New approximate-analytical solutions for the nonlinear fractional Schrödinger equation with second-order spatio-temporal dispersion via double Laplace transform method
Mohammed K. A. Kaabar, Francisco Martínez, José Francisco Gómez-Aguilar, Behzad Ghanbari, Melike Kaplan
NNew approximate-analytical solutions for the nonlinear fractionalSchr¨odinger equation with second-order spatio-temporal dispersionvia double Laplace transform method
Mohammed K. A. Kaabar ∗ , Francisco Mart´ınez , Jos´e Francisco G´omez-Aguilar , BehzadGhanbari and Melike Kaplan Department of Mathematics and Statistics, Washington State University, Pullman, WA,USA Department of Applied Mathematics and Statistics, Technological University ofCartagena, Cartagena, Spain CONACyT-Tecnol´ogico Nacional de M´exico/CENIDET, Interior Internado Palmira S/N,Col. Palmira, C.P. 62490, Cuernavaca Morelos, Mexico Department of Engineering Science, Kermanshah University of Technology, Kermanshah,Iran Department of Mathematics, Art-Science Faculty, Kastamonu University, Kastamonu,TurkeyOctober 22, 2020
Abstract
In this paper, a modified nonlinear Schr¨odinger equation with spatio-temporal dispersion is formulatedin the senses of Caputo fractional derivative and conformable derivative. A new generalized doubleLaplace transform coupled with Adomian decomposition method has been defined and applied to solvethe newly formulated nonlinear Schr¨odinger equation with spatio-temporal dispersion. The approximateanalytical solutions using the proposed generalized method in the sense of Caputo fractional derivativeand conformable derivatives are obtained and compared with each other graphically.
Fractional derivatives (FDs) which are generalized forms of fractional calculus (FC) have been recentlyapplied in various modeling scenarios arising from phenomena in science and engineering. FDs are usedparticularly in modeling various complex engineering systems in mechanics. Various systems in science andengineering have been investigated in [
ATT ; AG ; YZ ; AMDT ; MK ; SSR ] using analytical and approx-imate analytical methods to find solutions to the fractional equations in the sense of fractional derivativesand fractional integrals such as Caputo, Riemann-Liouville, and Gr¨unwald–Letnikov [
Ekm ]. All fractionalderivatives such as Caputo, Riemann-Liouville, and Gr¨unwald–Letnikov have shown the property of linearity,but the usual derivative properties such as the derivatives of constant, quotient rule, product rule, and chainrule can not be satisfied using any of those fractional derivatives [
AKH ].Khalil et al. introduced a new generalized local fractional derivative known as conformable derivativewhich is basically an extension of the usual limit-based derivative [
KHH ; MLT ]. Conformable deriva-tive (CD) occurs naturally and satisfies all properties of usual derivatives [
KHH ; MLT ]. The first author ∗ Corresponding author e-mail:[email protected] a r X i v : . [ m a t h . G M ] O c t iscussed in [ mkaabar ] three different analytical methods for solving two-dimensional wave equation involv-ing conformable derivative. Introducing CD formulations to (linear/nonlinear) partial differential equationscan easily transform them into simpler versions [ MLT ] than other commonly used fractional derivativesformulations such as Riemann-Liouville, Caputo, and Gr¨unwald–Letnikov because the difficulty of findinganalytical solutions using those fractional definitions makes CD formulation a good option for certain cases.The physical interpretation for conformable derivative can be simply presented as modified version of theusual derivative in magnitude and direction [
SiLa ] (see also [ coZ ]). Khalil et al. described in [ gKH ] the ge-ometrical meaning of conformable derivative using the concept of fractional cords. For a comparative reviewand analysis of all definitions of fractional derivatives and fractional operators, we refer to [ mRev23 ].From this newly defined derivative, many essential elements of the mathematical analysis of functionsof a real variable have been successfully developed, among which we can mention: Mean Value Theorem,Rolle’s Theorem, chain rule, conformable integration by parts formulas, conformable power series expan-sion and conformable single and double Laplace transform definitions, [
KHH ; AJ ; TO ; mkaabar ]. Theconformable partial derivative of the order γ ∈ (0 ,
1] of the real-valued functions of several variables andconformable gradient vector are defined, and a conformable Clairaut’s Theorem for partial derivatives inthe conformable sense is proven in [
ATN ]. In [
YzzY ], the conformable Jacobian matrix is introduced;chain rule for multivariable conformable derivative is defined; the relation between conformable Jacobianmatrix and conformable partial derivatives is investigated. In [
MarT ], two new results on homogeneousfunctions involving their conformable partial derivatives are introduced, specifically, the homogeneity of theconformable partial derivatives of a homogeneous function and conformable Euler’s Theorem.The partial differential equations (PDEs), particularly the Schr¨odinger equation, have been applied inseveral applications in physics and engineering due to the importance of this equation in nonlinear opticswhich can successfully explain the dynamics of optical soliton propagation in optical fibers. Various fractionalformulations have been introduced in [
YmMwRm ; JFmWO ] to obtain exact optical soliton solutions formodified nonlinear Schr¨odinger equation (MNLSE) with spatio-temporal dispersion. Finding analytical andapproximate analytical solutions for the modified forms of nonlinear fractional Schr¨odinger equation havebecome a common research interest for physicists and applied mathematicians in the field of optical solitonpropagation because of the applications of this equation in plasma, optics, electromagnetism, fluid dynamics,and optical communication [
JFmWO ; YmMwRm ]. The dynamics of optical soliton propagation in opticalfiber can be interpreted from the MNLSE with second-order spatio-temporal dispersion and group velocitydispersion coefficients [
JFmWO ]. Given Ψ( x, t ) as a complex-valued wave function that represents themacroscopic property of wave profile of the spatial and temporal variables which are expressed as x and t ,respectively. Then, MNLSE can be written as [ JFmWO ]: i (cid:18) ∂ Ψ ∂x + ω ∂ Ψ ∂t (cid:19) + ω ∂ Ψ ∂t + ω ∂ Ψ ∂x + | Ψ | Ψ = 0 , where [ JFmWO ] ω is proportional to the ratio of group speed; ω is a group velocity dispersion coefficient; ω is a spatial dispersion coefficient; (1)To formulate (1) in the sense of fractional derivatives, let us first define Caputo fractional derivative asfollows: Definition 1.
For ξ, γ >
0, given two functions: h ( x ) and h ( t ) such that for x, t >
0, the Caputo fractionalderivative (CpFD) of h of order ξ and γ , denoted by D ξx ( h )( x ) and D γt ( h )( t ), respectively where D ξx and D γt are Caputo derivative operators which can be simply expressed as [ CHinD ]: D ξx h ( x ) = 1 ξ (Ω − ξ ) (cid:90) x ( x − η ) Ω − ξ − h (Ω) ( η )d η ; Ω − < ξ ≤ Ω for Ω ∈ N , (2) D γt h ( t ) = 1Γ( w − γ ) (cid:90) t ( t − µ ) w − γ − h ( w ) ( µ )d µ ; w − < γ ≤ w for w ∈ N . (3)If ξ = Ω and γ = w where Ω , w ∈ N , then D ξx h ( x ) = d Ω dx Ω h ( x ) and D γt h ( t ) = d w dt w h ( t ). CpFD is very usefulin science and engineering due to their important properties such as the inclusion of initial and boundary2onditions in the fractional formulation of CpFD [ ZCpt ; ATT ]. Let us now define the Mittag-Lefflerfunction:
Definition 2.
The Mittag-Leffler function, denoted by E ξ,ζ ( t ), can be expressed as follows [ RdGw ]: E ξ,ζ ( t ) = ∞ (cid:88) m =0 t m Γ( ξm + ζ ) , where t, ζ ∈ C and (cid:60) ( ξ ) > . (4)From the above definition, the Mittag-Leffler function, denoted by E ( t, h, c ), can be written [ CHinD ]as: E ( t, h, c ) = t h E ,h +1 ( ct ), and the fractional derivative of Mittag-Leffler function can also be expressed[ CHinD ] as: ∂ δ ∂t δ ( t ζ − E ξ,ζ ( ct ξ )) = t ζ − δ − E ξ,ζ − δ ( ct ξ ) where δ ≥
0. let us now define the conformablederivative as follows:
Definition 3.
Given a function h : [0 , ∞ ) → (cid:60) such that for all t >
0, the conformable derivative (CD) oforder γ ∈ (0 ,
1] of h , denoted by M γ ( h )( t ), can be represented as: h ( γ ) ( t ) = M γ ( h )( t ) = lim ξ → h ( t + ξt − γ ) − h ( t ) ξ . (5)Suppose that h is γ -differentiable in some (0 , d ), d >
0, and the limit of h ( γ ) ( t ) exists as t −→ + , then fromCD definition, the following is obtained: h ( γ ) (0) = M γ ( h )(0) = lim t → + h ( γ ) ( t ) . (6)It is also important to define here the conformable integral (ComI) [ SiLa ] as follows:
Definition 4.
For γ ∈ (0 , h : [0 , ∞ ) → (cid:60) such that for all t ≥
0, the γ th order ComI of h from 0 to t can be expressed as: I γ ( h )( t ) = (cid:90) t h ( ψ ) d γ ψ = (cid:90) t h ( ψ ) ψ γ − dψ (7)If we suppose γ = 1, then we have I γ ( h )( t ) = I γ =1 ( t γ − h )( t ) which represents the classical improperRiemann integral of a function h ( t ). Given a continuous function, h , on (0 , ∞ ) and for γ ∈ (0 , M γ ( h )( t ) [ I γ ( h )( t )] = h ( t ). Lemma 1. [ AJ ] Given a function h : ( c, d ) → (cid:60) as a differentiable function and γ ∈ (0 , . Then for all c > , we have the following: I cγ M cγ ( h )( t ) = h ( t ) − h ( c ) . In addition, the following theorem [
KHH ; AKH ] shows that M γ satisfies all the standard properties ofbasic limit-based derivative as follows: Theorem 2.
For γ ∈ (0 , , given two functions say: h and v to be assumed γ -differentiable at a point t ,then the following is obtained:(1) M γ ( ch + ev ) = cM γ ( h ) + eM γ ( v ) , for all c, e ∈ (cid:60) .(2) M γ ( t r ) = rt r − γ , for all r ∈ (cid:60) .(3) M γ ( hv ) = hM γ ( v ) + vM γ ( h ) .(4) M γ ( hv ) = vM γ ( h ) − hM γ ( v ) v .(5) M γ ( λ ) = 0 , for all constant functions h ( t ) = λ .(6) If h is assumed to be a differentiable function, then M γ ( h )( t ) = t − γ dhdt . The conformable partial derivative of a real valued function with several variables is defined in [
ATN ; YzzY ] as follows: 3 efinition 5.
Let h be a real-valued function with n variables and c = ( c , c , ..., c n ) ∈ (cid:60) n be a point whose i th component is positive. Then, we have:lim ξ → h ( c , c , ..., c i + ξc − γi , ..., c n ) − h ( c , c , ..., c n ) ξ . If the limit exists, the i it conformable partial derivative of h of the order γ ∈ (0 ,
1] at c is denoted by ∂ γ ∂x γi h ( c ). Remark.
Let γ ∈ (0 , ], Λ ∈ Z + and h be a real-valued function with n variables defined on an open set D , such that for all ( x , x , ..., x n ) ∈ D , each x i >
0. The function, h , is said to be C Λ γ ( D, (cid:60) ) if all itsconformable partial derivatives of order less than or equal to Λ exist and are continuous on D , [ MarT ].For more information about other related fractional derivatives’ definitions and their physical and geomet-rical interpretations, we refer to [ mkaabar ].The main goal of this paper is to obtain approximate-analyticalsolutions for MNLSE in (1) using double Laplace transform method in the sense of Caputo and conformablederivatives. From definition 3, Let us first formulate the MNLSE in (1) in the sense of CD as follows: i (cid:18) D γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:19) + ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + ω D γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + | Ψ | Ψ = 0; iD γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + ω iD δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + ω D γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + | Ψ | Ψ = 0;where i = √−
1, 0 < γ, δ ≤ t, x >
0. (8)From definition 1, MNLSE in (1) can be similarly formulated in the sense of CpFD as follows: i (cid:18) ∂ γ Ψ( x, t ) ∂x γ + ω ∂ δ Ψ( x, t ) ∂t δ (cid:19) + ω ∂ δ Ψ( x, t ) ∂t δ + ω ∂ γ Ψ( x, t ) ∂x γ + | Ψ | Ψ = 0; i ∂ γ Ψ( x, t ) ∂x γ + ω i ∂ δ Ψ( x, t ) ∂t δ + ω ∂ δ Ψ( x, t ) ∂t δ + ω ∂ γ Ψ( x, t ) ∂x γ + | Ψ | Ψ = 0;where i = √−
1, 0 < γ, δ ≤ t, x >
0. (9)
Many differential equations can be easily solved by applying the method of Laplace transform (DT) for asingle-variable function. New generalized forms of the classical Laplace transform methods such as doubleLaplace transform and multiple Laplace transform have been first introduced in [
EHLAPLACE ] to solvepartial differential equations (PDEs). Recently, double Laplace transform (DLTr) has become an interestingtopic of research for many mathematicians and researchers [
Adam1 ; Adam2 ; Adam3 ] because not manyresearch studies have been done on this topic [
LDDL1 ] and the need to find an efficient method for solvingPDEs. Applying the DLTr in the sense of fractional derivatives has been rarely discussed and it is consideredas an open problem [
LABALN ]. DLTr has been successfully introduced in solving some fractional differentialequations (FDEs) such as the fractional heat equation and the fractional telegraph equation via the definitionof CpFD [
RdGw ; LABALN ]. According to our knowledge, the generalized DLTr method has never beenapplied before for solving the MNLSE in (1) in the senses of CpFD and CD. Therefore, the results in thiswork are new and worthy.The first author has defined in [ mkaabar ] the conformable double Laplace transform (CmDLTr) asfollows: 4 efinition 6.
Given a function, Ψ (cid:18) x γ γ , t δ δ (cid:19) : [0 , ∞ ) → (cid:60) such that for all x, t >
0, the CmDLTr of order γ, δ ∈ (0 ,
1] of Ψ (cid:18) x γ γ , t δ δ (cid:19) , denoted by (cid:96) xtγδ (cid:20) Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) , starting from 0 can be expressed as follows: (cid:96) xtγδ (cid:20) Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) = (cid:96) xγ (cid:96) tδ (cid:20) Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) = ˜Ψ xtγδ ( s , s )= (cid:90) ∞ (cid:90) ∞ e − ( s xγγ + s tδδ ) Ψ (cid:18) x γ γ , t δ δ (cid:19) x γ − t δ − dx dt. (10)where s , s ∈ C . If the integral in the above definition exists, then this definition holds true.To define the double Laplace transform in the sense of Caputo partial fractional derivatives, let’s assumethat ˜Ψ xt ( s , s ) = (cid:82) ∞ (cid:82) ∞ e − ( s x + s t ) Ψ( x, t ) dx dt. From [
RdGw ], theorems 3.1 and 3.3 in [
LABALN ], andtheorem 2 in [
Eurp ], the Caputo double Laplace transform can be defined as follows:
Definition 7.
Given a function, Ψ( x, t ) : [0 , ∞ ) → (cid:60) such that for all x, t >
0, the double Laplace transformof the Caputo partial fractional derivatives (CpDLTr) of Ψ( x, t ) of orders ξ and γ where ξ ∈ ( j − , j ] and γ ∈ ( b − , b ] such that ξ, γ > j, b ∈ N , denoted by ∂ ξ ∂x ξ Ψ( x, t ) and ∂ γ ∂t γ Ψ( x, t ), respectively can beexpressed as: (cid:96) xtξ (cid:20) ∂ ξ ∂x ξ Ψ( x, t ) (cid:21) = s ξ ˜Ψ xtξ ( s , s ) − j − (cid:88) i =0 s ξ − − i (cid:96) t (cid:20) ∂ i Ψ(0 , t ) ∂x i (cid:21) . (11) (cid:96) xtγ (cid:20) ∂ γ ∂t γ Ψ( x, t ) (cid:21) = s γ ˜Ψ xtγ ( s , s ) − b − (cid:88) a =0 s γ − − a (cid:96) x (cid:20) ∂ a Ψ( x, ∂t a (cid:21) . (12)The double Laplace transform in the sense of conformable partial fractional derivatives can be similarlydefined [ mkaabar ] as follows: Definition 8.
Given a function, Ψ (cid:18) x γ γ , t δ δ (cid:19) : [0 , ∞ ) → (cid:60) such that for all x, t >
0, the double Laplacetransform of the conformable partial fractional derivatives (CmDLTr) of Ψ (cid:18) x γ γ , t δ δ (cid:19) of orders γ and δ where γ, δ ∈ (0 , ∂ γ ∂x γ Ψ (cid:18) x γ γ , t δ δ (cid:19) and ∂ δ ∂t δ Ψ (cid:18) x γ γ , t δ δ (cid:19) , respectively can be written as: (cid:96) xtγ (cid:20) ∂ γ ∂x γ Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) = s γ ˜Ψ xtγ ( s , s ) − j − (cid:88) i =0 s γ − − i (cid:96) t (cid:20) ∂ i Ψ(0 , t ) ∂x i (cid:21) . (13) (cid:96) xtδ (cid:20) ∂ δ ∂t δ Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) = s δ ˜Ψ xtδ ( s , s ) − b − (cid:88) a =0 s δ − − a (cid:96) x (cid:20) ∂ a Ψ( x, ∂t a (cid:21) . (14)The first author has proved the existence and uniqueness of CmDLTr in [ mkaabar ], while the existenceand uniqueness of CpDLTr have been discussed in [ LABALN ]. It is obvious that the formulas of the doubleLaplace transform in definition 7 and definition 8 are the same when j, b = 1. Therefore, the general definitionof CpDLTr coincides with the general defintion of CmDLTr when j, b = 1. The properties of CmDLTr andCpDLTr have been discussed in [
Ozzz ] and [
Adam4 ], respectively. For γ, δ ∈ (0 , LDDL1 ; RdGw ] for both conformable andCaputo fractional derivatives, denoted by ( (cid:96) xtγδ ) − [ ˜Ψ xtγδ ( s , s )], as follows: Definition 9.
Given an analytic function: ˜Ψ xtγδ ( s , s ), for all s , s ∈ C and for γ, δ ∈ (0 ,
1] such that Re { s ≥ η } and Re { s ≥ σ } , where η, σ ∈ (cid:60) , then, the inverse fractional double Laplace transform (IFDLT)can be expressed [ mkaabar ] as follows:( (cid:96) xtγδ ) − [ ˜Ψ xtγδ ( s , s )] = ( (cid:96) xγ ) − ( (cid:96) tδ ) − [ ˜Ψ xtγ ( s , s )]= − π (cid:90) (cid:37) + i ∞ (cid:37) − i ∞ (cid:90) ς + i ∞ ς − i ∞ e s x e s t ˜Ψ xtγδ ( s , s ) ds ds (15)5o solve the MNLSE in (1) in the senses of CpFD and CD (see equations (9) and (8)) by the methodsof CpDLTr and CmDLTr. respectively, let us first re-write both Equation (9) and Equation (8) as follows: ∂ γ Ψ( x, t ) ∂x γ = − ω ω ∂ δ Ψ( x, t ) ∂t δ − iω ∂ γ Ψ( x, t ) ∂x γ − ω ω i ∂ δ Ψ( x, t ) ∂t δ − ω | Ψ | Ψ . subject to the following initial and boundary conditions:Ψ( x,
0) = a ( x ) and ∂ Ψ( x, ∂t = a ( x ) . Ψ(0 , t ) = b ( t ) and ∂ Ψ(0 , t ) ∂x = b ( t ) . where i = √−
1, 0 < γ, δ ≤ t, x > x, t ∈ (cid:60) + , and a , a , b , b ∈ C ( (cid:60) + , (cid:60) + ). (16) D γx Ψ (cid:18) x γ γ , t δ δ (cid:19) = − ω ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) − iω D γx Ψ (cid:18) x γ γ , t δ δ (cid:19) − ω ω iD δt Ψ (cid:18) x γ γ , t δ δ (cid:19) − ω | Ψ | Ψ . subject to the following initial and boundary conditions:Ψ (cid:18) x γ γ , (cid:19) = n (cid:18) x γ γ (cid:19) and D t Ψ (cid:18) x γ γ , (cid:19) = n (cid:18) x γ γ (cid:19) .Ψ (cid:18) , t δ δ (cid:19) = m (cid:18) t δ δ (cid:19) and D x Ψ (cid:18) , t δ δ (cid:19) = m (cid:18) t δ δ (cid:19) .where i = √−
1, 0 < γ, δ ≤ t, x > x, t ∈ (cid:60) + , and n , n , m , m ∈ C ( (cid:60) + , (cid:60) + ). (17)By applying the single Laplace transform to initial and boundary conditions in (16) and (17), respectively,we obtain the following: (cid:96) [Ψ( x, (cid:96) [ a ( x )] = ˜ a ( s ); (cid:96) (cid:20) ∂ Ψ( x, ∂t (cid:21) = ˜ a ( s ) .(cid:96) [Ψ(0 , t )] = (cid:96) [ b ( t )] = ˜ b ( s ); (cid:96) (cid:20) ∂ Ψ(0 , t ) ∂x (cid:21) = ˜ b ( s ) . (18) (cid:96) [Ψ( x, (cid:96) [ n ( x )] = ˜ n ( s ); (cid:96) (cid:20) D t Ψ (cid:18) x γ γ , (cid:19)(cid:21) = (cid:96) (cid:20) n (cid:18) x γ γ (cid:19)(cid:21) = ˜ n ( s ) .(cid:96) [Ψ(0 , t )] = (cid:96) [ m ( t )] = ˜ m ( s ); (cid:96) (cid:20) D x Ψ (cid:18) , t δ δ (cid:19)(cid:21) = (cid:96) (cid:20) m (cid:18) t δ δ (cid:19)(cid:21) = ˜ m ( s ) . (19)Let us now apply the CpDLTr (definition 7) to both left-hand and right-hand sides of Equation (16), weobtain: (cid:96) x (cid:96) t (cid:20) ∂ γ Ψ( x, t ) ∂x γ (cid:21) = ˜Ψ( s , s ) = s γ − s γ (cid:96) x (cid:96) t [ b ( t )] + s sγ − s γ (cid:96) x (cid:96) t [ b ( t )] − s γ (cid:20) (cid:96) x (cid:96) t (cid:20) ω ω ∂ δ Ψ( x, t ) ∂t δ + iω ∂ γ Ψ( x, t ) ∂x γ + i ω ω ∂ δ Ψ( x, t ) ∂t δ (cid:21) + (cid:96) x (cid:96) t (cid:20) ω | Ψ | Ψ (cid:21)(cid:21) . By simplifying the above, we obtain: (cid:96) x (cid:96) t (cid:20) ∂ γ Ψ( x, t ) ∂x γ (cid:21) = ˜Ψ( s , s ) = 1 s (cid:96) x (cid:96) t [ b ( t )] + 1 s (cid:96) x (cid:96) t [ b ( t )] − s γ (cid:20) (cid:96) x (cid:96) t (cid:20) ω ω ∂ δ Ψ( x, t ) ∂t δ + iω ∂ γ Ψ( x, t ) ∂x γ + i ω ω ∂ δ Ψ( x, t ) ∂t δ (cid:21) + (cid:96) x (cid:96) t (cid:20) ω (cid:2) Ψ Ψ ∗ (cid:3)(cid:21)(cid:21) . where | Ψ | Ψ = Ψ Ψ ∗ such that Ψ ∗ is the conjugate of Ψ. (20)6rom the Adomian decomposition method (ADcM) (see [ Kunm ; Pal ] for more information aboutADcM), Equation (20) is written according to the following standard operator form for nonlinear partialdifferential equations (NPDEs): L Ψ( x, t ) + R Ψ( x, t ) + N Ψ( x, t ) = S ( x, t ) where N represents the nonlineardifferential operator, L represents the 2nd-order partial differential operator, R represents the remaininglinear operator, and S ( x, t ) represents a source term. This method was first introduced by G. Adomian in1980s where ADcM is well-known for obtaining a series solution whose each term is obtained recursively[ Pal ]. Therefore, by applying the method of CpDLTr coupled with ADcM, the decomposition infinite seriescan be expressed for both linear and nonlinear terms in Equation (2), respectively, as follows:Ψ( x, t ) = ∞ (cid:88) i =0 Ψ i ( x, t ) N (Ψ( x, t )) = ∞ (cid:88) i =0 φ i (Ψ( x, t )) . (21)where the above nonlinear term, denoted by N (Ψ( x, t )), is represented by infiniteseries of the Adomian polynomials, donated by φ i , which can be expressed [ Pal ]as follows: φ i = 1 i ! d i d Ω i N ∞ (cid:88) j =0 Ω j φ j Ω=0 , i = 0 , , , , ... so, we can write some of those terms as follows: φ = N (Ψ ); φ = Ψ N (cid:48) (Ψ ); φ = Ψ N (cid:48) (Ψ ) + 12! Ψ N (cid:48)(cid:48) (Ψ ). (22)By applying the standard NPDEs operator form and (21) to Equation (20), we obtain the following: (cid:96) x (cid:96) t (cid:34) ∞ (cid:88) i =0 Ψ i ( x, t ) (cid:35) = 1 s (cid:96) x (cid:96) t [ b ( t )] + 1 s (cid:96) x (cid:96) t [ b ( t )] − s γ (cid:34) (cid:96) x (cid:96) t (cid:34) R (cid:34) ∞ (cid:88) i =0 Ψ i ( x, t ) (cid:35)(cid:35) + (cid:96) x (cid:96) t (cid:34) ∞ (cid:88) i =0 φ i (cid:35)(cid:35) . where R [Ψ( x, t )] = ω ω ∂ δ Ψ( x, t ) ∂t δ + iω ∂ γ Ψ( x, t ) ∂x γ + i ω ω ∂ δ Ψ( x, t ) ∂t δ and φ i [Ψ( x, t )] = 1 ω Ψ Ψ ∗ . (23)Let us now write some of the Adomian polynomials, φ (cid:48) i s using the formula in (22) as follows: φ = 1 ω Ψ Ψ ∗ , φ = 2 ω Ψ Ψ Ψ ∗ + 1 ω Ψ Ψ ∗ , φ = 2 ω Ψ Ψ Ψ ∗ + 1 ω Ψ Ψ ∗ + 2 ω Ψ Ψ Ψ ∗ + 1 ω Ψ Ψ ∗ .By applying the inverse double Laplace transform to the left-hand and right-hand sides of Equation (23),7e obtain the following general solution to Equation (16) recursively:Ψ ( x, t ) = b ( t ) + xb ( t ) , Ψ ( x, t ) = − ( (cid:96) x ) − ( (cid:96) t ) − (cid:34) s γ (cid:20) (cid:96) x (cid:96) t (cid:20) ω ω ∂ δ Ψ ( x, t ) ∂t δ + iω ∂ γ Ψ ( x, t ) ∂x γ + i ω ω ∂ δ Ψ ( x, t ) ∂t δ (cid:21) + (cid:96) x (cid:96) t [ φ (Ψ( x, t ))] (cid:21)(cid:35) , ...Ψ i +1 ( x, t ) = − ( (cid:96) x ) − ( (cid:96) t ) − (cid:34) s γ (cid:96) x (cid:96) t [ R [Ψ i ( x, t )]] + (cid:96) x (cid:96) t [ φ i (Ψ( x, t ))] (cid:35) , for i ≥ . (24)Similarly, we can apply the CmDLTr (definition 8) to both sides of Equation (17), we have: (cid:96) x (cid:96) t (cid:20) D γx Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) = ˜Ψ( s , s ) = s γ − s γ (cid:96) x (cid:96) t [ m ( t )] + s sγ − s γ (cid:96) x (cid:96) t (cid:20) m (cid:18) t δ δ (cid:19)(cid:21) − s γ (cid:20) (cid:96) x (cid:96) t (cid:20) ω ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + iω D γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + i ω ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) + (cid:96) x (cid:96) t (cid:20) ω | Ψ | Ψ (cid:21)(cid:21) . After simplifications, we have: (cid:96) x (cid:96) t (cid:20) D γx Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) = ˜Ψ( s , s ) = 1 s (cid:96) x (cid:96) t [ m ( t )] + 1 s (cid:96) x (cid:96) t (cid:20) m (cid:18) t δ δ (cid:19)(cid:21) − s γ (cid:20) (cid:96) x (cid:96) t (cid:20) ω ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + iω D γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + i ω ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) + (cid:96) x (cid:96) t (cid:20) ω (cid:2) Ψ Ψ ∗ (cid:3)(cid:21)(cid:21) . where | Ψ | Ψ = Ψ Ψ ∗ such that Ψ ∗ is the conjugate of Ψ. (25)Let us now apply the standard NPDEs operator form and (21) to Equation (25), we have: (cid:96) x (cid:96) t (cid:34) ∞ (cid:88) i =0 Ψ i ( x, t ) (cid:35) = 1 s (cid:96) x (cid:96) t [ m ( t )] + 1 s (cid:96) x (cid:96) t (cid:20) m (cid:18) t δ δ (cid:19)(cid:21) − s γ (cid:34) (cid:96) x (cid:96) t (cid:34) R (cid:34) ∞ (cid:88) i =0 Ψ i ( x, t ) (cid:35)(cid:35) + (cid:96) x (cid:96) t (cid:34) ∞ (cid:88) i =0 φ i (cid:35)(cid:35) . where R [Ψ( x, t )] = (cid:20) ω ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + iω D γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + i ω ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) and φ i [Ψ( x, t )] = 1 ω Ψ Ψ ∗ . (26)We apply the inverse double Laplace transform to both sides of Equation (26) to obtain the general solutionto Equation (17) recursively as follows:Ψ ( x, t ) = m (cid:18) t δ δ (cid:19) + x γ γ m (cid:18) t δ δ (cid:19) , Ψ ( x, t ) = − ( (cid:96) x ) − ( (cid:96) t ) − × (cid:34) s γ (cid:20) (cid:96) x (cid:96) t (cid:20) ω ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + iω D γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + i ω ω D δt Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) + (cid:96) x (cid:96) t [ φ (Ψ( x, t ))] (cid:21)(cid:35) , ...Ψ i +1 ( x, t ) = − ( (cid:96) x ) − ( (cid:96) t ) − (cid:34) s γ (cid:96) x (cid:96) t [ R [Ψ i ( x, t )]] + (cid:96) x (cid:96) t [ φ i (Ψ( x, t ))] (cid:35) , for i ≥ . (27)8 umerical Experiment 1: By applying definitions and properties of Caputo fractional derivative and double Laplace transform, thefollowing numerical experiment will solve Equation (16) analytically: Let ω = ω = ω = 1, and b ( t ) = e it ; b ( t ) = 0; a ( x ) = a ( x ) = 0 in (16), we have: ∂ γ Ψ( x, t ) ∂x γ = − ∂ δ Ψ( x, t ) ∂t δ − i ∂ γ Ψ( x, t ) ∂x γ − i ∂ δ Ψ( x, t ) ∂t δ − | Ψ | Ψ . subject to the following initial and boundary conditions:Ψ( x,
0) = 0 and ∂ Ψ( x, ∂t = 0 . Ψ(0 , t ) = e it and ∂ Ψ(0 , t ) ∂x = 0 . where i = √−
1, 0 < γ, δ ≤ t, x > x, t ∈ (cid:60) + . (28)To solve Equation (28), we use our result in (24) as follows:Ψ ( x, t ) = e it , Ψ ( x, t ) = − ( (cid:96) x ) − ( (cid:96) t ) − (cid:34) s γ (cid:20) (cid:96) x (cid:96) t (cid:20) ∂ δ Ψ ( x, t ) ∂t δ + i ∂ γ Ψ ( x, t ) ∂x γ + i ∂ δ Ψ ( x, t ) ∂t δ (cid:21) + (cid:96) x (cid:96) t [ φ (Ψ( x, t ))] (cid:21)(cid:35) = − ( (cid:96) x ) − ( (cid:96) t ) − (cid:34) s γ (cid:20) (cid:96) x (cid:96) t (cid:20) ∂ δ Ψ ( x, t ) ∂t δ + i ∂ γ Ψ ( x, t ) ∂x γ + i ∂ δ Ψ ( x, t ) ∂t δ (cid:21) + (cid:96) x (cid:96) t [Ψ Ψ ∗ ] (cid:21)(cid:35) = − x γ Γ(2 γ + 1) (cid:2)(cid:2) it − δ E , − δ ( it ) − t − δ E , − δ ( it ) (cid:3) + e it (cid:3) = − x γ Γ(2 γ + 1) (cid:2) [ iE ( t, − δ, i ) − E ( t, − δ, i )] + e it (cid:3) , Ψ ( x, t ) = − ( (cid:96) x ) − ( (cid:96) t ) − (cid:34) s γ (cid:20) (cid:96) x (cid:96) t (cid:20) ∂ δ Ψ ( x, t ) ∂t δ + i ∂ γ Ψ ( x, t ) ∂x γ + i ∂ δ Ψ ( x, t ) ∂t δ (cid:21) + (cid:96) x (cid:96) t [ φ (Ψ( x, t ))] (cid:21)(cid:35) = − ( (cid:96) x ) − ( (cid:96) t ) − (cid:34) s γ (cid:20) (cid:96) x (cid:96) t (cid:20) ∂ δ Ψ ( x, t ) ∂t δ + i ∂ γ Ψ ( x, t ) ∂x γ + i ∂ δ Ψ ( x, t ) ∂t δ (cid:21) + (cid:96) x (cid:96) t [2Ψ Ψ Ψ ∗ + Ψ Ψ ∗ ] (cid:21)(cid:35) = − x γ Γ(2 γ + 1) (cid:20) − x γ Γ(2 γ + 1) ∂ δ ∂t δ (cid:0) [ iE ( t, − δ, i ) − E ( t, − δ, i )] + e it (cid:1)(cid:21) − i x γ Γ(2 γ + 1) (cid:20)(cid:2)(cid:0) [ iE ( t, − δ, i ) − E ( t, − δ, i )] + e it (cid:1)(cid:3) ∂ γ ∂x γ (cid:18) − x γ Γ(2 γ + 1) (cid:19)(cid:21) − i x γ Γ(2 γ + 1) (cid:20) − x γ Γ(2 γ + 1) ∂ δ ∂t δ (cid:0) [ iE ( t, − δ, i ) − E ( t, − δ, i )] + e it (cid:1)(cid:21) − x γ Γ(2 γ + 1) (cid:20) (2 e it ) (cid:18) − x γ Γ(2 γ + 1) (cid:19) (cid:0) [ iE ( t, − δ, i ) − E ( t, − δ, i )] + e it (cid:1) ( e − it ) (cid:21) − x γ Γ(2 γ + 1) (cid:20) ( e it ) (cid:18) − x γ Γ(2 γ + 1) (cid:19) (cid:0) [ − iE ( t, − δ, i ) − E ( t, − δ, i )] + e − it (cid:1) ( e − it ) (cid:21) x γ Γ(4 γ + 1) [ iE ( t, − δ, i ) − E ( t, − δ, i ) + iE ( t, − δ, i )]+ x γ Γ(3 γ + 1) (cid:2) E ( t, − δ, i ) − iE ( t, − δ, i ) + ie it (cid:3) + x γ Γ(4 γ + 1) [ − E ( t, − δ, i ) − iE ( t, − δ, i ) − E ( t, − δ, i )]+ x γ Γ(4 γ + 1) (cid:2) − iE ( t, − δ, i ) − E ( t, − δ, i ) + e it (cid:3) + x γ Γ(4 γ + 1) (cid:2) − ie it E ( t, − δ, i ) − e it E ( t, − δ, i ) + e it (cid:3) = x γ Γ(3 γ + 1) (cid:2) E ( t, − δ, i ) − iE ( t, − δ, i ) + ie it (cid:3) + x γ Γ(4 γ + 1) { iE ( t, − δ, i ) − E ( t, − δ, i ) + iE ( t, − δ, i ) − E ( t, − δ, i ) − iE ( t, − δ, i ) − E ( t, − δ, i ) − iE ( t, − δ, i ) − E ( t, − δ, i ) + e it − ie it E ( t, − δ, i ) − e it E ( t, − δ, i ) + e it } . ...and so on.By using all above obtained results, the general approximate-analytical solution to Equation (28) can bewritten as follows:Ψ( x, t ) = e it − x γ Γ(2 γ + 1) (cid:2) [ iE ( t, − δ, i ) − E ( t, − δ, i )] + e it (cid:3) + x γ Γ(3 γ + 1) (cid:2) E ( t, − δ, i ) − iE ( t, − δ, i ) + ie it (cid:3) + x γ Γ(4 γ + 1) { iE ( t, − δ, i ) − E ( t, − δ, i ) + iE ( t, − δ, i ) − E ( t, − δ, i ) − iE ( t, − δ, i ) − E ( t, − δ, i ) − iE ( t, − δ, i ) − E ( t, − δ, i ) + e it − ie it E ( t, − δ, i ) − e it E ( t, − δ, i ) + e it } . + . . . (29)Hence, the approximate-analytical solution for the MNLSE in (1) in the sense of Caputo fractional derivativehas been easily obtained via the double Laplace transform coupled with the Adomian decomposition method. Numerical Experiment 2:
By applying definitions and properties of conformable derivative in [
SiLa ; AJ ] and double Laplace transform,the following numerical experiment will solve Equation (17) analytically: Let ω = ω = ω = 1, and m ( t ) = e i tδδ ; m ( t ) = 0; n ( x ) = n ( x ) = 0 in (17), we have: D γx Ψ (cid:18) x γ γ , t δ δ (cid:19) = − D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) − iD γx Ψ (cid:18) x γ γ , t δ δ (cid:19) − iD δt Ψ (cid:18) x γ γ , t δ δ (cid:19) − | Ψ | Ψ . subject to the following initial and boundary conditions:Ψ (cid:18) x γ γ , (cid:19) = 0 and D t Ψ (cid:18) x γ γ , (cid:19) = 0.Ψ (cid:18) , t δ δ (cid:19) = e i tδδ and D x Ψ (cid:18) , t δ δ (cid:19) = 0.where i = √−
1, 0 < γ, δ ≤ t, x > x, t ∈ (cid:60) + . (30)10o solve Equation (30), we use our result in (27) as follows:Ψ ( x, t ) = e i tδδ , Ψ ( x, t ) = − ( (cid:96) x ) − ( (cid:96) t ) − × (cid:34) s γ (cid:20) (cid:96) x (cid:96) t (cid:20) D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + iD γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + iD δt Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) + (cid:96) x (cid:96) t [ φ (Ψ( x, t ))] (cid:21)(cid:35) = − ( (cid:96) x ) − ( (cid:96) t ) − (cid:34) s γ (cid:20) (cid:96) x (cid:96) t (cid:20) D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + iD γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + iD δt Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) + (cid:96) x (cid:96) t [Ψ Ψ ∗ ] (cid:21)(cid:35) = − x γ − γ γ − Γ(2 γ ) (cid:104)(cid:104) − e i tδδ − e i tδδ (cid:105) + e i tδδ (cid:105) = − x γ − γ γ − Γ(2 γ ) (cid:104) − e i tδδ (cid:105) Ψ ( x, t ) = − ( (cid:96) x ) − ( (cid:96) t ) − × (cid:34) s γ (cid:20) (cid:96) x (cid:96) t (cid:20) D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + iD γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + iD δt Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) + (cid:96) x (cid:96) t [ φ (Ψ( x, t ))] (cid:21)(cid:35) = − ( (cid:96) x ) − ( (cid:96) t ) − × (cid:34) s γ (cid:20) (cid:96) x (cid:96) t (cid:20) D δt Ψ (cid:18) x γ γ , t δ δ (cid:19) + iD γx Ψ (cid:18) x γ γ , t δ δ (cid:19) + iD δt Ψ (cid:18) x γ γ , t δ δ (cid:19)(cid:21) + (cid:96) x (cid:96) t [2Ψ Ψ Ψ ∗ + Ψ Ψ ∗ ] (cid:21)(cid:35) = − x γ − γ γ − Γ(2 γ ) { (cid:18) − x γ − γ γ − Γ(2 γ ) (cid:19) e i tδδ + (2 γ − x γ − (cid:32) i e i tδδ γ γ − Γ(2 γ ) (cid:33) − (cid:18) x γ − γ γ − Γ(2 γ ) (cid:19) e i tδδ − (cid:18) x γ − γ γ − Γ(2 γ ) (cid:19) e − i tδδ + (cid:18) x γ − γ γ − Γ(2 γ ) (cid:19) e i tδδ } = x γ − γ γ − Γ(4 γ ) e i tδδ − i (2 γ − x γ − γ γ − Γ(4 γ ) e i tδδ + x γ − γ γ − Γ(4 γ ) e i tδδ + 2 x γ − γ γ − Γ(4 γ ) e − i tδδ − x γ − γ γ − Γ(4 γ ) e i tδδ = − i (2 γ − x γ − γ γ − Γ(4 γ ) e i tδδ + x γ − γ γ − Γ(4 γ ) (cid:104) e i tδδ + e i tδδ + 2 e − i tδδ − e i tδδ (cid:105) = − i (2 γ − x γ − γ γ − Γ(4 γ ) e i tδδ + x γ − γ γ − Γ(4 γ ) (cid:104) e i tδδ + 2 e − i tδδ (cid:105) . ...and so on.By using the above obtained results, the general approximate-analytical solution to Equation (30) can bewritten as follows:Ψ( x, t ) = e i tδδ − x γ − γ γ − Γ(2 γ ) (cid:104) − e i tδδ (cid:105) − i (2 γ − x γ − γ γ − Γ(4 γ ) e i tδδ + x γ − γ γ − Γ(4 γ ) (cid:104) e i tδδ + 2 e − i tδδ (cid:105) + . . . (31)Hence, the approximate-analytical solution for the MNLSE in (1) in the sense of conformable derivativehas also been easily obtained via the double Laplace transform coupled with the Adomian decompositionmethod. In this section, the obtained approximate solutions in both (29) and (31) have been graphically comparedfor various values of γ and δ (see figures 1 to 10 ) where each graph shows both real part and imaginary11 igure 1:
3D Plot of the real part (a) and imaginary part (b) of the Approximate Analytical Solution in(29) for γ = δ = 0 . Figure 2:
3D Plot of the real part (a) and imaginary part (b) of the Approximate Analytical Solution in(29) for γ = δ = 0 . igure 3:
3D Plot of the real part (a) and imaginary part (b) of the Approximate Analytical Solution in(29) for γ = 0 . δ = 0 . Figure 4:
3D Plot of the real part (a) and imaginary part (b) of the Approximate Analytical Solution in(29) for γ = 0 . δ = 0 . γ = δ = 0 . γ = δ = 0 . γ = δ = 1 with the exact solution from13 igure 5:
3D Plot of the real part (a) and imaginary part (b) of the Approximate Analytical Solution in(29) for γ = δ = 1 Figure 6:
3D Plot of the real part (a) and imaginary part (b) of the Approximate Analytical Solution in(31) for γ = δ = 0 . CHinD ]. According to table 1, at x = t = 0 . x = t = 0 . x = t = 0 .
5, the absolute error14 igure 7:
3D Plot of the real part (a) and imaginary part (b) of the Approximate Analytical Solution in(31) for γ = δ = 0 . Figure 8:
3D Plot of the real part (a) and imaginary part (b) of the Approximate Analytical Solution in(31) for γ = 0 . δ = 0 . igure 9:
3D Plot of the real part (a) and imaginary part (b) of the Approximate Analytical Solution in(31) for γ = 0 . δ = 0 . Figure 10:
3D Plot of the real part (a) and imaginary part (b) of the Approximate Analytical Solution in(31) for γ = δ = 1and the approximate solution from CmDLTr. At x = t = 0 .
7, for γ = δ = 0 .
25; 0 .
75 the absolute error value16 able 1:
Comparison of the absolute approximate solutions from CpDLTr, CmDLTr, and the exact solutionof the space-time fractional nonlinear Schr¨odinger equation (Example 9) in [
CHinD ]. Note:
Error cp = | Exact − CpDLT r | and Error cm = | Exact − CmDLT r | .( x, t ) Exact γ δ CpDLTr CmDLTr
Error cp Error cm (0.1,0.1) 0.108060 0.25 0.25 0.034850 0.627680 0.073309 0.5196200.75 0.75 0.052744 0.972022 0.055315 0.8639621 1 0.060835 0.058760 0.047224 0.049299(0.3,0.3) 0.324180 0.25 0.25 0.061912 0.983617 0.262268 0.6594360.75 0.75 0.142055 0.857462 0.182126 0.5332811 1 0.223340 0.199797 0.100841 0.124384(0.5,0.5) 0.540300 0.25 0.25 0.054601 0.975461 0.485698 0.4351610.75 0.75 0.198301 0.701852 0.341999 0.1615521 1 0.440288 0.361382 0.100012 0.178918(0.7,0.7) 0.756420 0.25 0.25 0.021153 0.869221 0.735269 0.1127980.75 0.75 0.211584 0.523044 0.544839 0.2333791 1 0.711680 0.530550 0.044743 0.225873(0.9,0.9) 0.972540 0.25 0.25 0.034267 0.728667 0.938276 0.2438770.75 0.75 0.173940 0.332328 0.798597 0.6402161 1 1.037520 0.694332 0.064976 0.278212from exact and the approximate solution from CmDLTr is less than the absolute error value from exact andthe approximate solution from CpDLTr, while at x = t = 0 . γ = δ = 1, the the approximate solutionfrom CpDLTr converges to the exact solution better than the one from CmDLTr. Similarly, at x = t = 0 . γ = δ = 1, the approximate solution from CpDLTr converges to the exact solution better than the onefrom CmDLTr. Therefore, the obtained approximate solution in the sense of Caputo fractional derivative ismuch better than the obtained solution in the sense of conformable derivatives. On one hand, the Caputofractional derivative is a nonlocal fractional operator which provides a good interpretation to the physicalbehavior of systems, while the conformable derivative is a type of local fractional derivative which is basicallya generalized form of usual limit-based derivative which lacks some of the important properties to be classifiedas a fractional derivative. As a result, solving systems of nonlinear partial differential equations in the senseof Caputo fractional derivatives is highly recommended. However, exploring the definition of conformablederivative is also interesting because as the authors believe that any new mathematical definition deservesto be explored and investigated. Nonlinear Schr¨odinger equation has been an interesting field of research for many mathematicians andscientists due to the important applications of this equation in physics and engineering. This researchstudy provides a powerful mathematical tool to solve the nonlinear Schr¨odinger equation involving bothCaputo fractional derivative and conformable derivative. Therefore, the generalized double Laplace transformmethod can be efficiently applied in solving nonlinear fractional Schr¨odinger equation and all other nonlinearfractional partial differential equations.
Disclosure statement
The authors declare no conflict of interests.
Acknowledgments
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. 17 eferences [AJ] T. Abdeljawad,
On conformable fractional calculus , Journal of computational and Applied Mathemat-ics. (2015), 57–66.[AKH] M. Abu Hammad and R. Khalil,
Conformable fractional heat differential equation , Int. J. Pure Appl.Math. (2014), no.2, 215–221.[AG] O. P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems ,Journal ofMathematical Analysis and Applications. (2002), no. 1, 368–379.[ATT] R. Almeida, D. Tavares, and D. F. Torres,
The variable-order fractional calculus of variations , arXivpreprint:1805.00720l, Springer International Publishing, 2018.[Ekm] E. Amoupour, E. A. Toroqi, and H. S. Najafi
Numerical experiments of the Legendre polynomialby generalized differential transform method for solving the Laplace equation , Communications of theKorean Mathematical Society. (2018), no. 2, 639–650.[LABALN] A. M. O. Anwar, F. Jarad, D. Baleanu, and F. Ayaz, Fractional Caputo heat equation within thedouble Laplace transform , Rom. Journ. Phys., (2013), 15–22.[ATN] A. Atangana, D. Baleanu, and A. Alsaedi, New properties of conformable derivative , Open Math., (2015), 57–63.[LDDL1] L. Debnath, The double Laplace transforms and their properties with applications to functional,integral and partial differential equations , International Journal of Applied and Computational Mathe-matics, (2016), no. 2, 223–241.[RdGw] R. R. Dhunde and G. L. Waghmare, Double Laplace transform method for solving space and timefractional telegraph equations , International Journal of Mathematics and Mathematical Sciences, (2016), 1414595.[Adam2] H. Eltayeb and A. Kılı¸cman,
A note on solutions of wave, Laplace’s and heat equations withconvolution terms by using a double Laplace transform , Applied Mathematics Letters, (2008), no.12, 1324–1329.[Adam1] H. Eltayeb and A. Kılı¸cman, On double Sumudu transform and double Laplace transform , Malaysianjournal of mathematical sciences, (2010), no. 1, 17–30.[EHLAPLACE] T. A. Estrin and T. J. Higgins, The solution of boundary value problems by multiple Laplacetransformations , Journal of the Franklin Institute, (1951), no. 2, 153–167.[JFmWO] B. Ghanbari and J. F. G´omez-Aguilar,
New exact optical soliton solutions for nonlinearSchr¨odinger equation with second-order spatio-temporal dispersion involving M-derivative , ModernPhysics Letters B. (2019), no.20, 1950235.[YzzY] N. Y. G¨oz¨utok and U. G¨oz¨utok, Multivariable conformable fractional calculus , arXivpreprint:1701.00616v1, 2017.[Kunm] H. G¨undo˘gdu and ¨O. G¨oz¨ukızıl,
Double Laplace Decomposition Method and Exact Solutions ofHirota, Schr¨odinger and Complex mKdV Equations , Konuralp Journal of Mathematics, (2019), no. 1,7–15.[CHinD] S. H. M. Hamed, E. A. Yousif, and A. I. Arbab, Analytic and approximate solutions of the space-time fractional Schr¨odinger equations by homotopy perturbation Sumudu transform method , Abstractand Applied Analysis, (2014), 13.[TO] O.S. Iyiola and E.R. Nwaeze,
Some new results on the new conformable fractional calculus with appli-cation using D’Alambert approach , Progr. Fract. Differ. Appl., (2016), no. 2, 1–7.18mkaabar] M. Kaabar, Novel Methods for Solving the Conformable Wave Equation, Journal of New Theory,31 (2020) 56–85.[gKH] R. Khalil, M. Al Horani, and M. Abu Hammad, Geometric meaning of conformable derivative viafractional cords , Journal of Mathematics and Computer Science. (2019), no.4, 241–245.[KHH] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative ,Journal of Computational and Applied Mathematics. (2014), 65–70.[Eurp] A. Khan, T. S. Khan, M. I. Syam, and H. Khan,
Analytical solutions of time-fractional wave equationby double Laplace transform method , The European Physical Journal Plus, (2019), no. 4, 163–167.[MK] M. Klimek,
Fractional sequential mechanics—models with symmetric fractional deriva-tive ,Czechoslovak Journal of Physics. (2001), no.12, 1348–1354.[Adam3] A. Kılı¸cman and H. E. Gadain, On the applications of Laplace and Sumudu transforms , Journal ofthe Franklin Institute, (2010), no. 5, 848–862.[MLT] M. J. Lazo and D. F. Torres,
Variational calculus with conformable fractional derivatives , IEEE/CAAJournal of Automatica Sinica. (2017), no.2, 340–352.[AMDT] A. B. Malinowska and D. F. Torres, Fractional variational calculus in terms of a combined Caputoderivative ,arXiv preprint:1007.0743, 2010.[MarT] F. Mart´ınez, I. Mart´ınez, and S. Paredes,
Conformable Euler’s theorem on homogeneous functions ,Computational and Mathematical Methods, (2019), no. 5, 1–11.[Pal] R. I. Nuruddeen, L. Muhammad, A. M. Nass, and T. A. Sulaiman, A review of the integral transforms-based decomposition methods and their applications in solving nonlinear PDEs , Palestine Journal ofMathematics, (2018), no. 7, 262–280.[ZCpt] Z. Odibat, S. Momani, and A. Alawneh, Analytic study on time-fractional Schr¨odinger equations:exact solutions by GDTM , Journal of Physics: Conference Series, (2008), no. 1, 012066.[Adam4] M. Omran and A. Kili¸cman, Fractional double Laplace transform and its properties , AIP ConferenceProceedings, (2017), no. 1, 020021.[Ozzz] O. ¨Ozkan, and A. Kurt,
Conformable Double Laplace Transform For Fractional Partial DiferentialEquations Arising in Mathematical Physics , Mathematical Studies and Applications. (2018), 471–476.[SSR] S. G. Samko and B. Ross,
Integration and differentiation to a variable fractional order , IntegralTransforms and Special Functions. (1993), no.4, 277–300.[SiLa] F. Silva, D. M. Moreira, and M. A. Moret, Conformable Laplace Transform of Fractional DifferentialEquations , Axioms, (2018), no. 3, 55.[mRev23] G. S. Teodoro, J. A. T, Machado, and E. C. De Oliveira, A review of definitions of fractionalderivatives and other operators , Journal of Computational Physics. (2019), 195–208.[YmMwRm] E. Ya¸sar and E. Ya¸sar,
Optical solitons of conformable space-time fractional NLSE with Spatio-temporal dispersion , New Trends in Mathematical Sciences. (2018), no.3, 116–127.[YZ] Z. Yi, Fractional differential equations of motion in terms of combined Riemann—Liouville deriva-tives ,Chinese Physics B. (2012), no. 8, 084502.[coZ] D. Zhao and M. Luo, General conformable fractional derivative and its physical interpretation , Calcolo54