution of real part at  𝑡 = 2 

-5

0

5

-5

0

5
-1

-0.5

0

0.5

1

x

w (Real part)

y

w 
(R

ea
l p

ar
t)

-5

0

5

-5

0

5
-0.4

-0.2

0

0.2

0.4

x

w (Imaginary part)

y

w
 (I

m
ag

in
ar

y 
pa

rt)

-2 -1 0 1 2-2
0

2

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

x

w (real part)

y

w 
(re

al
 p

ar
t)



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Figure 8: Physical behavior of the solution of imaginary part at  𝑡 = 2 

Figures 5 & 6 show the real & imaginary part solutions physical behavior of Example 2 at 𝑡 = 0.5 

respectively.  Figures 7 & 8 show the real & imaginary part solutions physical behavior of Example 2 

at 𝑡 = 2 respectively.   

6. Conclusion 

The numerical data suggests that the Elzaki transform-based Homotopy analysis technique provides 

accurate solutions for solving (2+1)-D and (3+1)-D nonlinear fractional Schrodinger eqns. In the 

future, this approach will be valid for various applications of sciences and engineering.    

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-1

-0.5

0

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1

x

w (Imaginary part)

y

w
(Im

ag
in

ar
y 

pa
rt)



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