Analytical solution of time-fractional Schr<i>ö</i>dinger equations via Shehu Adomian Decomposition Method

Mamta Kapoor; Nehad Ali Shah; Wajaree Weera; Mamta Kapoor; Nehad Ali Shah; Wajaree Weera

doi:10.3934/math.20221074

AIMS Mathematics

2022, Volume 7, Issue 10: 19562-19596. doi: 10.3934/math.20221074

Previous Article Next Article

Research article

Analytical solution of time-fractional Schrödinger equations via Shehu Adomian Decomposition Method

1.
Department of Mathematics, Lovely Professional University, Phagwara, Punjab, India
2.
Department of Mechanical Engineering Sejong University, Seoul 05006, South Korea
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
^†These authors contributed equally to this work and are co-first authors

Received: 29 June 2022 Revised: 27 July 2022 Accepted: 08 August 2022 Published: 05 September 2022
MSC : 49M27, 35R11

Present research deals with the time-fractional Schrödinger equations aiming for the analytical solution via Shehu Transform based Adomian Decomposition Method [STADM]. Three types of time-fractional Schrödinger equations are tackled in the present research. Shehu transform ADM is incorporated to solve the time-fractional PDE along with the fractional derivative in the Caputo sense. The developed technique is easy to implement for fetching an analytical solution. No discretization or numerical program development is demanded. The present scheme will surely help to find the analytical solution to some complex-natured fractional PDEs as well as integro-differential equations. Convergence of the proposed method is also mentioned.

Keywords:

Citation: Mamta Kapoor, Nehad Ali Shah, Wajaree Weera. Analytical solution of time-fractional Schrödinger equations via Shehu Adomian Decomposition Method[J]. AIMS Mathematics, 2022, 7(10): 19562-19596. doi: 10.3934/math.20221074

Related Papers:

[1]	Maysaa Al-Qurashi, Saima Rashid, Fahd Jarad, Madeeha Tahir, Abdullah M. Alsharif . New computations for the two-mode version of the fractional Zakharov-Kuznetsov model in plasma fluid by means of the Shehu decomposition method. AIMS Mathematics, 2022, 7(2): 2044-2060. doi: 10.3934/math.2022117
[2]	Rasool Shah, Abd-Allah Hyder, Naveed Iqbal, Thongchai Botmart . Fractional view evaluation system of Schrödinger-KdV equation by a comparative analysis. AIMS Mathematics, 2022, 7(11): 19846-19864. doi: 10.3934/math.20221087
[3]	Sajjad Ali Khan, Kamal Shah, Poom Kumam, Aly Seadawy, Gul Zaman, Zahir Shah . Study of mathematical model of Hepatitis B under Caputo-Fabrizo derivative. AIMS Mathematics, 2021, 6(1): 195-209. doi: 10.3934/math.2021013
[4]	Azzh Saad Alshehry, Humaira Yasmin, Rasool Shah, Roman Ullah, Asfandyar Khan . Numerical simulation and analysis of fractional-order Phi-Four equation. AIMS Mathematics, 2023, 8(11): 27175-27199. doi: 10.3934/math.20231390
[5]	Ihsan Ullah, Aman Ullah, Shabir Ahmad, Hijaz Ahmad, Taher A. Nofal . A survey of KdV-CDG equations via nonsingular fractional operators. AIMS Mathematics, 2023, 8(8): 18964-18981. doi: 10.3934/math.2023966
[6]	Yousef Jawarneh, Humaira Yasmin, M. Mossa Al-Sawalha, Rasool Shah, Asfandyar Khan . Numerical analysis of fractional heat transfer and porous media equations within Caputo-Fabrizio operator. AIMS Mathematics, 2023, 8(11): 26543-26560. doi: 10.3934/math.20231356
[7]	Muammer Ayata, Ozan Özkan . A new application of conformable Laplace decomposition method for fractional Newell-Whitehead-Segel equation. AIMS Mathematics, 2020, 5(6): 7402-7412. doi: 10.3934/math.2020474
[8]	Zonghu Xiu, Shengjun Li, Zhigang Wang . Existence of infinitely many small solutions for fractional Schrödinger-Poisson systems with sign-changing potential and local nonlinearity. AIMS Mathematics, 2020, 5(6): 6902-6912. doi: 10.3934/math.2020442
[9]	Azzh Saad Alshehry, Naila Amir, Naveed Iqbal, Rasool Shah, Kamsing Nonlaopon . On the solution of nonlinear fractional-order shock wave equation via analytical method. AIMS Mathematics, 2022, 7(10): 19325-19343. doi: 10.3934/math.20221061
[10]	Ahmed M. Zidan, Adnan Khan, Rasool Shah, Mohammed Kbiri Alaoui, Wajaree Weera . Evaluation of time-fractional Fisher's equations with the help of analytical methods. AIMS Mathematics, 2022, 7(10): 18746-18766. doi: 10.3934/math.20221031

Abstract

1. Introduction

Fractional calculus is a modification of the notion of differentiation and integration of arbitrary orders ^{[1,2,3,4,5,6,7]}. As there exist various prototypes in engineering and sciences, fractional calculus has attained the researcher's interest. Multiple models in numerous aspects such as; physics, biology, and engineering are tackled as per fractional differential and integral calculus. Some examples are electrochemistry, signal processing, diffusion, finance, acoustic, plasma physics, image processing, and others ^{[8,9,10,11,12,13]}.

Integral transforms are notified as one of the most suitable approaches to tackle models regarding applied mathematics, mathematical physics, engineering, and some other branches as well. The primary motivation is to deal with the provided mathematical model via any suitable integral transform and to retrieve the associated outcome in the best possible approach.

Via an appropriate selection of integral transform, the differential and integral equations can be transformed into an algebraic equations system, which can be easily tackled.

Various integral transforms and integral transform-based approaches are generated and incorporated for this purpose, such as; Laplace transform ^[14], Sumudu transform ^[15], Elzaki transform ^[16], and Natural transform ^[17], and many others.

Definition 1: Shehu transform of the function $\theta \left(\tau \right)$ is defined over the following functions ^[18,19,20].

$A = \left\{\epsilon \left(\tau \right):there\;exists\;N, {k}_{1}, {k}_{2} > 0\;s.t.\;\left|\theta \left(\tau \right)\right| < Nexp\left(\frac{\left|\tau \right|}{{k}_{i}}\right),\; where\;\tau \in {\left(-1\right)}^{j}\times [0, \infty )\right\}.$

(1)

Definition 2: Shehu transform of function $\epsilon \left(\tau \right)$ is defined as follows ^[18,19,20]:

$S\left[\epsilon \left(\tau \right)\right] = F\left[s, u\right] = {\int }_{0}^{\infty }exp\left[-\frac{s\tau }{u}\right]\epsilon \left(\tau \right)d\tau .$

(2)

Definition 3: Inverse Shehu transform is defined as follows ^[18,19,20]:

${S}^{-1}\left[F\left(s, u\right)\right] = \epsilon \left(\tau \right).$

(3)

Where $s$ , $u$ are the Shehu transform variables. $\alpha \in R$ .

Definition 4: ^[18,19,20]

$S\left[{\epsilon }^{{'}}\left(\tau \right)\right] = \frac{s}{u}F\left[s, u\right]-\epsilon \left(0\right) ,$

(4)

$S\left[{\epsilon }^{{'}{'}}\left(\tau \right)\right] = \frac{{s}^{2}}{{u}^{2}}F\left(s, u\right)-\frac{s}{u}\epsilon \left(0\right)-\epsilon {'}\left(0\right) ,$

(5)

$S\left[{\epsilon }^{{'}{'}{'}}\left(\tau \right)\right] = \frac{{s}^{3}}{{u}^{3}}F\left[s, u\right]-\frac{{s}^{2}}{{u}^{2}}\epsilon \left(0\right)-\frac{s}{u}{\epsilon }^{{'}}\left(0\right)-\epsilon {'}{'}\left(0\right) .$

(6)

Definition 5: Linearity property of Shehu transform ^[18,19,20]:

$S\left[{c}_{1}{\epsilon }_{1}\left(\tau \right)+{c}_{2}{\epsilon }_{2}\left(\tau \right)\right] = {c}_{1}S\left[{\epsilon }_{1}\left(\tau \right)\right]+{c}_{2}S\left[{\epsilon }_{2}\left(\tau \right)\right] .$

(7)

Definition 6: Linearity property of inverse Shehu transform ^[18,19,20]:

${\epsilon }_{1}\left(\tau \right) = {S}^{-1}\left[{F}_{1}\right(s, u\left)\right]\; {\rm{and}} \; {\epsilon }_{2}\left(\tau \right) = {S}^{-1}\left[{F}_{2}\right(s, u\left)\right] ,$

then

${S}^{-1}\left[{c}_{1}{F}_{1}(s, u)+{c}_{2}{F}_{2}(s, u)\right] = {c}_{1}{S}^{-1}\left[{F}_{1}(s, u)\right]+{c}_{2}{S}^{-1}\left[{F}_{2}(s, u)\right],$

${S}^{-1}\left[{c}_{1}{F}_{1}(s, u)+{c}_{2}{F}_{2}(s, u)\right] = {c}_{1}{\epsilon }_{1}\left(\tau \right)+{c}_{2}{\epsilon }_{2}\left(\tau \right) .$

Definition 7: Shehu transform of Caputo fractional derivative [C.F.D.] ^[20,21]:

$S\left[{D}_{t}^{\alpha }\epsilon \left(\mu , \tau \right)\right] = \frac{{s}^{\alpha }}{{\nu }^{\alpha }}S\left[\epsilon \left(\mu , \tau \right)\right]-{\sum }_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\epsilon }^{r}\left(\mu , 0\right).$

(8)

Definition 8: Mittag-Leffler function considered for two parameters was given in ^[22,23,24].

${E}_{\mu , \nu }\left(n\right) = {\sum }_{k = 0}^{\infty }\frac{{n}^{k}}{\mathrm{\Gamma }(k\mu +\nu )} .$

(9)

Where ${E}_{\mathrm{1, 1}}\left(n\right) = \mathrm{e}\mathrm{x}\mathrm{p}\left(n\right)$ and ${E}_{\mathrm{2, 1}}\left({n}^{2}\right) = \mathrm{c}\mathrm{o}\mathrm{s}\left(n\right)$ .

In Tables 1 and 2 basic formulae regarding Shehu transform and inverse Shehu transform are provided.

Table 1. Chart regarding to the Shehu transform ^[20].

	$\mathit{\boldsymbol{\epsilon }}\left(\mathit{\boldsymbol{\tau }}\right)$	$\mathit{\boldsymbol{S}}\left[\mathit{\boldsymbol{\epsilon }}\right(\mathit{\boldsymbol{\tau }}\left)\right]=\mathit{\boldsymbol{F}}(\mathit{\boldsymbol{s}}, \mathit{\boldsymbol{\nu }})$
1.	$1$	$\frac{\nu }{s}$
2.	$\tau$	$\frac{{\nu }^{2}}{{s}^{2}}$
3.	${\tau }^{m}, m\in N$	$\angle m{\left(\frac{\nu }{s}\right)}^{m+1}$
4.	${\tau }^{m}, m > -1$	$\mathrm{\Gamma }(m+1){\left(\frac{\nu }{s}\right)}^{m+1}$
5.	${e}^{a\tau }$	$\frac{\nu }{s-a\nu }$
6.	$\mathrm{s}\mathrm{i}\mathrm{n}\left(m\tau \right)$	$\frac{m{\nu }^{2}}{{s}^{2}+{m}^{2}{\nu }^{2}}$
7.	$cos\left(m\tau \right)$	$\frac{s{\nu }^{2}}{{s}^{2}+{m}^{2}{\nu }^{2}}$
8.	$sinh\left(m\tau \right)$	$\frac{m{\nu }^{2}}{{s}^{2}-{m}^{2}{\nu }^{2}}$
9.	$cosh\left(m\tau \right)$	$\frac{s{\nu }^{2}}{{s}^{2}-{m}^{2}{\nu }^{2}}$

| Show Table

DownLoad: CSV

Table 2. Chart regarding to the inverse Shehu transform ^[20].

	$\mathit{\boldsymbol{F}}(\mathit{\boldsymbol{s}}, \mathit{\boldsymbol{\nu }})$	$\mathit{\boldsymbol{\epsilon }}\left(\mathit{\boldsymbol{\tau }}\right)={\mathit{\boldsymbol{S}}}^{{\bf{-1}}}\left[\mathit{\boldsymbol{F}}\right(\mathit{\boldsymbol{s}}, \mathit{\boldsymbol{\nu }}\left)\right]$
1.	$\frac{\nu }{s}$	$1$
2.	$\frac{{\nu }^{2}}{{s}^{2}}$	$\tau$
3.	${\left(\frac{\nu }{s}\right)}^{m+1}$	$\frac{{\tau }^{m}}{\angle m}$
4.	$\mathrm{\Gamma }(m+1){\left(\frac{\nu }{s}\right)}^{m+1}$	$\frac{{\tau }^{m}}{\mathrm{\Gamma }(m+1)}$
5.	$\frac{\nu }{s-a\nu }$	${e}^{a\tau }$
6.	$\frac{m{\nu }^{2}}{{s}^{2}+{m}^{2}{\nu }^{2}}$	$\mathrm{s}\mathrm{i}\mathrm{n}\left(m\tau \right)$
7.	$\frac{s{\nu }^{2}}{{s}^{2}+{m}^{2}{\nu }^{2}}$	$cos\left(m\tau \right)$
8.	$\frac{m{\nu }^{2}}{{s}^{2}-{m}^{2}{\nu }^{2}}$	$sinh\left(m\tau \right)$
9.	$\frac{s{\nu }^{2}}{{s}^{2}-{m}^{2}{\nu }^{2}}$	$cosh\left(m\tau \right)$

| Show Table

DownLoad: CSV

${\bf{1}}\mathit{\boldsymbol{D}}$ Non-linear time-fractional Schrödinger equation.

$i{D}_{t}^{\alpha }\theta \left(\mu , \tau \right) = R\left[\theta \left(\mu , \tau \right)\right]+N\left[\theta \left(\mu , \tau \right)\right]+\phi \left(\mu , \tau \right).$

(10)

${\bf{2}}\mathit{\boldsymbol{D}}$ Non-linear time-fractional Schrödinger equation.

$i{D}_{t}^{\alpha }\theta ({\mu }_{1}, {\mu }_{2}, \tau ) = R\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+\phi ({\mu }_{1}, {\mu }_{2}, \tau ) .$

(11)

${\bf{3}}\mathit{\boldsymbol{D}}$ Non-linear time-fractional Schrödinger equation.

$i{D}_{t}^{\alpha }\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = R\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right).$

(12)

One of the well-known models in mathematical physics is Schr $\ddot{o}$ dinger equation model. There exist various implementations in numerous branches, such as; non-linear optics ^[25], mean-field theory of Bose-Einstein condensates ^[26,27], and plasma physics ^[28]. One emerging aspect of quantum physics is considered as fractional Schr $\ddot{o}$ dinger equation; which is associated with the notion of non-local quantum phenomena.

Naber ^[29] notified time-fractional Schr $\ddot{o}$ dinger equation regarding Caputo derivative. Wang and Xu ^[30] elaborated on the linear Schr $\ddot{o}$ dinger equation regarding the space-fractional and time-fractional aspects as well as tackled the models via integral transform approach. Due to the existence of numerous implementations of the time-fractional Schr $\ddot{o}$ dinger equation; various researchers have worked in this field. Regarding this, novel analytical and numerical regimes have been generated for the time-fractional Schr $\ddot{o}$ dinger equation ^{[31,32,33,34]}. Hemida et al. ^[35] implemented HAM to provide the approximated results for the space-time fractional Schr $\ddot{o}$ dinger equation. More work related to fractional Schr $\ddot{o}$ dinger equation is provided in ^[36,37,38]. Other noteworthy work in this regard is notified as ^{[39,40,41,42]}.

The main advantage of ADM is that it does not rely upon perturbation or linearization or any discretization. Therefore, the actual outcome of the model remains unchanged. Discretization of variables is not demanded, which is a difficult and challenging approach. It means that the obtained results are error-free, which occurred because of discretization. Furthermore, it is accurate in finding the approximated and exact solutions of the non-linear prototypes. Such methods can be implemented to the diversified differential equations such as; integro-differential equations, differential-algebraic equations, differential-difference equations, as well as some functional equations, eigenvalue problems, and Stochastic system problems.

The main idea of the present study is to concentrate on the implementation of Shehu ADM for attaining the exact solution to the Schr $\ddot{o}$ dinger equations in various dimensions. Some latest research regarding this field is provided on ^{[43,44,45,46,47]}.

Novelty and significance of the paper

There are several schemes observed in the literature that deal with fractional Schr $\ddot{o}$ dinger equation in one, two, or three dimensions, but rare methods are provided that tackles fractional Schr $\ddot{o}$ dinger equation in all one, two, and three dimensions. Therefore, the authors have focused on developing a technique that proves the validity of the approximated-analytical solution of the mentioned equations in one, two, and three dimensions.

An iterative scheme is developed in the present research regarding the solution of fractional Schr $\ddot{o}$ dinger equation in one, two, and three dimensions. The present scheme is easy to implement and needs no complex programming regarding numerical discretization. Developing the numerical programs for the fractional PDEs is not an easy task; therefore, developing such iterative schemes is the need of time to find the approximated-analytical solutions. There exist several transforms provided in the literature, but from the calculation aspect, some transforms are easy to implement, and some are not. Shehu ADM is noticed as one of the easiest methods to implement integral transform among all existing integral transforms; as in the case of Shehu ADM, no perturbation parameter is required. Via literature, it is observed that fractional Schrödinger equations have never been solved in one, two, and three dimensions with the aid of a single integral transform. Therefore, due to the importance of such equations, in this research, concentration is focused upon the solution for the same, which retains the novelty of the study. Furthermore, convergence analyses are also incorporated in the article.

Motivation of the study

In the present research, an iterative regime is developed and incorporated named Shehu ADM regarding the solution of fractional Schr $\ddot{o}$ dinger equation in one, two, and three dimensions. The present regime is easy to implement and needs no complex calculation in the process of numerical discretization. Generating the numerical programs to deal with the fractional PDEs is cumbersome; therefore, generating such novel iterative regimes is demanded to fetch the approximated-analytical solutions.

There exist numerous transforms in the literature, but as per the calculation aspect, some transforms are easy to incorporate, but some are not. Shehu transform ADM is notified as one of the easiest methods. From an exploration of the literature, it is noticed that fractional Schrödinger equations are rarely solved in one, two, and three dimensions with the aid of a single integral transform-based method. Therefore, in this research, the focus is on finding the solution for the same, which contains the novelty of the research. Moreover, error analysis and convergence analysis are also elaborated in this article.

In the present paper, the convergence of the method is checked numerically. Convergence is affirmed via Tables 3–. As per –, it is observed that on increasing the number of grid points the ${L}_{\infty }$ error norm got reduced rapidly, which is robust proof of the convergence of the generated semi-analytical techniques.

Table 3. Comparison of

${L}_{\infty }$ errors at different time levels regarding Example 1.

$\mathit{\boldsymbol{N}}$	${\mathit{\boldsymbol{L}}}_{{\infty }}$ at $\mathit{\boldsymbol{t}}$ =1	${\mathit{\boldsymbol{L}}}_{{\infty }}$ at $\mathit{\boldsymbol{t}}$ =2	${\mathit{\boldsymbol{L}}}_{{\infty }}$ at $\mathit{\boldsymbol{t}}$ =3
11	2.4979e-08	5.0711e-05	4.3236e-03
21	4.4755e-16	4.1023e-14	2.0302e-10
31	4.4755e-16	5.6610e-16	1.3911e-15
	Convergence up to ${{\bf{10}}}^{{\bf{-16}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-16}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-16}}}$

| Show Table

DownLoad: CSV

Table 4. Comparison of

${L}_{\infty }$ errors at different time levels regarding Example 2.

$\mathit{\boldsymbol{N}}$	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =0.1	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =0.2	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =0.3
11	7.8430e-09	1.5949e-05	1.3635e-03
21	2.4825e-16	4.8963e-15	2.2250e-11
31	3.7238e-16	5.5788e-16	1.1802e-15
	Convergence up to ${{\bf{10}}}^{{\bf{-16}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-16}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-15}}}$

| Show Table

DownLoad: CSV

Table 5. Comparison of

${L}_{\infty }$ errors at different time levels regarding Example 3.

$\mathit{\boldsymbol{N}}$	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =1.0	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =1.3	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =1.5
21	3.1906e-07	7.8034e-05	1.5627e-03
31	8.4843e-15	7.0869e-12	5.9702e-10
41	6.4393e-15	2.9407e-14	5.6576e-14
	Convergence up to ${{\bf{10}}}^{{\bf{-15}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-14}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-14}}}$

| Show Table

DownLoad: CSV

Table 6. Comparison of

${L}_{\infty }$ errors at different time levels regarding Example 4.

$\mathit{\boldsymbol{N}}$	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =1.0	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =1.3	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =1.5
11	1.8114e-06	3.2318e-05	1.5541e-04
21	1.3092e-16	2.0510e-14	4.0767e-13
31	2.2377e-16	2.4825e-16	2.4825e-16
	Convergence up to ${{\bf{10}}}^{{\bf{-16}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-16}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-16}}}$

| Show Table

DownLoad: CSV

Table 7. Comparison of

${L}_{\infty }$ errors at different time levels regarding Example 5.

$\mathit{\boldsymbol{N}}$	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =1	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =2	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =3
21	4.0910e-14	8.4807e-08	4.1536e-04
31	3.3766e-16	1.9860e-15	1.5697e-10
41	3.3766e-16	1.7342e-15	1.6577e-14
	Convergence up to ${{\bf{10}}}^{{\bf{-16}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-15}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-14}}}$

| Show Table

DownLoad: CSV

Table 8. Comparison of

${L}_{\infty }$ errors at different time levels regarding Example 6.

$\mathit{\boldsymbol{N}}$	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =1	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =2	${\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{\infty }}}$ at $\mathit{\boldsymbol{t}}$ =3
21	4.4168e-13	9.1039e-07	4.4156e-03
31	5.7220e-17	5.5268e-14	1.5682e-08
41	4.6548e-17	6.9573e-16	8.8374e-15
	Convergence up to ${{\bf{10}}}^{{\bf{-17}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-16}}}$	Convergence up to ${{\bf{10}}}^{{\bf{-15}}}$

| Show Table

DownLoad: CSV

2. Outline of the paper

● The present paper is divided into different sections and subsections.

● In Section 3, Implementation of the Shehu ADM is developed for various kinds of fractional Schr $\ddot{o}$ dinger equations.

● In Sub-section 3.1, the general formula is generated for $1D$ time-fractional Schrödinger equation.

● In Sub-section 3.2, the general formula is generated for $2D$ time-fractional Schrödinger equation.

● In Sub-section 3.3, the general formula is generated for $3D$ time-fractional Schrödinger equation.

● In Section 4, six examples are elaborated to validate the efficiency and efficacy of the developed regime.

● In Section 5, graphical analysis, error analysis, and convergence analysis is notified.

● Section 6 is provided as the concluding remarks.

3. Implementation of the proposed regime

3.1. Implementation of the scheme on $1{D}$ time-fractional Schr $\ddot{{o}}$ dinger equation

Applying Shehu transform upon Eq (1):

$\begin{array}{l} iS\left[{D}_{\tau }^{\alpha }\theta \left(\mu , \tau \right)\right] = S\left[R\left[\theta \left(\mu , \tau \right)\right]+N\left[\theta \left(\mu , \tau \right)\right]+\phi \left(\mu , \tau \right)\right] \\ \Rightarrow S\left[{D}_{\tau }^{\alpha }\theta \left(\mu , \tau \right)\right] = -iS\left[R\left[\theta \left(\mu , \tau \right)\right]+N\left[\theta \left(\mu , \tau \right)\right]+\phi \left(\mu , \tau \right)\right]\\ {\Rightarrow \left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta \left(\mu , \tau \right)\right]-\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right) = -iS\left[R\left[\theta \left(\mu , \tau \right)\right]+N\left[\theta \left(\mu , \tau \right)\right]+\phi \left(\mu , \tau \right)\right] \\ {\Rightarrow \left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta \left(\mu , \tau \right)\right] = \sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)-iS\left[R\left[\theta \left(\mu , \tau \right)\right]+N\left[\theta \left(\mu , \tau \right)\right]+\phi \left(\mu , \tau \right)\right] \\ \Rightarrow S\left[\theta \left(\mu , \tau \right)\right] = {\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)-i{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[\theta \left(\mu , \tau \right)\right]+N\left[\theta \left(\mu , \tau \right)\right]+\phi \left(\mu , \tau \right)\right] \\ \Rightarrow \theta \left(\mu , \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)\right]-i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[\theta \left(\mu , \tau \right)\right]+N\left[\theta \left(\mu , \tau \right)\right]+\phi \left(\mu , \tau \right)\right]\right] \\ \Rightarrow \sum\limits_{n = 0}^{\infty }{\theta }_{n}(\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)+\phi \left(\mu , \tau \right)\right] \\ \;\;\;\;\;\;\;\;\;\; -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}(\mu , \tau )\right]+N\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}(\mu , \tau )\right]\right]\right] . \end{array}$

(13)

Where,

${\theta }_{0}\left(\mu , \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }{\sum }_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)+\phi \left(\mu , \tau \right)\right].$

(14)

${\theta }_{n+1}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}\left(\mu , \tau \right)\right]+N\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}\left(\mu , \tau \right)\right]\right]\right], n = 0, 1, 2, 3, \dots$

(15)

3.2. Implementation of the scheme on $2{D}$ time-fractional Schr $\ddot{{o}}$ dinger equation

Applying Shehu transform upon Eq (2):

$\begin{array}{l} iS\left[{D}_{t}^{\alpha }\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right] = S\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]\\ \Rightarrow S\left[{D}_{t}^{\alpha }\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right] = -iS\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right] \\ {\Rightarrow \left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]-\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;= -iS\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right] \\ \Rightarrow {\left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;= \sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right)-iS\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right] \\ \Rightarrow S\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;= {\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right] \\ \Rightarrow \theta \left({\mu }_{1}, {\mu }_{2}, \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right)\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]\right] \\ \Rightarrow \sum\limits_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right)+\phi \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+N\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]\right]\right] .\end{array}$

(16)

Where,

$\begin{array}{l} {\theta }_{0}\left({\mu }_{1}, {\mu }_{2}, \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }{\sum }_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right)+\phi \left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]. \\ {\theta }_{n+1}\left({\mu }_{1}, {\mu }_{2}, \tau \right) \end{array}$

(17)

$= -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]+N\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]\right]\right], n = 0, 1, 2, 3, \dots$

(18)

3.3. Implementation of the scheme on $3{D}$ time-fractional Schr $\ddot{{o}}$ dinger equation

Applying Shehu transform upon Eq (3):

$\begin{array}{l} iS\left[{D}_{t}^{\alpha }\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right] = S\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+\phi \left(x, y, z, t\right)\right] \\ \Rightarrow S\left[{D}_{t}^{\alpha }\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right] = -iS\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+\phi \left(x, y, z, t\right)\right] \\ {\Rightarrow \left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]-\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;= -iS\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right] \\ {\Rightarrow \left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;= \sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;-iS\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right] \\ \Rightarrow S\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;= {\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right] \\ \Rightarrow \theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;= {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+N\left[\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+\phi \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]\right] \\ \Rightarrow \sum\limits_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)+\phi \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+N\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]\right]\right] . \end{array}$

(19)

Where,

$\begin{array}{l} {\theta }_{0}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }{\sum }_{r = 0}^{\theta -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)+\phi \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right].\\ {\theta }_{n+1}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) \end{array}$

(20)

$= -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[R\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]+N\left[{\sum }_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right]\right]\right], n = 0, 1, 2, 3, \dots$

(21)

4. Examples and derivation

In the present section, six examples are tested to ensure the validity of the proposed regime, Examples 1–4 are associated with $1D$ time-fractional Schr $\ddot{o}$ dinger. Example 5 is provided regarding $2D$ time-fractional Schrödinger. Example 6 is associated with $3D$ time-fractional Schrödinger equation. In all the provided cases, approximated and exact profiles are generated.

Example 1: Considered $1D$ non-linear time-fractional Schr $\ddot{o}$ dinger as follows ^[36]:

${iD}_{t}^{\alpha }\theta +{\theta }_{\mu \mu }+2{\left|\theta \right|}^{2}\theta = 0 .$

(22)

I.C.: $\theta \left(\mu , 0\right) = {e}^{i\mu }.$

Applying Shehu transform upon Eq (22):

$\begin{array}{l} iS\left[{D}_{\tau }^{\alpha }\theta (\mu , \tau )\right] = -S\left[{\theta }_{\mu \mu }\right(\mu , \tau )+2{\left|\theta (\mu , \tau )\right|}^{2}\theta (\mu , \tau \left)\right] \\ \Rightarrow S\left[{D}_{\tau }^{\alpha }\theta (\mu , \tau )\right] = iS\left[{\theta }_{\mu \mu }\right(\mu , \tau )+2{\left|\theta (\mu , \tau )\right|}^{2}\theta (\mu , \tau \left)\right] \\ \Rightarrow {\left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta (\mu , \tau )\right]-\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right) = iS\left[{\theta }_{\mu \mu }(\mu , \tau )+2{\left|\theta (\mu , \tau )\right|}^{2}\theta (\mu , \tau )\right] \\ \Rightarrow {\left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta (\mu , \tau )\right] = \sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)+iS\left[{\theta }_{\mu \mu }(\mu , \tau )+2{\left|\theta (\mu , \tau )\right|}^{2}\theta (\mu , \tau )\right] \\ \Rightarrow S\left[\theta (\mu , \tau )\right] = {\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)+i{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\theta }_{\mu \mu }\right(\mu , \tau )+2{\left|\theta (\mu , \tau )\right|}^{2}\theta (\mu , \tau \left)\right] \\ \Rightarrow \theta (\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)\right]+i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\theta }_{\mu \mu }(\mu , \tau )+2{\left|\theta (\mu , \tau )\right|}^{2}\theta (\mu , \tau )\right]\right] \\ \Rightarrow \sum\nolimits_{n = 0}^{\infty }{\theta }_{n}(\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\nolimits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)\right]+i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left({\sum }_{n = 0}^{\infty }{\theta }_{n}(\mu , \tau )\right)}_{\mu \mu }+2{\sum }_{n = 0}^{\infty }{A}_{n}\right]\right]. \end{array}$

(23)

Where

$F\left(\theta \right) = {\left|\theta (\mu , \tau )\right|}^{2}\theta (\mu , \tau ) = \sum\limits_{n = 0}^{\infty }{A}_{n}.$

${\theta }_{0}\left(\mu , \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)\right].$

${\theta }_{n+1}(\mu , \tau ) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left({\theta }_{n}(\mu , \tau )\right)}_{\mu \mu }+2{A}_{n}\right]\right], n = 0, 1, 2, 3, \dots$

Considered, $\xi = 1$ :

${\theta }_{0}(\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)\right]$

$\Rightarrow {\theta }_{0}(\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }{\left(\frac{s}{\nu }\right)}^{\alpha -1}\theta \left(\mu , 0\right)\right]$

$\Rightarrow {\theta }_{0}\left(\mu , \tau \right) = \theta \left(\mu , 0\right) = {e}^{i\mu }.$

Considered $n = 0$ :

${\theta }_{1}(\mu , \tau ) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left({\theta }_{0}(\mu , \tau )\right)}_{\mu \mu }+2{A}_{0}\right]\right] .$

Where,

${\left({\theta }_{0}\right)}_{\mu \mu }\left(\mu , \tau \right) = -{e}^{ix}, {A}_{0} = F\left({\theta }_{0}\right) = {\theta }_{0}^{2}\stackrel{-}{{\theta }_{0}} = {e}^{i\mu }.$

${\theta }_{1}\left(\mu , \tau \right) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[-{e}^{i\mu }+2{e}^{i\mu }\right]\right].$

$\Rightarrow {\theta }_{1}(\mu , \tau ) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{e}^{i\mu }\right]\right]$

$\Rightarrow {\theta }_{1}(\mu , \tau ) = i{e}^{i\mu }{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[1\right]\right]$

$\Rightarrow {\theta }_{1}\left(\mu , \tau \right) = i{e}^{i\mu }{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{2\alpha }\right]\Rightarrow {\theta }_{1}\left(\mu , \tau \right) = i{e}^{i\mu }\frac{{t}^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}.$

Considered $n = 1:$

${\theta }_{2}\left(\mu , \tau \right) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left({\theta }_{1}\right)}_{\mu \mu }+2{A}_{1}\right]\right].$

Where,

${\left({\theta }_{1}\right)}_{\mu \mu }(\mu , \tau ) = -i{e}^{i\mu }\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)} ,$

${A}_{1} = 2{\theta }_{0}{\theta }_{1}\stackrel{-}{{\theta }_{0}}+{\theta }_{0}^{2}\stackrel{-}{{\theta }_{1}} = i{e}^{i\mu }\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)} ,$

${\theta }_{2}(\mu , \tau ) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[-i{e}^{i\mu }\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}+2i{e}^{i\mu }\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right]\right]$

$\Rightarrow {\theta }_{2}(\mu , \tau ) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[i{e}^{i\mu }\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right]\right]$

$\Rightarrow {\theta }_{2}(\mu , \tau ) = \left({i}^{2}{e}^{i\mu }\right){S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }{\left(\frac{\nu }{s}\right)}^{2\alpha }\right]$

$\Rightarrow {\theta }_{2}(\mu , \tau ) = \left({i}^{2}{e}^{i\mu }\right){S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{3\alpha }\right]$

${\Rightarrow \theta }_{2}(\mu , \tau ) = {i}^{2}{e}^{i\mu }\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)} ,$

$\theta (\mu , \tau ) = {\theta }_{0}(\mu , \tau )+{\theta }_{1}(\mu , \tau )+{\theta }_{2}(\mu , \tau )+{\theta }_{3}(\mu , \tau )+\dots$

$\Rightarrow \theta (\mu , \tau ) = {e}^{i\mu }+i{e}^{i\mu }\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}+{i}^{2}{e}^{i\mu }\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}+\dots$

$\Rightarrow \theta (\mu , \tau ) = {e}^{i\mu }\left[1+i\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}+{i}^{2}\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}+\dots \right] .$

Considered $\alpha = 1$ :

$\Rightarrow \theta (\mu , \tau ) = {e}^{i\mu }\left[1+\frac{i\tau }{1!}+\frac{{\left(i\tau \right)}^{2}}{2!}+\dots \right]$

$\Rightarrow \theta (\mu , \tau ) = {e}^{i[\mu +t]} .$

Example 2: Considered $1D$ linear time-fractional Schr $\ddot{o}$ dinger equation as follows ^[37]:

${D}_{\tau }^{\alpha }\theta (\mu , \tau )+i{\theta }_{\mu \mu }(\mu , \tau ) = 0 .$

(23)

I.C.: $\theta \left(\mu , 0\right) = {e}^{3i\mu }$

Applying Shehu transform in Eq (23):

${\left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta (\mu , \tau )\right]-\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right) = -iS\left[{\theta }_{\mu \mu }(\mu , \tau )\right]$

$\Rightarrow {\left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta (\mu , \tau )\right] = \sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)-iS\left[{\theta }_{\mu \mu }(\mu , \tau )\right]$

$\Rightarrow S\left[\theta (\mu , \tau )\right] = {\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)-i{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\theta }_{\mu \mu }(\mu , \tau )\right]$

$\Rightarrow \theta (\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)\right]-i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\theta }_{\mu \mu }(\mu , \tau )\right]\right]$

$\Rightarrow \sum\limits_{n = 0}^{\infty }{\theta }_{n}\left(\mu , \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)\right]-i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left(\sum\limits_{n = 0}^{\infty }{\theta }_{n}\left(\mu , \tau \right)\right)}_{\mu \mu }\right]\right],$

${\theta }_{n+1}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left({\theta }_{n}(\mu , \tau )\right)}_{\mu \mu }\right]\right], n = 0, 1, 2, 3, \dots$

Considered $\xi = 1$ :

${\theta }_{0}(\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }{\left(\frac{s}{\nu }\right)}^{\alpha -1}\theta \left(\mu , 0\right)\right]$

${\Rightarrow \theta }_{0}(\mu , \tau ) = {S}^{-1}\left[\left(\frac{\nu }{s}\right)\theta \left(\mu , 0\right)\right]$

$\Rightarrow {\theta }_{0}(\mu , \tau ) = \theta \left(\mu , 0\right){S}^{-1}\left[\frac{\nu }{s}\right]$

$\Rightarrow {\theta }_{0}\left(\mu , \tau \right) = \theta \left(\mu , 0\right) = {e}^{3i\mu }.$

Considered $n = 0$ :

${\theta }_{1}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left({\theta }_{0}(\mu , \tau )\right)}_{\mu \mu }\right]\right] , {\left({\theta }_{0}\right)}_{\mu \mu } = {\left(3i\right)}^{2}{e}^{3i\mu }$

$\Rightarrow {\theta }_{1}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left(3i\right)}^{2}{e}^{3i\mu }\right]\right]$

$\Rightarrow {\theta }_{1}(\mu , \tau ) = -i{\left(3i\right)}^{2}{e}^{3i\mu }{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[1\right]\right]$

${\Rightarrow \theta }_{1}(\mu , \tau ) = -i{\left(3i\right)}^{2}{e}^{3i\mu }{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{2\alpha }\right]$

$\Rightarrow {\theta }_{1}\left(\mu , \tau \right) = 9i{e}^{3i\mu }\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}.$

Considered $n = 1$ :

${\theta }_{2}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left({\theta }_{1}\right)}_{\mu \mu }\right]\right] , {\left({\theta }_{1}(\mu , \tau )\right)}_{\mu \mu } = 81{i}^{3}{e}^{3i\mu }\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}$

$\Rightarrow {\theta }_{2}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[81{i}^{3}{e}^{3i\mu }\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right]\right]$

$\Rightarrow {\theta }_{2}(\mu , \tau ) = -i\left(81{i}^{3}{e}^{3i\mu }\right){S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right]\right]$

${\Rightarrow \theta }_{2}(\mu , \tau ) = -i\left(81{i}^{3}{e}^{3i\mu }\right){S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{3\alpha }\right]$

$\Rightarrow {\theta }_{2}(\mu , \tau ) = 81{i}^{2}{e}^{3i\mu }\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}$

$\Rightarrow \theta (\mu , \tau ) = {\theta }_{0}(\mu , \tau )+{\theta }_{1}(\mu , \tau )+{\theta }_{2}(\mu , \tau )+{\theta }_{3}(\mu , \tau )+\dots$

$\Rightarrow \theta (\mu , \tau ) = {e}^{3i\mu }+9i{e}^{3i\mu }\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}+81{i}^{2}{e}^{3i\mu }\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}+\dots$

Considered $\alpha = 1$ :

$\Rightarrow \theta (\mu , \tau ) = {e}^{3i\mu }\left[1+\frac{9i\tau }{1!}+\frac{{\left(9i\tau \right)}^{2}}{2!}+\dots \right]$

$\Rightarrow \theta \left(\mu , \tau \right) = {e}^{3i\left[\mu +3\tau \right]}.$

Example 3: Considered $1D$ linear time-fractional Schr $\ddot{o}$ dinger equation as follows ^[37]:

${D}_{\tau }^{\alpha }\theta (\mu , \tau )+i{\theta }_{\mu \mu }(\mu , \tau ) = 0 .$

(24)

I.C.: $\theta \left(\mu , 0\right) = 1+cosh\left(2\mu \right)$

Applying Shehu transform upon Eq (24):

$S\left[{D}_{\tau }^{\alpha }\theta \right] = -iS\left[{\theta }_{\mu \mu }\right]$

${\Rightarrow \left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta (\mu , \tau )\right]-\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right) = -iS\left[{\theta }_{\mu \mu }(\mu , \tau )\right]$

${\Rightarrow \left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta (\mu , \tau )\right] = \sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)-iS\left[{\theta }_{\mu \mu }(\mu , \tau )\right]$

Where

${\theta }_{n+1}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left({\theta }_{n}(\mu , \tau )\right)}_{\mu \mu }\right]\right], n = 0, 1, 2, 3, \dots$

Considered $\xi = 1$ :

$\Rightarrow {\theta }_{0}(\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }{\left(\frac{s}{\nu }\right)}^{\alpha -1}\theta \left(\mu , 0\right)\right]$

$\Rightarrow {\theta }_{0}(\mu , \tau ) = \theta \left(x, 0\right){S}^{-1}\left[\frac{\nu }{s}\right]$

$\Rightarrow {\theta }_{0}\left(\mu , \tau \right) = \theta \left(\mu , 0\right) = 1+\mathrm{cosh}\left(2\mu \right).$

Considered $n = 0$ :

${\theta }_{1}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left({\theta }_{0}(\mu , \tau )\right)}_{\mu \mu }\right]\right], {\left({\theta }_{0}\right)}_{\mu \mu } = 4\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(2\mu \right)$

${\Rightarrow \theta }_{1}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[4\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(2\mu \right)\right]\right]$

$\Rightarrow {\theta }_{1}(\mu , \tau ) = -4i\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(2\mu \right){S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{2\alpha }\right]$

$\Rightarrow {\theta }_{1}\left(\mu , \tau \right) = -4i\mathrm{cosh}\left(2\mu \right)\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}.$

Considered $n = 1$ :

${\theta }_{2}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[{\left({\theta }_{1}\right)}_{\mu \mu }\right]\right], {\left({\theta }_{1}(\mu , \tau )\right)}_{\mu \mu } = -16i\mathrm{cosh}\left(2\mu \right)\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}$

$\Rightarrow {\theta }_{2}(\mu , \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[-16i\mathrm{cosh}\left(2\mu \right)\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right]\right]$

$\Rightarrow {\theta }_{2}(\mu , \tau ) = 16{i}^{2}\mathrm{cosh}\left(2\mu \right){S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{3\alpha }\right]$

${\Rightarrow \theta }_{2}(\mu , \tau ) = 16{i}^{2}\mathrm{cosh}\left(2\mu \right)\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}$

$\Rightarrow \theta (\mu , \tau ) = {\theta }_{0}(\mu , \tau )+{\theta }_{1}(\mu , \tau )+{\theta }_{2}(\mu , \tau )+{\theta }_{3}(\mu , \tau )+\dots$

$\Rightarrow \theta (\mu , \tau ) = 1+\mathrm{cosh}\left(2\mu \right)-4i\mathrm{cosh}\left(2\mu \right)\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}+16{i}^{2}\mathrm{cosh}\left(2\mu \right)\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}-\dots$

Considered $\alpha = 1$ :

$\Rightarrow \theta (\mu , \tau ) = 1+\mathrm{cosh}\left(2\mu \right)\left[1-\frac{4i\tau }{1!}+\frac{{\left(4i\tau \right)}^{2}}{2!}-\dots \right]$

$\Rightarrow \theta \left(\mu , \tau \right) = 1+\mathrm{cosh}\left(2\mu \right)\;\mathrm{exp}\left[-4\mathrm{i}\tau \right].$

Example 4: Considered $1D$ non-linear time-fractional Schr $\ddot{o}$ dinger equation as follows ^[38]:

$i{D}_{t}^{\alpha }\theta (\mu , \tau )+\frac{1}{2}{\theta }_{\mu \mu }(\mu , \tau )-\theta \left(\mu , \tau \right)co{s}^{2}\mu -\theta (\mu , \tau ){\left|\theta (\mu , \tau )\right|}^{2} = 0 , \text{μ}\in [0, 1] .$

(25)

I.C.: $u\left(\mu , 0\right) = sin\mu$

Applying Shehu transform upon Eq (25):

$iS\left[{D}_{\tau }^{\alpha }\theta (\mu , \tau )\right] = S\left[-\frac{1}{2}{u}_{\mu \mu }(\mu , \tau )+\theta \left(\mu , \tau \right)co{s}^{2}\mu +\theta (\mu , \tau ){\left|\theta (\mu , \tau )\right|}^{2}\right]$

$\Rightarrow S\left[{D}_{\tau }^{\alpha }\theta (\mu , \tau )\right] = -iS\left[-\frac{1}{2}{u}_{\mu \mu }(\mu , \tau )+\theta \left(\mu , \tau \right)co{s}^{2}\mu +\theta (\mu , \tau ){\left|\theta (\mu , \tau )\right|}^{2}\right]$

$\begin{array}{l} \Rightarrow {\left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta (\mu , \tau )\right]-\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;= -iS\left[-\frac{1}{2}{u}_{\mu \mu }(\mu , \tau )+\theta \left(\mu , \tau \right)co{s}^{2}\mu +\theta (\mu , \tau ){\left|\theta (\mu , \tau )\right|}^{2}\right] \end{array}$

$\begin{array}{l} {\Rightarrow \left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta (\mu , \tau )\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;= \sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)-iS\left[-\frac{1}{2}{u}_{\mu \mu }(\mu , \tau )+\theta \left(\mu , \tau \right)co{s}^{2}\mu +\theta (\mu , \tau ){\left|\theta (\mu , \tau )\right|}^{2}\right] \end{array}$

$\begin{array}{l} \Rightarrow S\left[\theta (\mu , \tau )\right] = {\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[-\frac{1}{2}{u}_{\mu \mu }(\mu , \tau )+\theta \left(\mu , \tau \right)co{s}^{2}\mu +\theta (\mu , \tau ){\left|\theta (\mu , \tau )\right|}^{2}\right] \end{array}$

$\begin{array}{l} \Rightarrow \theta (\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[-\frac{1}{2}{u}_{\mu \mu }(\mu , \tau )+\theta \left(\mu , \tau \right)co{s}^{2}\mu +\theta (\mu , \tau ){\left|\theta (\mu , \tau )\right|}^{2}\right]\right] \end{array}$

$\begin{array}{l} \Rightarrow \sum\limits_{n = 0}^{\infty }{\theta }_{n}(\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(\mu , 0\right)\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;+i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[\frac{1}{2}{\left(\sum\limits_{n = 0}^{\infty }{u}_{n}(\mu , \tau )\right)}_{\mu \mu }-co{s}^{2}\mu \left(\sum\limits_{n = 0}^{\infty }{u}_{n}(\mu , \tau )\right)-\sum\limits_{n = 0}^{\infty }{A}_{n}\right]\right] \end{array}$

${\theta }_{n+1}(\mu , \tau ) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[\frac{1}{2}{\left({\theta }_{n}(\mu , \tau )\right)}_{\mu \mu }-co{s}^{2}\mu \left({\theta }_{n}(\mu , \tau )\right)-{A}_{n}\right]\right], n = 0, 1, 2, 3, \dots$

Considered $\xi = 1$ :

$\Rightarrow {\theta }_{0}(\mu , \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }{\left(\frac{s}{\nu }\right)}^{\alpha -1}\theta \left(\mu , 0\right)\right]$

$\Rightarrow {\theta }_{0}(\mu , \tau ) = \theta \left(\mu , 0\right){S}^{-1}\left[\frac{\nu }{s}\right]$

${\Rightarrow \theta }_{0} = u\left(\mu , 0\right) = sin\mu .$

Considered $n = 0$ :

${\theta }_{1}\left(\mu , \tau \right) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[\frac{1}{2}{\left({\theta }_{0}\right)}_{\mu \mu }-co{s}^{2}\mu \left({\theta }_{0}\right)-{A}_{0}\right]\right].$

Where,

${\left({\theta }_{0}\right)}_{\mu \mu } = -sin\mu , {A}_{0} = F\left({\theta }_{0}\right) = {\theta }_{0}^{2}\stackrel{-}{{\theta }_{0}} = {\mathrm{sin}}^{3}\mu$

$\Rightarrow {\theta }_{1}(\mu , \tau ) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[\frac{1}{2}(-sin\mu )-co{s}^{2}x\left(sin\mu \right)-{\mathrm{sin}}^{3}\mu \right]\right]$

$\Rightarrow {\theta }_{1}(\mu , \tau ) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[-\frac{3}{2}sin\mu \right]\right]$

$\Rightarrow {\theta }_{1}(\mu , \tau ) = -\frac{3i}{2}sin\mu {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{2\alpha }\right]$

$\Rightarrow {\theta }_{1}\left(\mu , \tau \right) = -\frac{3i}{2}sin\mu \frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}.$

Considered $n = 1$ :

${\theta }_{2}\left(\mu , \tau \right) = i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[\frac{1}{2}{\left({\theta }_{1}\left(\mu , \tau \right)\right)}_{\mu \mu }-co{s}^{2}\mu \left({\theta }_{1}\left(\mu , \tau \right)\right)-{A}_{1}\right]\right].$

Where

${\left({\theta }_{1}\right)}_{\mu \mu }(\mu , \tau ) = \left(\frac{3i}{2}\right)sin\mu \frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)} , {A}_{1} = 2{\theta }_{0}{\theta }_{1}\stackrel{-}{{\theta }_{0}}+{\theta }_{0}^{2}\stackrel{-}{{\theta }_{1}} = \left(-\frac{3i}{2}\right){\mathrm{sin}}^{3}\mu \frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}$

$\begin{array}{l} \Rightarrow {\theta }_{2}(\mu , \tau ) = i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[\frac{1}{2}\left(\left(\frac{3i}{2}\right)sin\mu \frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right)-co{s}^{2}\mu \left(-\frac{3i}{2}sin\mu \frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right)\\ \;\;\;\;\;\;\;\;\;-\left(\left(-\frac{3i}{2}\right){\mathrm{sin}}^{3}\mu \frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right)]] \end{array}$

${\Rightarrow \theta }_{2}(\mu , \tau ) = \left(\frac{9{i}^{2}}{4}\right)sin\mu {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right]\right]$

${\Rightarrow \theta }_{2}(\mu , \tau ) = \left(\frac{9{i}^{2}}{4}\right)sin\mu {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{3\alpha }\right]$

${\Rightarrow \theta }_{2}(\mu , \tau ) = \left(\frac{9{i}^{2}}{4}\right)sin\mu \frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}$

$\Rightarrow \theta (\mu , \tau ) = {\theta }_{0}(\mu , \tau )+{\theta }_{1}(\mu , \tau )+{\theta }_{2}(\mu , \tau )+{\theta }_{3}(\mu , \tau )+\dots$

$\Rightarrow \theta (\mu , \tau ) = sin\mu -\frac{3i}{2}sin\mu \frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}+\left(\frac{9{i}^{2}}{4}\right)sin\mu \frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}-\dots$

Considered $\alpha = 1$ :

$\Rightarrow \theta (\mu , \tau ) = sin\mu \left[1-\frac{\left(\frac{3i\tau }{2}\right)}{1!}+\frac{{\left(\frac{3i\tau }{2}\right)}^{2}}{2!}-\dots \right]$

$\Rightarrow \theta (\mu , \tau ) = sin\mu \;\mathrm{exp}\left[-\frac{3i\tau }{2}\right] .$

Example 5: Considered $2D$ non-linear time-fractional Schr $\ddot{o}$ dinger equation as follows ^[38]:

$i{D}_{\tau }^{\alpha }\theta = -\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}+{\theta }_{{\mu }_{2}{\mu }_{2}}\right]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right)\theta +\theta {\left|\theta \right|}^{2} ,$

(26)

where ${\mu }_{1}, \mu {\text{}}_{2}\in \left[0, 2\pi \right]\times [0, 2\pi ]$ .

I.C.: $\theta \left({\mu }_{1}, {\mu }_{2}, 0\right) = sin{\mu }_{1}sin{\mu }_{2}$

Applying Shehu transform in Eq (26):

$\begin{array}{l} iS\left[{D}_{\tau }^{\alpha }\theta ({\mu }_{1}, {\mu }_{2}, \tau )\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= S [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, \tau )\right]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right)\theta ({\mu }_{1}, {\mu }_{2}, \tau )\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\theta ({\mu }_{1}, {\mu }_{2}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, \tau )\right|}^{2}] \end{array}$

$\begin{array}{l} \Rightarrow S\left[{D}_{\tau }^{\alpha }\theta ({\mu }_{1}, {\mu }_{2}, \tau )\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= -iS [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, \tau )\right]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right)\theta ({\mu }_{1}, {\mu }_{2}, \tau )\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\theta ({\mu }_{1}, {\mu }_{2}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, \tau )\right|}^{2}] \end{array}$

$\begin{array}{l} \Rightarrow {\left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta ({\mu }_{1}, {\mu }_{2}, \tau )\right]-\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= -iS [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, \tau )\right]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right)\theta ({\mu }_{1}, {\mu }_{2}, \tau )\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\theta ({\mu }_{1}, {\mu }_{2}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, \tau )\right|}^{2}] \end{array}$

$\begin{array}{l} \Rightarrow {\left(\frac{s}{\nu }\right)}^{\alpha }S [\theta ({\mu }_{1}, {\mu }_{2}, \tau )] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-iS [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, \tau )\right]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right)\theta ({\mu }_{1}, {\mu }_{2}, \tau )\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\theta ({\mu }_{1}, {\mu }_{2}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, \tau )\right|}^{2}] \end{array}$

$\begin{array}{l} \Rightarrow S\left[\theta ({\mu }_{1}, {\mu }_{2}, \tau )\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= {\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{\left(\frac{\nu }{s}\right)}^{\alpha }S [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, \tau )\right]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right)\theta ({\mu }_{1}, {\mu }_{2}, \tau )\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\theta ({\mu }_{1}, {\mu }_{2}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, \tau )\right|}^{2}] \end{array}$

$\begin{array}{l} \Rightarrow \theta ({\mu }_{1}, {\mu }_{2}, \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right)\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{S}^{-1} [{\left(\frac{\nu }{s}\right)}^{\alpha }S [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, \tau )\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right)\theta ({\mu }_{1}, {\mu }_{2}, \tau )+\theta ({\mu }_{1}, {\mu }_{2}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, \tau )\right|}^{2}]] \end{array}$

$\begin{array}{l} \Rightarrow \sum\limits_{n = 0}^{\infty }{\theta }_{n}({\mu }_{1}, {\mu }_{2}, \tau ) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right)\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[-\frac{1}{2}\left[{\left(\sum\limits_{n = 0}^{\infty }{\theta }_{n}({\mu }_{1}, {\mu }_{2}, \tau )\right)}_{{\mu }_{1}{\mu }_{1}}+{\left(\sum\limits_{n = 0}^{\infty }{\theta }_{n}({\mu }_{1}, {\mu }_{2}, \tau )\right)}_{{\mu }_{2}{\mu }_{2}}\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right)\sum\limits_{n = 0}^{\infty }{u}_{n}({\mu }_{1}, {\mu }_{2}, \tau )+\sum\limits_{n = 0}^{\infty }{A}_{n}]] \end{array}$

${u}_{0}\left({\mu }_{1}, {\mu }_{2}, \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, 0\right)\right],$

$\begin{array}{l} {u}_{n+1}({\mu }_{1}, {\mu }_{2}, \tau ) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= -i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[-\frac{1}{2}\left[{\left({\theta }_{n}\right)}_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, \tau )+{\left({\theta }_{n}\right)}_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, \tau )\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right){\theta }_{n}({\mu }_{1}, {\mu }_{2}, \tau )+{A}_{n}]], n = 0, 1, 2, 3, \dots \end{array}$

Considered $\xi = 1$ :

$\Rightarrow {\theta }_{0}({\mu }_{1}, {\mu }_{2}, \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }{\left(\frac{s}{\nu }\right)}^{\alpha -1}\theta \left({\mu }_{1}, {\mu }_{2}, 0\right)\right]$

$\Rightarrow {\theta }_{0}\left({\mu }_{1}, {\mu }_{2}, \tau \right) = \theta \left({\mu }_{1}, {\mu }_{2}, 0\right),$

${\theta }_{0}\left({\mu }_{1}, {\mu }_{2}, \tau \right) = sin{\mu }_{1}sin{\mu }_{2}.$

Considered $n = 0:$

$\begin{array}{l} {u}_{1}\left({\mu }_{1}, {\mu }_{2}, \tau \right) = -i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[-\frac{1}{2}\left[{\left({\theta }_{0}\right)}_{{\mu }_{1}{\mu }_{1}}\left({\mu }_{1}, {\mu }_{2}, \tau \right)+{\left({\theta }_{0}\right)}_{{\mu }_{2}{\mu }_{2}}\left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]\\ \;\;\; +\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right){\theta }_{0}\left({\mu }_{1}, {\mu }_{2}, \tau \right)+{A}_{0}]]. \end{array}$

Where

${\left({\theta }_{0}\right)}_{{\mu }_{1}{\mu }_{1}}\left({\mu }_{1}, {\mu }_{2}, \tau \right) = {\left({\theta }_{0}\right)}_{{\mu }_{2}{\mu }_{2}}\left({\mu }_{1}, {\mu }_{2}, \tau \right) = -sin{\mu }_{1}sin{\mu }_{2},$

and

${A}_{0} = {\theta }_{0}^{2}\stackrel{-}{{\theta }_{0}} = {\mathrm{sin}}^{3}{\mu }_{1}{\mathrm{sin}}^{3}{\mu }_{2}$

$\Rightarrow {\theta }_{1}\left({\mu }_{1}, {\mu }_{2}, \tau \right)$

$= -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[-\frac{1}{2}\left[-2sin{\mu }_{1}sin{\mu }_{2}\right]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right)\left(sin{\mu }_{1}sin{\mu }_{2}\right)+{\mathrm{sin}}^{3}{\mu }_{1}{\mathrm{sin}}^{3}{\mu }_{2}\right]\right]$

$\Rightarrow {\theta }_{1}({\mu }_{1}, {\mu }_{2}, \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[2sin{\mu }_{1}sin{\mu }_{2}\right]\right]$

${\Rightarrow \theta }_{1}({\mu }_{1}, {\mu }_{2}, \tau ) = -i\left(2sin{\mu }_{1}sin{\mu }_{2}\right){S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[1\right]\right]$

$\Rightarrow {\theta }_{1}({\mu }_{1}, {\mu }_{2}, \tau ) = -2isin{\mu }_{1}sin{\mu }_{2}{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{2\alpha }\right]$

${\Rightarrow \theta }_{1}\left({\mu }_{1}, {\mu }_{2}, \tau \right) = -2isin{\mu }_{1}sin{\mu }_{2}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}.$

Considered $n = 1:$

$\begin{array}{c} {\theta }_{2}\left({\mu }_{1}, {\mu }_{2}, \tau \right) = -i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[-\frac{1}{2}\left[{\left({u}_{1}\right)}_{{\mu }_{1}{\mu }_{1}}\left({\mu }_{1}, {\mu }_{2}, \tau \right) +{\left({u}_{1}\right)}_{{\mu }_{2}{\mu }_{2}}\left({\mu }_{1}, {\mu }_{2}, \tau \right)\right]\\+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right){u}_{1}\left({\mu }_{1}, {\mu }_{2}, \tau \right)+{A}_{1}]]. \end{array}$

Where,

${\left({\theta }_{1}\right)}_{{\mu }_{1}{\mu }_{1}} = {\left({\theta }_{1}\right)}_{{\mu }_{2}{\mu }_{2}} = 2isin{\mu }_{1}sin{\mu }_{2}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)},$

${\left({\theta }_{1}\right)}_{{\mu }_{1}{\mu }_{1}}+{\left({\theta }_{1}\right)}_{{\mu }_{2}{\mu }_{2}} = 4isin{\mu }_{1}sin{\mu }_{2}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)},$

${A}_{1} = 2{\theta }_{0}{\theta }_{1}\stackrel{-}{{\theta }_{0}}+{\theta }_{0}^{2}\stackrel{-}{{\theta }_{1}} = -2i{\mathrm{sin}}^{3}{\mu }_{1}{\mathrm{sin}}^{3}{\mu }_{2}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}$

$\Rightarrow {\theta }_{2}({\mu }_{1}, {\mu }_{2}, \tau )$

$\begin{array}{l} = -i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[-\frac{1}{2}\left[4isin{\mu }_{1}sin{\mu }_{2}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}\right)\left(-2isin{\mu }_{1}sin{\mu }_{2}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(-2i{\mathrm{sin}}^{3}x{\mathrm{sin}}^{3}y\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right)]] \end{array}$

$\Rightarrow {\theta }_{2}({\mu }_{1}, {\mu }_{2}, \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left\{-4isin{\mu }_{1}sin{\mu }_{2}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right\}\right]$

$\Rightarrow {\theta }_{2}({\mu }_{1}, {\mu }_{2}, \tau ) = 4{i}^{2}sin{\mu }_{1}sin{\mu }_{2}{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{3\alpha }\right]$

${\Rightarrow \theta }_{2}({\mu }_{1}, {\mu }_{2}, \tau ) = 4{i}^{2}sin{\mu }_{1}sin{\mu }_{2}\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}$

$\Rightarrow \theta \left({\mu }_{1}, {\mu }_{2}, \tau \right) = {u}_{0}\left({\mu }_{1}, {\mu }_{2}, \tau \right)+{u}_{1}\left({\mu }_{1}, {\mu }_{2}, \tau \right)+{u}_{2}\left({\mu }_{1}, {\mu }_{2}, \tau \right)+{u}_{3}\left({\mu }_{1}, {\mu }_{2}, \tau \right)+\dots$

$\Rightarrow \theta ({\mu }_{1}, {\mu }_{2}, \tau ) = sin{\mu }_{1}sin{\mu }_{2}-2isin{\mu }_{1}sin{\mu }_{2}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}+4{i}^{2}sin{\mu }_{1}sin{\mu }_{2}\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}-\dots$

Considered $\alpha = 1:$

$\Rightarrow \theta ({\mu }_{1}, {\mu }_{2}, \tau ) = sin{\mu }_{1}sin{\mu }_{2}\left[1-\frac{2i\tau }{1!}+\frac{{\left(2i\tau \right)}^{2}}{2!}-\dots \right]$

$\Rightarrow \theta \left({\mu }_{1}, {\mu }_{2}, \tau \right) = sin{\mu }_{1}sin{\mu }_{2}\;\mathrm{exp}\left(-2i\tau \right).$

Example 6: Considered $2D$ non-linear time-fractional Schr $\ddot{o}$ dinger equation as follows ^[34]:

$\begin{array}{l} i{D}_{\tau }^{\alpha }\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = -\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)+{\theta }_{{\mu }_{2}{\mu }_{2}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)+{\theta }_{{\mu }_{3}{\mu }_{3}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right] \\ \;\;\;\;\;\; +\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right)\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right|}^{2} . \end{array}$

(27)

Where ${\mu }_{1}, {\mu }_{2}, {\mu }_{3}\in \left[0, 2\pi \right]\times \left[0, 2\pi \right]\times \left[0, 2\pi \right].$

I.C.: $\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right) = sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}$

Applying Shehu transform in Eq (27):

$\begin{array}{l} iS\left[{D}_{t}^{\alpha }\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= S [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{3}{\mu }_{3}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right)\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right|}^{2}] \end{array}$

$\begin{array}{l} \Rightarrow S\left[{D}_{t}^{\alpha }\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= -iS [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{3}{\mu }_{3}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right)\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right|}^{2}] \end{array}$

$\begin{array}{l} {\Rightarrow \left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right]-\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{u}^{r}\left(x, y, z, 0\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = -iS [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{3}{\mu }_{3}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right)\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right|}^{2}] \end{array}$

$\begin{array}{l} \Rightarrow {\left(\frac{s}{\nu }\right)}^{\alpha }S\left[\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left(x, y, z, 0\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-iS [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{3}{\mu }_{3}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right)\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right|}^{2}] \end{array}$

$\begin{array}{l} \Rightarrow S\left[\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right] \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= {\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{\left(\frac{\nu }{s}\right)}^{\alpha }S [-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{3}{\mu }_{3}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right)\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right|}^{2}] \end{array}$

$\begin{array}{l} \Rightarrow \theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[-\frac{1}{2}\left[{\theta }_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{{\mu }_{3}{\mu }_{3}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right)\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ){\left|\theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right|}^{2}]] \end{array}$

$\begin{array}{l} \Rightarrow \sum\limits_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[-\frac{1}{2}[{\left(\sum\limits_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right)}_{{\mu }_{1}{\mu }_{1}}+{\left(\sum\limits_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right)}_{{\mu }_{2}{\mu }_{2}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+{\left(\sum\limits_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\right)}_{{\mu }_{3}{\mu }_{3}}]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right)\sum\limits_{n = 0}^{\infty }{\theta }_{n}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)+\sum\limits_{n = 0}^{\infty }{A}_{n}]], \end{array}$

${\theta }_{0}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }\sum\limits_{r = 0}^{\xi -1}{\left(\frac{s}{\nu }\right)}^{\alpha -r-1}{\theta }^{r}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)\right],$

$\begin{array}{l} {\theta }_{n+1}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= -i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[-\frac{1}{2}\left[{\left({\theta }_{n}\right)}_{{\mu }_{1}{\mu }_{1}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\left({\theta }_{n}\right)}_{{\mu }_{2}{\mu }_{2}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\left({\theta }_{n}\right)}_{{\mu }_{3}{\mu }_{3}}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )\right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right){\theta }_{n}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{A}_{n}]], n = 0, 1, 2, 3, \dots \end{array}$

Considered $\xi = 1$ :

$\Rightarrow {\theta }_{0}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = {S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }{\left(\frac{s}{\nu }\right)}^{\alpha -1}\theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)\right]$

$\Rightarrow {\theta }_{0}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = \theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, 0\right)$

$\Rightarrow {\theta }_{0}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}.$

Considered $n = 0:$

$\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\theta }_{1}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) \\ = -i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S [-\frac{1}{2}[{\left({\theta }_{0}\right)}_{{\mu }_{1}{\mu }_{1}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)+{\left({\theta }_{0}\right)}_{{\mu }_{2}{\mu }_{2}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\\+{\left({\theta }_{0}\right)}_{{\mu }_{3}{\mu }_{3}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)] +\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right){\theta }_{0}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)+{A}_{0}]]. \end{array}$

Where

$\begin{array}{l} {\left({\theta }_{0}\right)}_{{\mu }_{1}{\mu }_{1}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = {\left({\theta }_{0}\right)}_{{\mu }_{2}{\mu }_{2}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = {\left({\theta }_{0}\right)}_{{\mu }_{3}{\mu }_{3}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= -sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}{\left({\theta }_{0}\right)}_{{\mu }_{1}{\mu }_{1}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)+{\left({\theta }_{0}\right)}_{{\mu }_{2}{\mu }_{2}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{+\left({\theta }_{0}\right)}_{{\mu }_{3}{\mu }_{3}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = -3sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}, \end{array}$

and

${A}_{0} = {\theta }_{0}^{2}\stackrel{-}{{\theta }_{0}} = {\mathrm{sin}}^{3}{\mu }_{1}{\mathrm{sin}}^{3}{\mu }_{2}{\mathrm{s}\mathrm{i}\mathrm{n}}^{3}{\mu }_{3}$

$\Rightarrow {\theta }_{1}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )$

$\begin{array}{l} = -i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[-\frac{1}{2}\left[-3sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\right]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right)\left(sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+{\mathrm{sin}}^{3}{\mu }_{1}{\mathrm{sin}}^{3}{\mu }_{2}{\mathrm{sin}}^{3}{\mu }_{3}]] \end{array}$

$\Rightarrow {\theta }_{1}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[\frac{5}{2}sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\right]\right]$

$\Rightarrow {\theta }_{1}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = -\frac{5i}{2}sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left[1\right]\right]$

${\Rightarrow \theta }_{1}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = -\frac{5i}{2}sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{2\alpha }\right]$

${\Rightarrow \theta }_{1}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = -\frac{5i}{2}sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}.$

Considered $n = 1:$

$\begin{array}{l} {\theta }_{2}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= -i{S}^{-1}[{\left(\frac{\nu }{s}\right)}^{\alpha }S[-\frac{1}{2}[{\left({\theta }_{1}\right)}_{{\mu }_{1}{\mu }_{1}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)+{\left({\theta }_{1}\right)}_{{\mu }_{2}{\mu }_{2}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+{\left({\theta }_{1}\right)}_{{\mu }_{3}{\mu }_{3}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)]+\left(1-si{n}^{2}{\mu }_{1}si{n}^{2}{\mu }_{2}si{n}^{2}{\mu }_{3}\right){\theta }_{1}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)+{A}_{1}]]. \end{array}$

Where,

${\left({\theta }_{1}\right)}_{{\mu }_{1}{\mu }_{1}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = {\left({\theta }_{1}\right)}_{{\mu }_{2}{\mu }_{2}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = {\left({\theta }_{1}\right)}_{{\mu }_{3}{\mu }_{3}}\left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right)$

$= \left(\frac{5i}{2}\right)sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}.$

${A}_{1} = 2{\theta }_{0}{\theta }_{1}\stackrel{-}{{\theta }_{0}}+{\theta }_{0}^{2}\stackrel{-}{{\theta }_{1}} = -\frac{5i}{2}{\mu }_{1}{\mathrm{sin}}^{3}{\mu }_{2}{\mathrm{sin}}^{3}{\mu }_{3}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}$

$\Rightarrow {\theta }_{2}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = -i{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{\alpha }S\left\{-\frac{25i}{4}sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}\right\}\right]$

$\Rightarrow {\theta }_{2}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = \frac{25{i}^{2}}{4}sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}{S}^{-1}\left[{\left(\frac{\nu }{s}\right)}^{3\alpha }\right]$

$\Rightarrow {\theta }_{2}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = \frac{25{i}^{2}}{4}sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}$

$\Rightarrow \theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = {\theta }_{0}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{1}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{2}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+{\theta }_{3}({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau )+\dots$

$\Rightarrow \theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}-\frac{5i}{2}sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\frac{{\tau }^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}+\frac{25{i}^{2}}{4}sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\frac{{\tau }^{3\alpha -1}}{\mathrm{\Gamma }\left(3\alpha \right)}-\dots$

Considered $\alpha = 1:$

$\Rightarrow \theta ({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau ) = sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\left[1-\frac{\left(\frac{5i\tau }{2}\right)}{1!}+\frac{{\left(\frac{5i\tau }{2}\right)}^{2}}{2!}-\dots \right]$

$\Rightarrow \theta \left({\mu }_{1}, {\mu }_{2}, {\mu }_{3}, \tau \right) = sin{\mu }_{1}sin{\mu }_{2}sin{\mu }_{3}\;\mathrm{exp}\left(-\frac{5i\tau }{2}\right).$

5. Graphical and tabular discussion

In the present section, the validity of the proposed regime is affirmed vias graphical and tabular analysis of the results. In , approx. and exact profiles are matched at $\tau$ = 1, 2, 3, 4, and 5 for Example 1. In , matching of approx. and exact profiles are provided at $\tau$ = 6, 7, 8, 9, and 10 for Example 1. A wide range of time levels is checked for compatibility. In , a match of approx. and exact profiles is provided at $\tau$ = 1.0, 1.5, 2.0, 2.5 and 3.0 for Example 2. is related to the approx. and exact profile compatibility at $\tau$ = 1, 2, 3, 4, and 5 for Example 3. In , compatibility of approx. and exact profiles are matched at $\tau$ = 1, 2, 3, 4, and 5 for Example 4. In , approx.-exact compatibility is mentioned at $\tau$ = 6, 7, 8, 9, and 10 for Example 4. and are related to the mesh-contour and surface-contour presentations at $\tau$ = 1 and $\tau$ = 2 respectively for Example 5. and are related to the mesh-contour and surface-contour presentations at $\tau$ = 1 and $\tau$ = 2, respectively for Example 6.

Figure 1. Approx. and exact profiles at

$t$ = 1, 2, 3, 4 and 5 regarding Example 1.

DownLoad: Full-Size Img PowerPoint

Figure 2. Approx. and exact profiles at

$t$ = 6, 7, 8, 9, and 10 where

$N$ = 101 regarding Example 1.

DownLoad: Full-Size Img PowerPoint

Figure 3. Approx. and exact profiles at

$t$ = 1.0, 1.5, 2.0, 2.5, and 3.0 where

$N$ = 101 regarding Example 2.

DownLoad: Full-Size Img PowerPoint

Figure 4. Approx. and exact profiles at

$t$ = 1, 2, 3, 4, and 5 where

$N$ = 101 regarding Example 3.

DownLoad: Full-Size Img PowerPoint

Figure 5. Approx. and exact profiles at

$t$ = 1, 2, 3, 4, and 5 where

$N$ = 101 regarding Example 4.

DownLoad: Full-Size Img PowerPoint

Figure 6. Approx. and exact profiles at

$t$ = 6, 7, 8, 9, and 10 where

$N$ = 101 regarding Example 4.

DownLoad: Full-Size Img PowerPoint

Figure 7. Approx. and exact profiles at

$t$ = 1 where

$N$ = 51 regarding Example 5.

DownLoad: Full-Size Img PowerPoint

Figure 8. Approx. and exact profiles at

$t$ = 2 where

$N$ = 51 regarding Example 5.

DownLoad: Full-Size Img PowerPoint

Figure 9. Approx. and exact profiles at

$t$ = 1, where

$N$ = 101,

$z$ = 0.1 regarding Example 6.

DownLoad: Full-Size Img PowerPoint

Figure 10. Approx. and exact profiles at

$t$ = 2 where

$N$ = 101,

$z$ = 0.1 regarding Example 6.

DownLoad: Full-Size Img PowerPoint

Via Tables 3–8, it can be affirmed that the approx. and exact profiles are matched for a wide range of time levels regarding the proposed scheme. Error analysis and convergence property are done by means of Tables 3–. In –, ${L}_{\infty }$ errors are provided at various grid points and time levels. Via –, it is reported that on increasing the number of grid points at different time levels, ${L}_{\infty }$ error got reduced, and in most of the cases, convergence is affirmed up to higher order.

6. Conclusions

In the present study, the motive is to deal with $1D$ , $2D$ , and $3D$ time-fractional Schr $\ddot{o}$ dinger equations regarding the approx.-analytical outcome. Shehu transform ADM is incorporated for this purpose. The prime key of a developed regime is it's easy-to-implement approach and accurate results. Furthermore, with the aid of the six mentioned examples, it is ensured that the retrieved approximated profiles are compatible with exact profiles. The graphical matching aspect between the approximated and exact results is also notified, which ensures that the developed technique is a good alternative to solve complex natured time-fractional PDEs. ${L}_{\infty }$ error is mentioned in–. Via mentioned tables, it is ensured that the proposed methodology is convergent, as on increasing the number of grid points, ${L}_{\infty }$ error got reduced. With the aid of the developed regime, various complex natured fractional PDEs can be easily solved.

Acknowledgments

This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation (grant number B05F640092).

Conflict of interest

The authors declare no conflict of interest.

References

[1]	K. Oldham, J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, 1 Ed., Elsevier, 1974.
[2]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, 1 Ed., Wiley, 1993.
[3]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, USA, 1993.
[4]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[5]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier, 2006.
[6]	M. D. Ortigueira, Fractional calculus for scientists and engineers, Vol. 84, Springer Dordrecht, 2011. https://doi.org/10.1007/978-94-007-0747-4
[7]	S. Das, Functional fractional calculus, Springer Berlin, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-20545-3
[8]	R. Hilfer, Applications of fractional calculus in physics, World Scientific, 2000. https://doi.org/10.1142/3779
[9]	B. J. West, M., Bologna, P. Grigolini, Physics of fractal operators, Vol. 35, New York: Springer, 2003. https://doi.org/10.1007/978-0-387-21746-8
[10]	L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci., 2003 (2003), 3413–3442. https://doi.org/10.1155/S0161171203301486 doi: 10.1155/S0161171203301486
[11]	F. Mainardi, Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models, World Scientific, 2010. https://doi.org/10.1142/p614
[12]	D. Baleanu, Z. B. Güvenç, J. T. Machado, New trends in nanotechnology and fractional calculus applications, New York: Springer, 2010. https://doi.org/10.1007/978-90-481-3293-5
[13]	R. Herrmann, Fractional calculus: An introduction for physicists, World Scientific, 2011.
[14]	A. Papoulis, A new method of inversion of the Laplace transform, Q. Appl. Math., 14 (1957), 405–414.
[15]	A. Kılıçman, H. E. Gadain, On the applications of Laplace and Sumudu transforms, J. Franklin Inst., 347 (2010), 848–862. https://doi.org/10.1016/j.jfranklin.2010.03.008 doi: 10.1016/j.jfranklin.2010.03.008
[16]	T. M. Elzaki, On the connections between Laplace and Elzaki transforms, Adv. Theor. Appl. Math., 6 (2011), 1–10.
[17]	M. S. Rawashdeh, S. Maitama, Solving coupled system of nonlinear PDE's using the natural decomposition method, Int. J. Pure Appl. Math., 92 (2014), 757–776. https://doi.org/10.12732/ijpam.v92i5.10 doi: 10.12732/ijpam.v92i5.10
[18]	S. Maitama, W. Zhao, New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations, Int. J. Nonlinear Anal. Appl., 17 (2019), 167–190. https://doi.org/10.28924/2291-8639-17-2019-167 doi: 10.28924/2291-8639-17-2019-167
[19]	D. Ziane, R. Belgacem, A. Bokhari, A new modified Adomian decomposition method for nonlinear partial differential equations, Open J. Math. Anal., 3 (2019), 81–90. https://doi.org/10.30538/psrp-oma2019.0041 doi: 10.30538/psrp-oma2019.0041
[20]	L. Akinyemi, O. S. Iyiola, Exact and approximate solutions of time‐fractional models arising from physics via Shehu transform, Math. Methods Appl. Sci., 43 (2020), 7442–7464. https://doi.org/10.1002/mma.6484 doi: 10.1002/mma.6484
[21]	R. Belgacem, D. Baleanu, A. Bokhari, Shehu transform and applications to Caputo-fractional differential equations, Int. J. Anal. Appl., 17 (2019), 917–927. https://doi.org/10.28924/2291-8639-17-2019-917 doi: 10.28924/2291-8639-17-2019-917
[22]	A. K. Shukla, J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336 (2007), 797–811. https://doi.org/10.1016/j.jmaa.2007.03.018 doi: 10.1016/j.jmaa.2007.03.018
[23]	O. S. Iyiola, E. O. Asante-Asamani, B. A. Wade, A real distinct poles rational approximation of generalized Mittag-Leffler functions and their inverses: Applications to fractional calculus, J. Comput. Appl. Math., 330 (2018), 307–317. https://doi.org/10.1016/j.cam.2017.08.020 doi: 10.1016/j.cam.2017.08.020
[24]	Y. S. Kivshar, G. P. Agrawal, Optical solitons: From fibers to photonic crystals, Academic Press, 2003.
[25]	F. Dalfovo, S. Giorgini, L. P. Pitaevskii, S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463. https://doi.org/10.1103/RevModPhys.71.463 doi: 10.1103/RevModPhys.71.463
[26]	J. Belmonte-Beitia, G. F. Calvo, Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials, Phys. Lett. A, 373 (2009), 448–453. https://doi.org/10.1016/j.physleta.2008.11.056 doi: 10.1016/j.physleta.2008.11.056
[27]	T. Xu, B. Tian, L. L. Li, X. Lü, C. Zhang, Dynamics of Alfvén solitons in inhomogeneous plasmas, Phys. Plasmas, 15 (2008), 102307. https://doi.org/10.1063/1.2997340 doi: 10.1063/1.2997340
[28]	M. Naber, Time fractional Schrödinger equation, J. Math. Phys., 45 (2004), 3339–3352. https://doi.org/10.1063/1.1769611 doi: 10.1063/1.1769611
[29]	S. Wang, M. Xu, Generalized fractional Schrödinger equation with space-time fractional derivatives, J. Math. Phys., 48 (2007), 043502. https://doi.org/10.1063/1.2716203 doi: 10.1063/1.2716203
[30]	S. Z. Rida, H. M. El-Sherbiny, A. A. M. Arafa, On the solution of the fractional nonlinear Schrödinger equation, Phys. Lett. A, 372 (2008), 553–558. https://doi.org/10.1016/j.physleta.2007.06.071 doi: 10.1016/j.physleta.2007.06.071
[31]	R. K. Saxena, R. Saxena, S. L. Kalla, Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum mechanics, Appl. Math. Comput., 216 (2010), 1412–1417. https://doi.org/10.1016/j.amc.2010.02.041 doi: 10.1016/j.amc.2010.02.041
[32]	J. R. Wang, Y. Zhou, W. Wei, Fractional Schrödinger equations with potential and optimal controls, Nonlinear Anal.: Real World Appl., 13 (2012), 2755–2766. https://doi.org/10.1016/j.nonrwa.2012.04.004 doi: 10.1016/j.nonrwa.2012.04.004
[33]	N. A. Khan, M. Jamil, A. Ara, Approximate solutions to time-fractional Schrödinger equation via homotopy analysis method, Int. Scholarly Res. Not., 2012 (2012), 1–11.
[34]	K. M. Hemida, K. A. Gepreel, M. S. Mohamed, Analytical approximate solution to the time-space nonlinear partial fractional differential equations, Int. J. Pure Appl. Math., 78 (2012), 233–243.
[35]	S. H. M. Hamed, E. A. Yousif, A. I. Arbab, Analytic and approximate solutions of the space-time fractional Schrödinger equations by homotopy perturbation Sumudu transform method, Abstr. Appl. Anal., 2014 (2014), 863015. https://doi.org/10.1155/2014/863015 doi: 10.1155/2014/863015
[36]	S. O. Edeki, G. O. Akinlabi, S. A. Adeosun, Analytic and numerical solutions of time-fractional linear Schrödinger equation, Commun. Math. Appl., 7 (2016), 1–10. https://doi.org/10.26713/cma.v7i1.327 doi: 10.26713/cma.v7i1.327
[37]	A. Mohebbi, M. Abbaszadeh, M. Dehghan, The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics, Eng. Anal. Boundary Elem., 37 (2013), 475–485. https://doi.org/10.1016/j.enganabound.2012.12.002 doi: 10.1016/j.enganabound.2012.12.002
[38]	K. Shah, M. Junaid, N. Ali, Extraction of Laplace, Sumudu, Fourier and Mellin transform from the natural transform, J. Appl. Environ. Biol. Sci., 5 (2015), 108–115.
[39]	A. N. Malik, O. H. Mohammed, Two efficient methods for solving fractional Lane–Emden equations with conformable fractional derivative, J. Egypt. Math. Soc., 28 (2020), 1–11.
[40]	S. Ali, S. Bushnaq, K. Shah, K. M. Arif, Numerical treatment of fractional order Cauchy reaction diffusion equations, Chaos, Solitons Fract., 103 (2017), 578–587. https://doi.org/10.1016/j.chaos.2017.07.016 doi: 10.1016/j.chaos.2017.07.016
[41]	K. Shah, H. Naz, M. Sarwar, T. Abdeljawad, On spectral numerical method for variable-order partial differential equations, AIMS Math., 7 (2022), 10422–10438. https://doi.org/10.3934/math.2022581 doi: 10.3934/math.2022581
[42]	A. Bashan, N. M. Yagmurlu, Y. Ucar, A. Esen, An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method, Chaos, Solitons Fract., 100 (2017), 45–56. https://doi.org/10.1016/j.chaos.2017.04.038 doi: 10.1016/j.chaos.2017.04.038
[43]	N. A. Shah, P. Agarwal, J. D. Chung, E. R. El-Zahar, Y. S. Hamed, Analysis of optical solitons for nonlinear Schrödinger equation with detuning term by iterative transform method, Symmetry, 12 (2020), 1850. https://doi.org/10.3390/sym12111850 doi: 10.3390/sym12111850
[44]	N. A. Shah, I. Dassios, E. R. El-Zahar, J. D. Chung, S. Taherifar, The variational iteration transform method for solving the time-fractional Fornberg-Whitham equation and comparison with decomposition transform method, Mathematics, 9 (2021), 141. https://doi.org/10.3390/math9020141 doi: 10.3390/math9020141
[45]	M. Kapoor, N. A. Shah, S. Saleem, W. Weera, An analytical approach for fractional hyperbolic telegraph equation using Shehu transform in one, two and three dimensions, Mathematics, 10 (2022), 1961. https://doi.org/10.3390/math10121961 doi: 10.3390/math10121961
[46]	M. Kapoor, A. Majumder, V. Joshi, An analytical approach for Shehu transform on fractional coupled 1D, 2D and 3D Burgers' equations, Nonlinear Eng., 11 (2022), 268–297. https://doi.org/10.1515/nleng-2022-0024 doi: 10.1515/nleng-2022-0024
[47]	M. Kapoor, Sumudu transform HPM for Klein-Gordon and Sine-Gordon equations in one dimension from an analytical aspect, J. Math. Comput. Sci., 12 (2022), 1–25. https://doi.org/10.28919/jmcs/6979 doi: 10.28919/jmcs/6979

This article has been cited by:

1.	Nourhane Attia, Ali Akgül, Djamila Seba, Abdelkader Nour, Manuel De la Sen, Mustafa Bayram, An Efficient Approach for Solving Differential Equations in the Frame of a New Fractional Derivative Operator, 2023, 15, 2073-8994, 144, 10.3390/sym15010144
2.	Mamta Kapoor, Samanyu Khosla, Series Solution to Fractional Telegraph Equations Using an Iterative Scheme Based on Yang Transform, 2024, 0971-3514, 10.1007/s12591-024-00679-w
3.	Mamta Kapoor, Varun Joshi, A comparative study of Sumudu HPM and Elzaki HPM for coupled Burgers’ equation, 2023, 9, 24058440, e15726, 10.1016/j.heliyon.2023.e15726
4.	M. L. Rupa, K. Aruna, Optical solitons of time fractional Kundu–Eckhaus equation and massive Thirring system arises in quantum field theory, 2024, 56, 0306-8919, 10.1007/s11082-023-05914-2

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1971) PDF downloads(133) Cited by(4)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(10) / Tables(8)

AIMS Mathematics

Analytical solution of time-fractional Schrödinger equations via Shehu Adomian Decomposition Method

Related Papers:

Abstract

1. Introduction

2. Outline of the paper

3. Implementation of the proposed regime

3.1. Implementation of the scheme on $1{D}$ time-fractional Schr $\ddot{{o}}$ dinger equation

3.2. Implementation of the scheme on $2{D}$ time-fractional Schr $\ddot{{o}}$ dinger equation

3.3. Implementation of the scheme on $3{D}$ time-fractional Schr $\ddot{{o}}$ dinger equation

4. Examples and derivation

5. Graphical and tabular discussion

6. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

Analytical solution of time-fractional Schrödinger equations via Shehu Adomian Decomposition Method

Related Papers:

Abstract

1. Introduction

2. Outline of the paper

3. Implementation of the proposed regime

3.1. Implementation of the scheme on 1D 1{D} time-fractional Schr¨o \ddot{{o}} dinger equation

3.2. Implementation of the scheme on 2D 2{D} time-fractional Schr¨o \ddot{{o}} dinger equation

3.3. Implementation of the scheme on 3D 3{D} time-fractional Schr¨o \ddot{{o}} dinger equation

4. Examples and derivation

5. Graphical and tabular discussion

6. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

3.1. Implementation of the scheme on $1{D}$ time-fractional Schr $\ddot{{o}}$ dinger equation

3.2. Implementation of the scheme on $2{D}$ time-fractional Schr $\ddot{{o}}$ dinger equation

3.3. Implementation of the scheme on $3{D}$ time-fractional Schr $\ddot{{o}}$ dinger equation