# A Highly Accurate Technique to Obtain Exact Solutions to Time-Fractional Quantum Mechanics Problems with Zero and Nonzero Trapping Potential

Authors: Muhammad Imran Liaqat; Adnan Khan; Md. Ashraful Alam; M. K. Pandit
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, the highly accurate analytical Aboodh transform decomposition method (ATDM) in the sense of Caputo fractional derivative is used to determine the approximate and exact solutions of both linear and nonlinear time-fractional Schrodinger differential equations (SDEs) with zero and nonzero trapping potential that describe the nonrelativistic quantum mechanical activity. The Adomian decomposition method (ADM) and the Aboodh transform of Caputo’s fractional derivative are combined in this method. The recurrence and absolute error of the four problems are analyzed to evaluate the efficiency and consistency of the presented method. In addition, numerical results are also compared with other methods such as the fractional reduced differential transform method (FRDTM), the homotopy analysis method (HAM), and the homotopy perturbation method (HPM). The results obtained by the proposed method show excellent agreement with these methods, which indicates its effectiveness and reliability. This technique has the benefit of not requiring any minor or major physical parameter assumptions in the problem. As a result, it may be used to solve both weakly and strongly nonlinear problems, overcoming some of the inherent constraints of classic perturbation approaches. To solve nonlinear fractional-order differential equations, just a few computations are necessary. As a consequence, it outperforms homotopy analysis and homotopy perturbation approaches significantly. The procedure is quick, precise, and easy to implement. Convergence analysis of the series solution is also offered.


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## Body

## 1. Introduction

The shortcoming of classical mechanics to explain several physical processes, including those on microscopic scales, such as the photoelectric effect, black body radiation, and atomic stability, led to the development of modern quantum mechanics. It is explained by the fact that all physical quantities of a bound system are confined to discrete value quantization. Quantum mechanics may successfully describe various modern physics processes in atomic and nuclear physics, as well as other fields of modern physics, where the Schrödinger equation can be used to describe the behavior of electrons in atomic physics and nucleons in nuclear physics [1]. This equation was developed by Austrian physicist Erwin Schrodinger in late 1925 and published in 1926.Fractional partial differential equations (PDEs), which are an extension of integer-order PDEs, have subsequently received much interest. They can be used to extract memory and hereditary qualities from a variety of materials and processes. Fractional PDEs such as the Boussinesq equation, Korteweg-de Vries equation, Schrodinger equation, Burger’s equation, and others are frequently used to describe varied nonlinear wave processes in mechanics, physics, biology, chemistry, and other areas [2]. The time-fractional SDE is the fundamental physics equation for characterizing nonrelativistic quantum mechanical activity. Electromagnetic waves, quantitative finance, quantum development of complex systems, and dielectric polarization have all been pulled into the time-fractional SDE in recent years [3, 4].We examined the exact and numerical approaches to comprehend the physical mechanism of such a natural phenomenon. As a result, we are looking for a mathematical solution to PDEs that is both precise and numerical. Many papers have focused on constructing solutions to PDEs through well-known methods. Lie symmetry analysis [5], inverse scattering approach [6], spectral collocation method [7], Hirota method [8], Backlund transformation method [9], Modified Kudryashov method [10], Laplace transform coupled with Adomian decomposition method [11], Elzaki residual power series method [12], adaptation on power series method with conformable operator [13], Legendre wavelet method [14] and modified conformable Shehu transform decomposition method [15] are some of the most effective and efficient methods.Differential equations (DEs), partial integrodifferential equations (PIDs), and delayed differential equations (DDEs) are all solved by employing integral transforms, which are among the most valuable techniques in mathematics. The conversion of DEs and integral equations into terms of a simple algebraic equation is enabled by the appropriate selection of integral transforms. The origins of integral transforms can be traced back to P. S. Laplace’s work in the 1780s and Joseph Fourier’s work in 1822. In the beginning, ordinary and PDEs were solved using the Laplace transform and the Fourier transform which are two well-known transforms. These modifications were then applied to fractional-order DEs in the domain of fractional calculus [16–19]. In recent years, researchers have proposed lots of new different transformations to solve a variety of mathematical problems. Fractional-order DEs are solved using the Laplace transform [20], fractional complex transform [21], travelling wave transform [22], Elzaki transform [23], Sumudu transforms [24], and ZZ transforms [25], among others. These transformations are paired with additional analytical, numerical, or homotopy-based techniques to handle fractional-order DEs.The general state of quantum mechanics equations, which are commonly described as fractional-order SDEs, will be solved in this study using an appealing and effective analytical technique, the Aboodh transform decomposition method (ATDM). The Aboodh transform was established in 2013 by Khalid Aboodh to facilitate solving ordinary DEs and PDEs in the time domain [26]. The Aboodh transform is generated using the traditional Fourier integral. The Elzaki transform and the Laplace transform are intimately related to this integral transform. The A-T, which has been used by several researchers for fractional-order DEs [27–31], has recently caught the interest of many mathematicians.We analyze then on linear time fractional-order SDE with zero trapping potential in its more general version, which is represented by the complex-valued functionξϖ,τ of the form [32].(1)iDτνξϖ,τ+Dϖϖξϖ,τ+ηξϖ,τ2ƛξϖ,τ=0,where η∈R,0<ν≤1,i=−1,Dτν indicates Caputo fractional derivative of order ν, ξϖ,τ is the unknown complex-valued function to be determined, ϖ∈R,τ≥0, and ξϖ,τ represent the modulus of ξϖ,τ; with the following initial and boundary conditions given by ξϖ,0=ϒϖ,ξ0,τ=Aϖ, and ξϖ0,τ=Bϖ.The time-fractional nonlinear SDE with nonzero trapping potential has the following form [33]:(2)iDτνξϖ,τ=−12ℑϖξϖ,τ+Θϖξϖ,τ+HdQξϖ,τ,τ≥0,0<ν≤1,ϖ∈R,with the initial condition:(3)ξϖ,0=ϒϖ,where ϖ∈R,Θdϖ is the trapping potential and Hd is a real constant ℑϖ is a linear operator, Qξϖ,τ is a nonlinear function, and Dτν is the Caputo fractional differential operator.The physical model (2) and its generalized forms arise in various areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics [34].The time-fractional SDE has been investigated through various methods, such as the homotopy perturbation method [35], exponential rational function method [36], residual power series method [37], modified transformation method [38], two-dimensional differential transform method [39], extended simple equation method [40], trigonometric B-spline method [41], fractional reduced differential transform method [42], and homotopy analysis method [43]. All these methods have their own specific limits and deficiencies. These methods require enormous computational work and high running times. In this study, we used the simple and efficient technique known as the ATDM to solve the SDE of the fractional derivative in the sense of Caputo. The recommended method is simple to use and can be applied to both linear and nonlinear problems. It also has the ability to reduce the complexity of the computational effort. The set of rules of ATDM depends on converting the SDE into Aboodh transform space and, after converting the SDE into an algebraic equation, applying inverse A-T and then introducing a series of solutions to the obtained algebraic equation, at the final step, obtaining the target result through the iteration process.The structure of the paper is as follows: we will employ various fundamental definitions and results from fractional calculus theory in the next section. The primary idea of the ATDM is investigated in Section3 in order to establish fractional SDE solutions. Section 4 demonstrates the method’s potential, capability, and simplicity by obtaining approximate and exact solutions to four SDE problems. The proposed method is illustrated numerically and graphically in Section 5, and the numerical results are evaluated in Section 5. The conclusion is covered in Section 6.


## 2. Basic Concepts and Representations in Fractional Calculus Theory

Fractional calculus is a modified version of classical calculus. Fractional calculus can explain a wide range of complex phenomena, including memory and heredity. This subject has drawn many researchers because of its worldwide aspect and numerous applications in several domains of science, such as physics, signal processing, modeling, control theory, economics, and chemistry [44, 45]. In this section, we covered some definitions and basic features of fractional calculus theory, as well as the fundamentals of the Aboodh transformation, which will be used later in this paper.Definition 1.
(see [43]). The Aboodh transform (A-T) for function ξτ of exponential order over the set of functions is defined as(4)R=ξτ|∃M,u1,u2>0,ξτ<Me−ℏτ,where M is a finite number and u1,u2 may be finite or infinite. A-T is denoted by the operator A. and defined as(5)Aξτ=Qℏ=1ℏ∫0∞ξτe−τℏdτ,τ≥0,u1≤ℏ≤u2.Definition 2.
(see [46]). The Inverse A-T of function ξτ is denoted by A−1Qℏ and defined as(6)A−1Qℏ=ξτ=12πi∫w−i∞w+i∞ℏeℏτQℏdℏ.
A-T of several functions can be seen in Table1 [27–29]. A-T for some elementary functions is given as(7)ξτQℏ=Aξτ11ℏτ1ℏ2ταα!ℏα+2,α=1,2,…τνΓν+1ℏν+2.Table 1
The absolute error in ATDM and FRDTM for Example1 at ν = 1.
τReal part [ATDM] Abs. errorImg. part [ATDM] Abs. errorReal part [FRDTM] Abs. errorImg. part [FRTDM] Abs. error0.202.775557561562891 × 10−1702.775557561562891 × 10−170.43.508304757815494 × 10−146.661338147750939 × 10−163.508304757815494 × 10−146.66133814775093 × 10−160.64.540257059204578 × 10−121.743050148661495 × 10−144.540257059204578 × 10−121.743050148661495 × 10−140.81.43222100845719 × 10−101.645683589401869 × 10−121.43222100845719 × 10−101.645683589401869 × 10−121.02.081645966711676 × 10−95.585931717178028 × 10−112.081645966711676 × 10−95.585931717178028 × 10−11Definition 3.
(see [47]). The Caputo fractional derivative of order ν>0 is defined by(8)Dτνξτ=dαdταξτ,α=ν∈N,1Γα−ν∫0ττ−ρα−ν−1ξαρdρ,α−1<ν<α.
We have a few properties of Caputo’s fractional derivative.(9)Dτν1Dτν2ξτ=Dτν1+ν2ξτ,DτνC=0,C∈R,Dτντ−ηβ=Γβ+1Γβ+1−ντ−ηβ−ν,α−1<ν≤α,β>α−1,α∈N,β∈R,DτνC1ξ1τ+C2ξ2τ=C1Dτνξ1τ+C2Dτνξ2τ,C1,C2∈R.i. DτνJτνξτ=ξτ,Jτν is the R-L integral of ξτ order ν.Lemma 1 (see [27–31]).
Ifξ1τ and ξ2τ are piecewise continuous on 0,∞ and are of exponential order Aξ1τ=Q1ℏ,Aξ2τ=Q2ℏ, and C1,C2 are constants, then the properties mentioned below are valid:(10)AC1ξ1τ+C2ξ2τ=C1Q1ℏ+C2Q2ℏ,A−1C1Q1ℏ+C2Q2ℏ=C1ξ1τ+C2ξ2τ,ADτνξτ=ℏνQℏ−∑κ=0α−1ξκ0ℏκ−ν+2,α−1<ν≤α,α∈N.


## 3. Analysis of the Aboodh Transform Decomposition Method

In this section, we derive the main algorithms of the ATDM separately for the time-fractional Schrodinger differential equation with zero and nonzero trapping potential. The convergence analysis of the expansion solution is also presented.To present the ATDM on the general SDE with zero trapping potential given in equation (1), we first rewrite equation (1) as(11)Dτνξϖ,τ=iDϖϖξϖ,τ+iηξϖ,τ2ƛξϖ,τ.Now, applying the A-T to equation (11), we have(12)ADτνξϖ,τ=AiDϖϖξϖ,τ+iηξϖ,τ2ƛξϖ,τ.Using the differentiation property of the A-T and the initial condition of equation (12), we get(13)ℏνAξϖ,τ−ξϖ,0ℏ2−ν=AiDϖϖξϖ,τ+Aiηξϖ,τ2ƛξϖ,τ.(14)Aξϖ,τ=ξϖ,0ℏ2−ν+1ℏνAiDϖϖξϖ,τ+1ℏνAiηξϖ,τ2ƛξϖ,τ.Taking the inverse A-T on equation (14), we get as(15)ξϖ,τ=A−1ξϖ,0ℏ2+A−11ℏνAiDϖϖξϖ,τ+A−11ℏνAiηξϖ,τ2ƛξϖ,τ.So, according to the ATDM, we can acquire the solutionξϖ,τ to equation (15) as follows:(16)ξϖ,τ=∑α=0∞ξαϖ,τ.The nonlinear operator is decomposed as(17)ℵξϖ,τ=∑α=0∞Wαξ0,ξ1,ξ2,…,where(18)Wα=1α!dαdƛαℵ∑κ=0αƛκξκϖ,τƛ=0.The few terms of the decomposed nonlinear terms which are calculated from equation (18) are given as(19)W0=ξ02ϖ,τξϖ,τ¯,W1=ξ02ϖ,τξ1ϖ,τ¯+2ξ0ϖ,τξ1ϖ,τξ0ϖ,τ¯,W2=ξ12ϖ,τξ0ϖ,τ¯+2ξ0ϖ,τξ2ϖ,τξ0ϖ,τ¯+2ξ0ϖ,τξ1ϖ,τξ1ϖ,τ¯+ξ02ϖ,τξ0ϖ,τ¯.Now, by replacing equations (18) and (16) with equation (15), we attain as follows:(20)∑α=0∞ξαϖ,τ=A−1ξϖ,0ℏ2+A−11ℏνAiDϖϖ∑α=0∞ξαϖ,τ+A−11ℏνAiη∑α=0∞Wα.Equating the like terms on both sides of equation (20) yields the general solution of equation (11), which is recursively expressed as(21)ξ0ϖ,τ=A−1ξϖ,0ℏ2,ξ1ϖ,τ=A−11ℏνAiDϖϖ∑α=0∞ξ0ϖ,τ+A−11ℏνAηiW0,ξ2ϖ,τ=A−11ℏνAiDϖϖ∑α=0∞ξ1ϖ,τ+A−11ℏνAηiW1,ξ3ϖ,τ=A−11ℏνAiDϖϖ∑α=0∞ξ2ϖ,τ+A−11ℏνAηiW2,ξ4ϖ,τ=A−11ℏνAiDϖϖ∑α=0∞ξ3ϖ,τ+A−11ℏνAηiW3,ξα+1ϖ,τ=A−11ℏνAiDϖϖ∑α=0∞ξαϖ,τ+A−11ℏνAiη∑α=0∞Wα.where α=0,1,2,…Now, to present the ATDM on the general SDE with nonzero trapping potential given in equation (2), we first rewrite equation (2) as(22)Dτνξϖ,τ=−i−12ℑϖξϖ,τ+Θdϖξϖ,τ+HdQξϖ,τ.Now, applying the A-T to equation (22), we have:(23)ADτνξϖ,τ=−iA−12ℑϖξϖ,τ+Θdϖξϖ,τ+HdQξϖ,τ.Using the differentiation property of the A-T and the initial condition on equation (23), we get(24)ℏνAξϖ,τ−ξϖ,0ℏ2−ν=−iA−12ℑϖξϖ,τ+Θdϖξϖ,τ+HdQξϖ,τ.(25)Aξϖ,τ=ξϖ,0ℏ2−1ℏνiA−12ℑϖξϖ,τ+Θdϖξϖ,τ+HdQξϖ,τ.Taking the inverse A-T on equation (25), we get as(26)ξϖ,τ=A−1ξϖ,0ℏ2−A−11ℏνiA−12ℑϖξϖ,τ+Θdϖξϖ,τ+HdQξϖ,τ.So, according to the ATDM, we can acquire the solutionξϖ,τ to equation (26) as follows:(27)ξϖ,τ=∑α=0∞ξαϖ,τ.(28)Qξϖ,τ=∑α=0∞Wα.(29)Wα=1α!dαdƛαℵ∑κ=0αƛκξκϖ,τƛ=0.The few terms of the decomposed nonlinear terms which are calculated from equation (29) are given as(30)W0=ξ02ϖ,τξϖ,τ¯,W1=ξ02ϖ,τξ1ϖ,τ¯+2ξ0ϖ,τξ1ϖ,τξ0ϖ,τ¯,W2=ξ12ϖ,τξ0ϖ,τ¯+2ξ0ϖ,τξ2ϖ,τξ0ϖ,τ¯+2ξ0ϖ,τξ1ϖ,τξ1ϖ,τ¯+ξ02ϖ,τξ0ϖ,τ¯.Now, by replacing equations (27) and (29) with equation (26), we attain as follows:(31)∑α=0∞ξαϖ,τ=A−1ξϖ,0ℏ2−A−11ℏνiA−12ℑϖ∑α=0∞ξαϖ,τ+Θdϖ∑α=0∞ξαϖ,τ+Hd∑α=0∞Wα.Equating the like terms on both sides of equation (20), we finally obtain the general solution of equation (22) given recursively as(32)ξ0ϖ,τ=ξϖ,0,ξ1ϖ,τ=−A−11ℏνiA−12ℑϖξ0ϖ,τ+Θdϖξ0ϖ,τ+HdW0,ξ2ϖ,τ=−A−11ℏνiA−12ℑϖξ1ϖ,τ+Θdϖξ1ϖ,τ+HdW1,ξ3ϖ,τ=−A−11ℏνiA−12ℑϖξ2ϖ,τ+Θdϖξ2ϖ,τ+HdW2,ξα+1ϖ,τ=−A−11ℏνiA−12ℑϖξαϖ,τ+Θdϖξαϖ,τ+HdWα,ξϖ,τ=ξ0ϖ,τ+ξ1ϖ,τ+ξ2ϖ,τ+⋯,where α=0,1,2,…The following theorem describes the criteria for the expansion solution to converge.Theorem 1.
LetM be a Banach space with an appropriate norm . and a series of partial sums ∑α=0∞ξαϖ,τ defined over it. Assume that the initial guess w0=ξ0ϖ,τ remains inside the ball Brξ of the solution ξϖ,τ. Then, the series solution ∑α=0∞ξαϖ,τ converges if ∃σ>0 such that ξα+1ϖ,τ≤σξαϖ,τ.Proof.
The following is the description of a sequence of partial sums:(33)Θ0=ξ0ϖ,τ,Θ1=ξ0ϖ,τ+ξ1ϖ,τ,Θ2=ξ0ϖ,τ+ξ1ϖ,τ+ξ2ϖ,τ,Θ3=ξ0ϖ,τ+ξ1ϖ,τ+ξ2ϖ,τ+ξ3ϖ,τ,⋮⋮⋮Θα=ξ0ϖ,τ+ξ1ϖ,τ+ξ2ϖ,τ+ξ3ϖ,τ+⋯+ξαϖ,τ.
Then, we would have to demonstrate thatΘαα=0∞ is a Cauchy sequence in M. To demonstrate this, consider the relationship that(34)Θα+1−Θα=ξα+1ϖ,τ≤σξαϖ,τ≤σ2ξα−1ϖ,τ≤σ3ξα−2ϖ,τ≤⋯σα+1ξ0ϖ,τ,where α=0,1,2,3,…
For everyℓ,α∈Ν,α≥ℓ, we have(35)Θα−Θℓ=Θα−Θα−1+Θα−1−Θα−2+Θα−2−Θα−3+Θα−3−Θα−4+σ4ξα−3ϖ,τ≤⋯+Θℓ+1−Θℓ.
We get the following from the triangle inequality:(36)Θα−Θα−1+Θα−1−Θα−2+Θα−2−Θα−3+Θα−3−Θα−4+⋯+Θℓ+1−Θℓ≤Θα−Θα−1+Θα−1−Θα−2+Θα−2−Θα−3+Θα−3−Θα−4+⋯+Θℓ+1−Θℓ,Θα−Θα−1+Θα−1−Θα−2+Θα−2−Θα−3+Θα−3−Θα−4+⋯+Θℓ+1−Θℓ≤σαξ0ϖ,τ+σα−1ξ0ϖ,τ+σα−2ξ0ϖ,τ+⋯+σℓξ0ϖ,τ=1−σα−ℓ1−σσℓ+1ξ0ϖ,τ.
As a result, we have the following inequality:(37)Θα−Θℓ≤1−σα−ℓ1−σσα+1ξ0ϖ,τ.
Demonstrating that the sequence is bounded and we can attain for0<σ<1, that,(38)limℓ,α⟶∞Θα−Θℓ=0.
As a consequence, the sequence of partial sums of the ATDM is Cauchy and so convergent.


## 4. Approximate and Exact Solutions to SDEs with Zero and Nonzero Trapping Potential

In this part, we determined the exact solution to linear and nonlinear time-fractional SDEs with zero and nonzero potential by using the ATDM.Example 1.
We consider the following linear SDE with zero trapping potential:(39)iDτνξϖ,τ+Dϖϖξϖ,τ=0,0<ν≤1,τ≥0,ϖ∈R,with the initial condition:(40)ξϖ,0=beiaϖ.
By using the A-T on both sides of equation (39), we get as follows:(41)AiDτνξϖ,τ+Dϖϖξϖ,τ=0.
Using the third part of Lemma1, equation (41) is transformed as follows:(42)Aξϖ,τ=ξϖ,0ℏ2−1ℏνDϖϖAξϖ,τ.
By using the inverse A-T, equation (42) becomes as(43)ξϖ,τ=A−1ξϖ,0ℏ2−A−11ℏνDϖϖAξϖ,τ.
By using the procedure of ATDM, as explained in Section3, the expansion solution of equation (39) can be represented by the expansion form as follows:(44)ξϖ,τ=∑α=0∞ξαϖ,τ.
We get as by substituting equation (44) into equation (43).(45)∑α=0∞ξαϖ,τ=A−1ξϖ,0ℏ2−A−11ℏνDϖϖA∑α=0∞ξαϖ,τ.
Using the approach outlined in Section3, we can get the following from equation (45):(46)ξ0ϖ,τ=beiaϖ,(47)ξα+1ϖ,τ=−A−11ϕνDϖϖQξαϖ,τ,α=0,1,2,….
By repeating the iteration process in equation (47), we obtain the following results:(48)ξ1ϖ,τ=−iτνa2beiaϖΓν+1,ξ2ϖ,τ=−τ2νa4beiaϖΓ2ν+1,ξ3ϖ,τ=iτ3νa6beiaϖΓ3ν+1,ξ4ϖ,τ=τ4νa8beiaϖΓ4ν+1,ξ5ϖ,τ=−iτ5νa10beiaϖΓ5ν+1.
As a result, we can find the series solution as(49)ξϖ,τ=beiaϖ−iτνa2beiaϖΓν+1−τ2νa4beiaϖΓ2ν+1+iτ3νa6beiaϖΓ3ν+1+τ4νa8beiaϖΓ4ν+1−iτ5νa10beiaϖΓ5ν+1+⋯.
When we useν=1 in equation (49), we get the following precise solution to equation (39).(50)ξϖ,τ=beiaϖ−aτ.Example 2.
Consider the following one-dimensional nonlinear SDE with zero trapping potential:(51)iDτνξϖ,τ+Dϖϖξϖ,τ+2ξϖ,τ2ξϖ,τ=0,0<ν≤1,τ≥0,ϖ∈R,with the initial condition:(52)ξϖ,0=eiϖ.
By using A-T on both sides of equation (52), we get as(53)Aξϖ,τ=ξϖ,0ℏ2−ν+1ℏνAiDϖϖξϖ,τ+1ℏνA2iξϖ,τ2ξϖ,τ.
By following the process stated in Section3, we achieve the following result:(54)ξϖ,τ=A−1eiϖℏ2+A−11ℏνAiDϖϖξϖ,τ+A−11ℏνA2iξϖ,τ2ξϖ,τ.(55)ξϖ,τ=∑α=0∞ξαϖ,τ,ξϖ,τ2ξϖ,τ=ℵξϖ,τ,ℵξϖ,τ=∑α=0∞Qαξ0,ξ1,ξ2,…,W0=ξ02ϖ,τξϖ,τ¯,W1=ξ02ϖ,τξ1ϖ,τ¯+2ξ0ϖ,τξ1ϖ,τξ0ϖ,τ,¯W2=ξ12ϖ,τξ0ϖ,τ¯+2ξ0ϖ,τξ2ϖ,τξ0ϖ,τ¯+2ξ0ϖ,τξ1ϖ,τξ1ϖ,τ¯+ξ02ϖ,τξ0ϖ,τ¯.
By using equation (55) in equation (54), we obtain as follows:(56)∑α=0∞ξαϖ,τ=A−1eiϖℏ2+A−11ℏνAiDϖϖ∑α=0∞ξαϖ,τ+A−11ℏνA2i∑α=0∞Qα.
By following the process stated in Section3, we achieve the following result:(57)ξ0ϖ,τ=eiϖ,ξ1ϖ,τ=eiϖiτν1Γν+1,ξ2ϖ,τ=eiϖiτν2Γ2ν+1,ξ3ϖ,τ=eiϖiτν3Γ3ν+1,ξ4ϖ,τ=eiϖiτν4Γ4ν+1,ξ5ϖ,τ=eiϖiτν5Γ5ν+1.
As a result, we get the following solution in series form for equation (51).(58)ξϖ,τ=eiϖ+eiϖiτν1Γν+1+eiϖiτν2Γ2ν+1+eiϖiτν3Γ3ν+1+eiϖ9iτν4Γ4ν+1+eiϖiτν5Γ5ν+1+⋯.
When we useν=1 in equation (58), we get the following precise solution to equation (51).(59)ξϖ,τ=eiϖ+τ.Example 3.
Consider the following one-dimensional nonlinear SDE with trapping potential:(60)Dτνξϖ,τ=−i−12Dϖξϖ,τ+cos2ϖξϖ,τ+ξϖ,τ2ξϖ,τ,0<ν≤1,τ≥0,ϖ∈R.with the initial condition:(61)ξϖ,0=sinϖ.
By using the A-T on both sides of equation (69), we obtain as follows:(62)ADτνξϖ,τ=A−i−12Dϖξϖ,τ+cos2ϖξϖ,τ+ξϖ,τ2ξϖ,τ.
Using the third part of Lemma1, equation (62) is transformed as follows:(63)Aξϖ,τ=ξϖ,0ℏ2+1ℏνA−i−12Dϖξϖ,τ+cos2ϖξϖ,τ+ξϖ,τ2ξϖ,τ.
On both sides of equation (63), consider the inverse of the A-T.(64)ξϖ,τ=A−1ξϖ,0ℏ2+A−11ℏνA−i−12Dϖξϖ,τ+cos2ϖξϖ,τ+ξϖ,τ2ξϖ,τ.
By following the process stated in Section3, we achieve the following result:(65)ξϖ,τ=∑α=0∞ξαϖ,τ,Qξϖ,τ=ξϖ,τ2ξϖ,τ=∑α=0∞Wα.
Using equation (65) in equation (64), we get as follows:(66)∑α=0∞ξαϖ,τ=A−1ξϖ,0ℏ2+A−11ℏνA−i−12Dϖ∑α=0∞ξαϖ,τ+cos2ϖ∑α=0∞ξαϖ,τ+∑α=0∞Wα.
By equating the like terms on both sides of equation (66), we get as follows:(67)ξ0ϖ,τ=A−1ξϖ,0ℏ2,ξα+1ϖ,τ=A−11ℏνA−i−12Dϖξαϖ,τ+cos2ϖξαϖ,τ+Wα.
The following results are obtained from equation (67) using the iteration procedure stated in Section 3.(68)ξ0ϖ,τ=sinϖ,ξ1ϖ,τ=−3iτν2Γν+1sinϖ,ξ2ϖ,τ=−9τ2ν4Γ2ν+1sinϖ,ξ3ϖ,τ=27iτ3ν8Γ3ν+1sinϖ,ξ4ϖ,τ=81τ4ν16Γ4ν+1sinϖ,ξ5ϖ,τ=−243iτ5ν32Γ5ν+1sinϖ.
As a result, we get the following solution in series form for equation (60).(69)ξϖ,τ=sinϖ−3iτν2Γν+1sinϖ−9τ2ν4Γ2ν+1sinϖ+27iτ3ν8Γ3ν+1sinϖ+81τ4ν16Γ4ν+1sinϖ−243iτ5ν32Γ5ν+1sinϖ+⋯.
When we useν=1 in equation (69), we get the following precise solution to equation (60).(70)ξϖ,τ=sinϖe−3i/2τ.Example 4.
Consider the nonlinear three-dimensional SDE with trapping potential:(71)Dτνξϖ,β,μ,τ=−i−12∂2∂ϖ2+∂2∂β2+∂2∂μ2ξϖ,β,μ,τ+Kϖ,β,μξϖ,β,μ,τ+ξϖ,β,μ,τ2ξϖ,β,μ,τ,where 0<ν≤1,τ≥0,ϖ,β,μ∈R×R×R, and Kϖ,β,μ=1−sin2ϖsin2βsin2μ, with the initial condition:(72)ξϖ,β,μ,0=sinϖsinβsinμ.
Using the A-T on equation (71), we get:(73)ADτνξϖ,β,μ,τ=A−i−12∂2∂ϖ2+∂2∂β2+∂2∂μ2ξϖ,τ+Kϖ,β,μξϖ,β,μ,τ+ξϖ,β,μ,τ2ξϖ,β,μ,τ.
Using the third part of Lemma1, equation (73) is transformed as follows:(74)Aξϖ,β,μ,τ=ξϖ,β,μ,0ℏ2+1ℏνA−i−12∂2∂ϖ2+∂2∂β2+∂2∂μ2ξϖ,β,μ,τ+Kϖ,β,μξϖ,β,μ,τ+ξϖ,β,μ,τ2ξϖ,β,μ,τ.
On both sides of equation (74), consider the inverse of the A-T.(75)ξϖ,β,μ,τ=A−1ξϖ,β,μ,0ℏ2+A−11ℏνA−i−∂2∂ϖ2+∂2∂β2+∂2∂μ212ξϖ,β,μ,τ+Kϖ,β,μξϖ,β,μ,τ+ξϖ,β,μ,τ2ξϖ,β,μ,τ.
By following the process stated in Section3, we achieve the following result:(76)ξϖ,β,μ,τ=∑α=0∞ξαϖ,β,μ,τ,Qξϖ,β,μ,τ=ξϖ,β,μ,τ2ξϖ,β,μ,τ=∑α=0∞Wα.
Using equation (76) in equation (75), we get as follows:(77)∑α=0∞ξαϖ,β,μ,τ=A−1ξϖ,β,μ,0ℏ2+A−11ℏνA−i−12∂2∂ϖ2+∂2∂β2+∂2∂μ2∑α=0∞ξαϖ,β,μ,τ+Kϖ,β,μ∑α=0∞ξαϖ,β,μ,τ+∑α=0∞Wα.
By equating the like terms on both sides of equation (77), we get as follows:(78)ξ0ϖ,β,μ,τ=A−1ξϖ,β,μ,0ℏ2,ξα+1ϖ,β,μ,τ=A−11ℏνA−i−12∂2∂ϖ2+∂2∂β2+∂2∂μ2ξαϖ,β,μ,τ+Kϖ,β,μξαϖ,β,μ,τ+Wα.
The following results are obtained from equation (78) using the iteration procedure stated in Section 3.(79)ξ0ϖ,β,μ,τ=sinϖsinβsinμ,ξ1ϖ,β,μ,τ=−5iτν2Γν+1sinϖsinβsinμ,ξ2ϖ,β,μ,τ=25i2τ2ν4Γ2ν+1sinϖsinβsinμ,ξ3ϖ,β,μ,τ=−125i3τ3ν8Γ3ν+1sinϖsinβsinμ,ξ4ϖ,β,μ,τ=625i4τ4ν16Γ4ν+1sinϖsinβsinμ,ξ5ϖ,β,μ,τ=−3125i5τ5ν32Γ5ν+1sinϖsinβsinμ.
As a result, we get the following solution in series form for equation (78).(80)ξϖ,β,μ,τ=sinϖsinβsinμ−5iτν2Γν+1sinϖsinβsinμ+25i2τ2ν4Γ2ν+1sinϖsinβsinμ−125i3τ3ν8Γ3ν+1sinϖsinβsinμ+625i4τ4ν16Γ4ν+1sinϖsinβsinμ−3125i5τ5ν32Γ5ν+1sinϖsinβsinμ+⋯.
When we useν=1 in equation (80), we get the following precise solution to equation (78).(81)ξϖ,β,μ,τ=sinϖsinβsinμe−i5/2τ.


## 5. Numerical Simulation and Discussion

In this section, we discuss and evaluate the graphic and numerical results of the approximate and exact solutions to the models discussed in Examples1–4. Figures 1–8 represent the 2D graphs of the 5th approximate solution obtained by ATDM at ν=0.6,0.7,0.8,0.9,1.0 and the exact solution. These figures show that the approximate solutions obtained by the ATDM approach the exact solutions. The approximate result corresponds with the precise result at ν=1, and this proves the effectiveness and precision of the suggested method. Figures 9–16 demonstrate the 2D graph of absolute error over the 5th term, with approximate and exact solutions to Examples 1–4. As for the figure, approximate and exact solutions are in very good agreement. Tables 1–4 show comparisons of the absolute error of the 5th approximate solution obtained by ATDM of Examples 1–4 at ν=1 with the absolute error of approximate solutions obtained by FRDTM [42], HPM [35], and HAM [43]. The results obtained from the suggested method are extremely similar to those obtained by FRDTM, HPM, and HAM. The convergence of the approximate solution to the exact solution for Examples 1–4 has been shown numerically as in Tables 5–12. The results show that the proposed method is a useful and efficient algorithm for solving certain classes of fractional-order differential equations with fewer calculations and iteration steps. The 3D graphs of these solutions are also sketched to show the behavior of the exact solutions in Figures 17–24.Figure 1
The approximate and exact solutions to the real part of Example1.Figure 2
The approximate and exact solutions to the imaginary part of Example1.Figure 3
The approximate and exact solutions to the real part of Example2.Figure 4
The approximate and exact solutions to the imaginary part of Example2.Figure 5
The approximate and exact solutions to the real part of Example3.Figure 6
The approximate and exact solutions to the imaginary part of Example3.Figure 7
The approximate and exact solutions to the real part of Example4.Figure 8
The approximate and exact solutions to the imaginary part of Example4.Figure 9
The absolute error for the real part of Example1.Figure 10
The absolute error for the imaginary part of Example1.Figure 11
The absolute error for the real part of Example2.Figure 12
The absolute error for the imaginary part of Example2.Figure 13
The absolute error for the real part of Example3.Figure 14
The absolute error for the imaginary part of Example3.Figure 15
The absolute error for the real part of Example4.Figure 16
The absolute error for the imaginary part of Example4.Table 2
The absolute error in ATDM and FRDTM for Example2 at ν = 2.
τReal part [ATDM] Abs. errorImg. part [ATDM] Abs. errorReal part [FRDTM] Abs. errorImg. part [FRTDM] Abs. error0.202.775557561562891 × 10−1702.775557561562891 × 10−170.43.508304757815494 × 10−142.886579864025407 × 10−153.508304757815494 × 10−142.886579864025407 × 10−150.64.519384866341625 × 10−124.35984581770299 × 10−134.519384866341625 × 10−124.35984581770299 × 10−130.81.423423601210061 × 10−101.593580822856211 × 10−111.423423601210061 × 10−101.593580822856211 × 10−111.02.065669746365017 × 10−92.633981921462691 × 10−102.065669746365017 × 10−92.633981921462691 × 10−10Table 3
The absolute error in ATDM and HPM for Example3 at ν = 1.
τReal part [ATDM] Abs. errorImg. part [ATDM] Abs. errorReal part [HPM] Abs. errorImg. part [HPM] Abs. error0.25.551115123125783 × 10−1705.551115123125783 × 10−1700.42.266797860528413 × 10−131.047079090099600 × 10−142.266797860528413 × 10−131.047079090099600 × 10−140.62.933806669824790 × 10−112.032304879939772 × 10−122.933806669824790 × 10−112.032304879939772 × 10−120.89.229918426778560 × 10−108.528888012504510 × 10−119.229918426778560 × 10−108.528888012504510 × 10−111.01.337196644298627 × 10−81.545451835949229 × 10−91.337196644298627 × 10−81.5454518359492290 × 10−9Table 4
The absolute error in ATDM and HAM for Example4 at ν = 1.
τReal part [ATDM] Abs. errorImg. part [ATDM] Abs. errorReal part [HAM] Abs. errorImg. part [HAM] Abs. error0.23.794707603699265 × 10−161.463672932855431 × 10−173.794707603699265 × 10−161.463672932855431 × 10−170.41.548127132131732 × 10−121.191737680761306 × 10−131.548127132131732 × 10−121.191737680761306 × 10−130.61.994952888051838 × 10−102.305647122877174 × 10−111.994952888051838 × 10−102.305647122877174 × 10−110.86.238188888683330 × 10−99.625127946185849 × 10−106.238188888683330 × 10−99.625127946185849 × 10−101.08.967831111224030 × 10−81.732387384885186 × 10−88.967831111224030 × 10−81.732387384885186 × 10−8Table 5
The recurrence error of the 5th approximate solution of the real part in Example1.
τν = 0.7ν = 0.8ν = 0.9ν = 1.00.11.9863230258832510 × 10−112.480476538225475 × 10−132.754480547804426 × 10−152.753540058639898 × 10−170.22.5462310564404910 × 10−96.356503419032404 × 10−111.41126628375623 × 10−122.820907153618413 × 10−140.34.3559264601363860 × 10−81.630598080005387 × 10−95.428928326273314 × 10−111.6274165548755540 × 10−120.43.2669475564040040 × 10−71.630154704745158 × 10−87.234972435809250 × 10−102.8912347348589850 × 10−110.51.5594317330665020 × 10−69.724388066839424 × 10−85.393791208135125 × 10−92.6938949149435820 × 10−100.65.5932228619978110 × 10−64.184575219876433 × 10−72.784754999984136 × 10−81.6687458133498650 × 10−90.71.6470066797307933 × 10−51.437319768581204 × 10−61.115745763781643 × 10−77.7992900939752010 × 10−90.84.1978532985731196 × 10−54.186076646645796 × 10−63.713180897405904 × 10−72.9660020264616400 × 10−80.99.5822726035593689 × 10−51.0748242434163506 × 10−61.072429735204339 × 10−69.6359264875560130 × 10−81.02.0050812724929500 × 10−42.4986212656296714 × 10−52.769704199336456 × 10−62.7648088107301450 × 10−7Table 6
The recurrence error of the 5th approximate solution of the imaginary part in Example1.
τν = 0.7ν = 0.8ν = 0.9ν = 1.00.15.729282854280857 × 10−145.566602532673186 × 10−159.391428658269753 × 10−171.127083924407682 × 10−180.26.735588070529998 × 10−131.29434773662193 × 10−132.86554558790132 × 10−148.979209489138208 × 10−160.32.240327760588405 × 10−92.657335380224294 × 10−113.928763432275554 × 10−133.7004114535955 × 10−140.42.415220588436187 × 10−85.443488739768476 × 10−103.891048192098205 × 10−123.947468488238387 × 10−130.51.477344106931362 × 10−74.825025026815447 × 10−99.526894455781187 × 10−111.232929037808270 × 10−120.66.392078149330569 × 10−72.727397229393246 × 10−88.267292729705227 × 10−107.495149197961302 × 10−120.72.187803743850105 × 10−61.152748324472957 × 10−74.630213360876863 × 10−91.0569208657930970 × 10−100.86.320873774992085 × 10−63.96750649205628 × 10−71.972744306783386 × 10−86.7041439426968450 × 10−100.91.606257206934282 × 10−51.171284786793358 × 10−66.92782295622906 × 10−83.0494721017930660 × 10−91.03.691229597776269 × 10−53.069260602197251 × 10−62.103049742344222 × 10−71.1247880551985750 × 10−8Table 7
The recurrence error of the the approximate solution of the real part in Example2.
τν = 0.7ν = 0.8ν = 0.9ν = 1.00.11.976971658223765 × 10−112.473641816908283 × 10−132.750095402344202 × 10−152.751035891508731 × 10−170.22.52678613921649 × 10−96.326039570110015 × 10−111.407076602719864 × 10−122.815778619333783 × 10−140.34.311799014005643 × 10−81.619798972797868 × 10−95.405728516404219 × 10−111.622980497927706 × 10−120.43.226514454036917 × 10−71.616576300475306 × 10−87.19494314294836 × 10−102.880731496644064 × 10−110.51.536892238892099 × 10−69.627636757901811 × 10−85.357333694498901 × 10−92.681667536373432 × 10−100.65.501465744903981 × 10−64.136441235299919 × 10−72.762589303466765 × 10−81.659660768720673 × 10−90.71.616936914302743 × 10−51.418630876198142 × 10−61.105549182157846 × 10−77.749774527601226 × 10−90.84.113778074791275 × 10−54.125554726718593 × 10−63.674935878917996 × 10−72.944491394597482 × 10−80.99.374043008389243 × 10−51.0577612629278684 × 10−51.060155771139881 × 10−69.557343069541972 × 10−81.01.9582134116140592 × 10−42.45549708950409 × 10−52.734871750812168 × 10−62.739767139418478 × 10−7Table 8
The recurrence error of the 5th approximate solution of the imaginary part in Example2.
τν = 0.7ν = 0.8ν = 0.9ν = 1.00.11.926007539341256 × 10−121.922465206587756 × 10−141.815440978456445 × 10−161.627499919875739 × 10−180.23.212183277944602 × 10−106.217126403566797 × 10−121.123792369482977 × 10−131.922772907632402 × 10−150.36.577805665150846 × 10−91.892287752233342 × 10−105.028971037475495 × 10−121.256513068851356 × 10−130.45.664659911177431 × 10−82.169068535347471 × 10−97.610081091968145 × 10−112.493643660279493 × 10−120.53.026797519340784 × 10−71.450910885437432 × 10−86.332736016444167 × 10−102.566730381652197 × 10−110.61.194404986717049 × 10−66.891376017773192 × 10−83.602715131890486 × 10−91.740543007331589 × 10−100.73.821136878718447 × 10−62.581914817482758 × 10−71.574595275095308 × 10−88.837938438846447 × 10−100.81.0480156108100884 × 10−58.126788825165077 × 10−75.66988415843565 × 10−83.628126275591497 × 10−90.92.5548636246620873 × 10−52.238467006737139 × 10−61.759964515566688 × 10−71.265411208128710 × 10−81.05.674529965660330 × 10−55.548386062052328 × 10−64.85763358662764 × 10−73.879371899481992 × 10−8Table 9
The recurrence error of the 5th approximate solution of the real part in Example3.
τν = 0.7ν = 0.8ν = 0.9ν = 1.00.15.718354659335265 × 10−117.147943324169061 × 10−137.942159249076738 × 10−157.942159249076745 × 10−170.27.319493963949132 × 10−91.829873490987278 × 10−104.066385535527289 × 10−128.132771071054586 × 10−140.31.250604163996619 × 10−74.689765614987320 × 10−91.563255204995775 × 10−104.689765614987315 × 10−120.49.368952273854868 × 10−74.684476136927436 × 10−82.081989394189970 × 10−98.327957576759896 × 10−110.54.467464577605666 × 10−62.792165361003536 × 10−71.551202978335300 × 10−87.756014891676498 × 10−100.61.6007733299156725 × 10−51.200579997436754 × 10−68.003866649578365 × 10−84.802319989747010 × 10−90.74.709310951212926 × 10−54.120647082311310 × 10−63.204947730686570 × 10−72.243463411480599 × 10−80.81.1992258910534259 × 10−41.1992258910534254 × 10−51.065978569825267 × 10−68.527828558602135 × 10−80.92.735071306660605 × 10−43.076955219993183 × 10−53.076955219993184 × 10−62.769259697993865 × 10−71.05.718354659335249 × 10−47.147943324169061 × 10−57.942159249076736 × 10−67.942159249076734 × 10−7Table 10
The recurrence error of the 5th approximate solution of the imaginary part in Example3.
τν = 0.7ν = 0.8ν = 0.9ν = 1.00.14.044352360290241 × 10−122.95593353802917 × 10−141.896521881939167 × 10−161.083021715783192 × 10−180.28.40968933466814 × 10−101.317524278577765 × 10−111.811985904661213 × 10−132.218028473923979 × 10−150.31.908458178643264 × 10−84.670482042212528 × 10−101.003363456757901 × 10−111.918540478858446 × 10−130.41.748682320562088 × 10−73.375983249079892 × 10−81.731218052350544 × 10−104.542522314596309 × 10−120.59.748056090119807 × 10−74.184376006926562 × 10−81.576742960965881 × 10−95.288191971597613 × 10−110.63.968383306108540 × 10−72.081736075628826 × 10−79.586393166413172 × 10−93.929170900702099 × 10−100.71.3004806485065536 × 10−54.646598110881215 × 10−74.409897082301801 × 10−82.141487801867844 × 10−90.83.636150797676397 × 10−52.617499157554769 × 10−61.654050363788450 × 10−79.303085700293241 × 10−90.99.005675804590360 × 10−57.379530772932580 × 10−65.308339629215292 × 10−73.398636902083380 × 10−81.02.0269777712677892 × 10−41.8650679756148277 × 10−51.506460878596710 × 10−61.083021715783191 × 10−7Table 11
The recurrence error of the 5th approximate solution of the real part in Example4.
τν = 0.7ν = 0.8ν = 0.9ν = 1.00.11.410899077403071 × 10−101.763623846753835 × 10−121.959582051948707 × 10−141.959582051948706 × 10−160.21.80595081907593 × 10−84.514877047689823 × 10−101.003306010597737 × 10−112.006612021195475 × 10−130.33.085636282280515 × 10−71.15711360585519 × 10−83.857045352850636 × 10−101.157113605855191 × 10−110.42.311617048417188 × 10−61.155808524208595 × 10−75.136926774260415 × 10−92.054770709704167 × 10−100.51.1022649042211474 × 10−56.889155651382168 × 10−73.827308695212317 × 10−81.913654347606158 × 10−90.63.949614441319053 × 10−52.96221083098929 × 10−61.974807220659528 × 10−71.184884332395716 × 10−80.71.1619360589017545 × 10−41.0166940515390365 × 10−57.907620400859165 × 10−75.535334280601419 × 10−80.82.958869821974002 × 10−42.958869821974005 × 10−50.0000026301065084213382.104085206737067 × 10−70.96.748286549347473 × 10−47.591822368015921 × 10−57.591822368015916 × 10−66.832640131214322 × 10−71.01.410899077403069 × 10−31.7636238467538362 × 10−41.959582051948707 × 10−51.959582051948706 × 10−6Table 12
The recurrence error of the 5th approximate solution of the imaginary part in Example4.
τν = 0.7ν = 0.8ν = 0.9ν = 1.00.11.663116214880405 × 10−111.215537262646501 × 10−137.79886620339425 × 10−164.453595572610698 × 10−180.23.458227535246130 × 10−95.417915641366231 × 10−117.451237851492589 × 10−139.120963732706710 × 10−150.37.847950572968780 × 10−81.920592896894366 × 10−94.126025345211919 × 10−117.889410949012672 × 10−130.47.190921222780890 × 10−72.414883590902916 × 10−87.119104761067955 × 10−101.867973372458334 × 10−110.54.008589930546624 × 10−61.720696657048121 × 10−76.483873192722287 × 10−92.174607213188817 × 10−100.61.6318762648010684 × 10−58.560502928660110 × 10−73.942110997503766 × 10−81.615751362357795 × 10−90.75.347828925356589 × 10−53.323770499911620 × 10−61.813435301914099 × 10−78.806213628229532 × 10−90.81.495255806774449 × 10−41.076366474421812 × 10−56.801776242977975 × 10−73.825609466794668 × 10−80.93.703308743231145 × 10−43.0346063333097793 × 10−52.182892321184059 × 10−66.832640131214322 × 10−71.08.335326149365687 × 10−47.669521633561604 × 10−56.194859624193012 × 10−64.453595572610698 × 10−7Figure 17
The real part of the exact solution to Example1.Figure 18
The imaginary part of the exact solution to Example1.Figure 19
The real part of the exact solution to Example2.Figure 20
The imaginary part of the exact solution to Example2.Figure 21
The real part of the exact solution to Example3.Figure 22
The imaginary part of the exact solution to Example3.Figure 23
The real part of the exact solution to Example4.Figure 24
The imaginary part of the exact solution to Example4.The following 2D graphs show the real and imaginary parts of approximate and exact solutions to Example1:Figures1 and 2 show the behavior of real imaginary in the interval τ∈0,1 between the 5th step iteration approximate and exact solutions of equation (39) at several values of ν when ϖ=0.05, a=1 and b=1. The approximate result corresponds with the precise result at ν=1 and this proves the effectiveness and precision of the suggested method.The graphs of absolute error for the real and imaginary parts of the 5th approximation and exact solutions to Example1 are as follows:Figures9 and 10 demonstrate the 2D graph of real and imaginary parts of absolute error in the intervals τ∈0,1 when ϖ=0.05, a=1, and b=1 are over the 5th terms, approximate and exact solutions of equation (39) at ν=1. As for the figures, approximate and exact solutions are in very good agreement.The accuracy and capability of the numerical method can be determined using error functions. To demonstrate the accuracy and capability of the ATDM, we used recurrence and absolute error functions.Table1 shows comparisons of the real and imaginary parts of the absolute error of the 5th approximate solution obtained by ATDM in Example 1 at ν=1 with the absolute error of approximate solutions obtained by FRDTM [42]. The results obtained from the suggested method are extremely similar to those obtained by FRDTM.The recurrence errorξ5ϖ,τ−ξ4ϖ,τ between the 5th and 4th approximate solutions of the real part with different values of ν, when ϖ=0.05,a=1, and b=1 in Example 1 are presented as follows:The recurrence errorξ5ϖ,τ−ξ4ϖ,τ between the 5th and 4th approximate solutions of the imaginary part with different values of ν, when ϖ=0.05,a=1, and b=1 in Example 1 are presented as follows:The convergence of the ATDM of real and imaginary of the approximate solution to the exact solution for equation (39) has been shown numerically as in Tables 5 and 6. The results show that the current technique is a useful and efficient algorithm for solving fractional-order differential equations with fewer calculations and iteration steps.The following are 3D graphs for the real and imaginary parts of the exact solution to Example1:The real and imaginary parts of the exact solution equation (39) at ν=1 are shown in Figures 17 and 18 respectively in the intervals τ∈0,2,ϖ∈−2π,2π with a=1 and b=1.The following 2D graphs show the real and imaginary parts of approximate and exact solutions to Example2:Figures3 and 4 show the behavior of real and imaginary parts in the interval τ∈0,1 between the 5th step iteration approximate and exact solutions of equation (51) at several values of ν when ϖ=0.05. The approximate result corresponds with the precise result at ν=1, and this proves the effectiveness and precision of the suggested method.The graphs of absolute error for the real and imaginary parts of the 5th approximation and exact solutions to Example2 are as follows:Figures11 and 12 demonstrate the 2D graph of real and imaginary parts of absolute error in the intervals τ∈0,1 when ϖ=0.05 are over the 5th terms approximate and exact solutions of equation (51) at ν=1. As for the figures, approximate and precise solutions are in very good agreement.Table2 shows comparisons of the real and imaginary parts of absolute error of the 5th approximate solution obtained by ATDM of Example 2 at ν=1 with the absolute error of approximate solutions obtained by FRDTM [42]. The results obtained from the suggested method are extremely similar to those obtained by FRDTM.The recurrence errorξ5ϖ,τ−ξ4ϖ,τ between the 5th and 4th approximate solutions of the real part with different values of ν, when ϖ=0.05 in Example 2 are presented as follows:The recurrence errorξ5ϖ,τ−ξ4ϖ,τ between the 5th and 4th approximate solutions of the imaginary part with different values of ν, when ϖ=0.05 in Example 2 are presented as follows:The convergence of the ATDM of real and imaginary of the approximate solution to the exact solution for equation (51) has been shown numerically as in Tables 7 and 8. The results show that the current technique is a useful and efficient algorithm for solving fractional-order differential equations with fewer calculations and iteration steps.The following are the 3D graphs for the real and imaginary parts of the exact solution to Example2:The real and imaginary parts of the exact solution equation (51) at ν=1 are shown in Figures 19 and 20 respectively in the intervals τ∈0,2, and ϖ∈−2π,2π.The following 2D graphs show the real and imaginary parts of approximate and exact solutions to Example3:Figures5 and 6 show the behavior of the real and imaginary parts in the interval τ∈0,1 between the 5th step iteration approximate and exact solutions of equation (60) at several values of ν when ϖ=0.05. The approximate result corresponds with the precise result at ν=1 and this proves the effectiveness and precision of the suggested method.The 2D graphs of absolute error for the real and imaginary parts of the 5th approximation and exact solutions to Example3 are as follows:Figures11 and 12 demonstrate the 2D graph of real and imaginary parts of absolute error in the intervals τ∈0,1 when ϖ=0.05 are over the 5th terms approximate and exact solutions of equation (60) at ν=1. As for the figures, approximate and precise solutions are in very good agreement.Table3 shows comparisons of the real and imaginary parts of the absolute error of the 5th approximate solution obtained by ATDM of Example 3 at ν=1 with the absolute error of approximate solutions obtained by HPM [35]. The results obtained from the suggested method are extremely similar to those obtained by HPM.The recurrence errorξ5ϖ,τ−ξ4ϖ,τ between the 5th and 4th approximate solution of the real part with different values of ν, when ϖ=0.05 for Example 3 are presented as follows:The recurrence errorξ5ϖ,τ−ξ4ϖ,τ between the 5th and 4th approximate solution of the imaginary part with different values of ν, when ϖ=0.05 for Example 3 are presented as follows:The convergence of the ATDM of real and imaginary of the approximate solution to the exact solution for equation (60) has been shown numerically as in Tables 9 and 10. The results show that the proposed technique is a useful and efficient algorithm for solving fractional-order differential equations with fewer calculations and iteration steps.The following are 3D graphs for the real and imaginary parts of the exact solution to Example3:The real and imaginary parts of the exact solution equation (60) at ν=1 are shown in Figures 21 and 22, respectively, in the intervals τ∈0,2,ϖ∈−3π,3π.The following 2D graphs show the real and imaginary parts of approximate and exact solutions to Example4:Figures7 and 8 show the behavior of the real and imaginary parts in the interval τ∈0,1 between the 5th step iteration approximate and exact solutions of equation (71) at several values of ν when ϖ=0.05,β=0.10,andμ=0.15. The approximate result corresponds with the precise result at ν=1 and this proves the effectiveness and precision of the recommended method.The 2D graphs of absolute error for the real and imaginary parts of the 5th approximation and exact solutions to Example4 are as follows:Figures15 and 16 demonstrate the 2D graph of real and imaginary parts of absolute error in the intervals τ∈0,1 when ϖ=0.05,β=0.10,andμ=0.15 are over the 5th terms approximate and exact solutions of equation (71) at ν=1. As for the figures, approximate and precise solutions are in very good agreement.Table4 shows comparisons of the real and imaginary parts of the absolute error of the 5th approximate solution obtained by ATDM of Example 4 at ν=1 with the absolute error of the approximate solution obtained by HAM [43]. The results obtained from the suggested method are extremely similar to those obtained by HAM.The recurrence errorξ5ϖ,β,μ,τ−ξ4ϖ,β,μ,τ between the 5th and 4th approximate solution of the real part with different values of ν, when ϖ=0.05, β=0.10, and μ=0.15 for Example 4 are presented.The recurrence errorξ5ϖ,β,μ,τ−ξ4ϖ,β,μ,τ between the 5th and 4th approximate solutions of the imaginary part with different values of ν, when ϖ=0.05, β=0.10, and μ=0.15 for Example 4 are presented.The convergence of the ATDM of real and imaginary of the approximate solution to the exact solution for equation (71) has been shown numerically as in Tables 11 and 12. The results show that the proposed technique is a useful and efficient algorithm for solving fractional-order differential equations with fewer calculations and iteration steps.The following are 3D graphs for the real and imaginary parts of the exact solution to Example4:The real and imaginary parts of the exact solution equation (71) at ν=1 are shown in Figures 23 and 24 respectively in the intervals τ∈0,2,ϖ∈−3π,3π with β=0.1 and μ=0.2.


## 6. Conclusion

The Aboodh transform decomposition method is effectively used in this study to obtain analytical approximate and exact solutions to time-fractional linear and nonlinear Schrodinger equations with zero and nonzero trapping potential that are regarded in the Caputo sense. The Aboodh transform is more closely related to the Laplace and Elzaki transforms. The Aboodh transform is a useful method for solving time-domain differential equations. The recurrence and absolute error of the four problems are analyzed to evaluate the efficiency and consistency of the presented method. In addition, numerical results are also compared with other methods such as the fractional reduced differential transform method (FRDTM), the homotopy analysis method (HAM), and the homotopy perturbation method (HPM). The results obtained by the proposed method show excellent agreement with these methods, which indicates its effectiveness and reliability. This method has the advantage of needing no assumptions regarding minor or important physical parameters in the problem. As a result, it can solve both weakly and strongly nonlinear problems, overcoming some of the drawbacks of traditional perturbation methods. Only a few computations are required to solve nonlinear fractional-order differential equations. As a result, it greatly improves homotopy analysis and homotopy perturbation techniques. The ATDM can construct expansion solutions for linear and nonlinear fractional-order differential equations without the requirement for perturbation, linearization, or discretization, unlike earlier analytic approximation methods.Therefore, we concluded that our proposed technique is simple to apply, accurate, and efficient according to the results. It is significant to consider that implementing the ATDM to solve other kinds of ordinary and partial DEs of noninteger order is actively attainable. For example, fractional kdv equations, fractional phi-4 equations, fractional Schrodinger equations, and many more.



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