Numerical Studies for the Fractional Schrödinger Equation with the Quantum Riesz-Feller Derivative

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1 Progr. Frac. Differ. Appl., No., - () Progress in Fracional Differeniaion and Applicaions An Inernaional Jornal hp://d.doi.org/.8/pfda/ Nmerical Sdies for he Fracional Schrödinger Eqaion wih he Qanm Riesz-Feller Derivaive Nasser Hassan Sweilam and Mner Msafa Abo Hasan Deparmen of Mahemaics, Facly of Science, Cairo Universiy, Giza, Egyp Received: Feb., Revised: May, Acceped: May Pblished online: Oc. Absrac: In his paper, we presen a nmerical mehod for solving he one-dimensional space fracional Schrödinger eqaion in he case of a paricle moving in a poenial field. The fracional derivaive is defined by he qanm Riesz-Feller fracional derivaive. A novel weighed average non-sandard finie difference mehod is presened o solve he nderline problem nmerically. The sabiliy analysis of he proposed mehod is given by a recenly proposed procedre similar o he sandard John von Nemann sabiliy analysis and he rncaion error is analyzed. Several nmerical eamples are inrodced for varios choices of derivaive order α, < α, and for varios choices of skewness θ o demonsrae iliy of he proposed mehod. We demonsrae ha he proposed echniqe is more accrae han he sandard weighed average finie difference mehod. Keywords: Space fracional Schrödinger eqaion, Riesz-Feller fracional derivaive, weighed average non-sandard finie difference mehods, von Nemann sabiliy analysis. Inrodcion The famos Schrödinger eqaion is one of he fndamenal eqaions in qanm mechanics ha describes he change of he qanm behavior of some physical sysems, I was formlaed in 9, by he Asrian physicis Erwin Schrödinger. I was shown in ] ha he Feynman pah inegral over he Lévy like qanm-mechanical pahs allows o develop a fracional generalizaion of he qanm mechanics. Whereas he Feynman pah inegral over Brownian rajecories leads o he well-known Schrödinger eqaion, he pah inegrals over Lévy rajecories lead o he fracional Schrödinger eqaion (FSE) wih he qanm Riesz derivaive. Nick Laskin ] discovered he fndamenal eqaion of FSE in he form: i h Ψ(r,) = C α (m)( ) α/ Ψ(r,)+V(r,)Ψ(r,),, r R, () for he wave fncion Ψ of a qanm paricle wih he mass m ha moves in a poenial field wih he poenial V. In (), h= π h, where h is he Plank consan. C α(m) is a posiive consan which eqals m h for α = ], and( )α/ was called in (], ]) he qanm Riesz fracional derivaive of order α. In he mahemaical lierare, ( ) α/ is sally referred o as he fracional Laplacian. For α =, he qanm Riesz fracional derivaive becomes he negaive Laplace operaor and Eq. () is redced o he classical Schrödinger eqaion for a qanm paricle wih he mass m ha moves in a poenial field wih he poenial V. The non-sandard finie difference (NSFD) schemes were firsly proposed by Mickens ], boh for ordinary differenial eqaions (ODEs) and parial differenial eqaions (PDEs) wih more accracy han sandard finie difference mehod (SFDM), Recenly Sweilam e al. (], ]) sed his echniqe o solve fracional and variable order fracional differenial eqaions, also hey sed o solve Two-dimensional fracional diffsion eqaion ]. The prpose of his work is o sdy nmerically he fracional Schrödinger eqaion wih he qanm Riesz-Feller derivaive for a paricle ha moves in a poenial field sing new echniqe called weighed average non-sandard finie difference mehod (WA-NSFDM) and o illsrae he behavior of he solions of FSE wih varios vales of α and θ. Corresponding ahor nsweilam@sci.c.ed.eg

2 N. H. Sweilam, M. M. Abo Hasan: Nmerical sdies for he fracional Schrödinger... Nmerical resls are given o highligh he high accracy of he presen mehod. Recenly, in, Al Saqabi ] showed ha his model wih he qanm Riesz-Feller derivaive can be considered from he mahemaical viewpoin, b i seems o have no physical applicaions in he case θ. Several analyical and nmerical mehods (see e.g. 8], ], ] o menion only few of hem) have been proposed for he one-dimensional space-fracional and space-ime-fracional Schrödinger eqaions wih some specific poenial fields inclding zero poenial (free paricle), he δ-poenial, he infinie poenial well, he Colomb poenial, and a recanglar barrier. In 9] he ahors inrodced he implici flly discree local disconinos Galerkin mehod (IFDLDGM) for a solion of he T-FSE, while Mohebbi e al. ] sed he meshless echniqe (MT) for approimaing is solion nmerically. Moreover, Bhrawy e al. ] proposed a new Jacobi specral collocaion mehod for solving fracional Schrödinger eqaions and fracional copled Schrödinger sysem. More recenly, Bhrawy e al. ] proposed a flly specral collocaion approimaion for mli-dimensional ime fracional Schrödinger eqaions. This paper is srcred as follows: In he ne secion we give some definiions on fracional calcls and some properies of non-sandard discreizaion. Secion is devoed o discreizaion of he Cachy-ype problem for fracional Schrödinger eqaion wih he qanm Riesz-Feller derivaive in he case of a free paricle sing weighed average nonsandard finie difference mehods. In Secion sabiliy analysis and rncaing error of he proposed mehod for solving he menion model were sdied. In Secion some nmerical reamens are esablishmen wih heir resls. Conclding remarks are given in Secion. Preliminaries and Noaions This secion gives some preliminary resls which are needed in sbseqen secions of his paper.. Fracional Calcls Definiions In he las years fracional derivaives have fond nmeros applicaions in many fields of physics, mechanical engineering, biology, elecrical engineering, conrol heory and finance (], ], ], ], ]). Fracional calcls in mahemaics is a naral eension of ineger-order calcls and gives a sefl mahemaical ool for modeling many processes in nare more han classic calcls. Indeed, many definiions of he fracional inegrals and derivaives were inrodced (see e.g. ]). The ime-fracional derivaives are ofen given in he Capo, Riemann-Lioville, or Grünwald-Lenikov sense. As o he space-fracional derivaive, i is sally defined as an operaor inverse o he Riesz poenial (see e.g. ], ], ]) and is referred o as he Riesz fracional derivaive. Podlbny menioned (in 8]) ha he complee heory of fracional differenial eqaions, especially he heory of bondary vale problems for fracional differenial eqaions, can be developed only wih he se of boh lef-and righ-sided derivaives. So he spaial derivaives discssed in his paper are all Riesz-Feller poenial operaor, which inclde he wo-sided Riemann-Lioville fracional derivaives. Recenly, he Riesz-Feller spacefracional derivaive of order α and skewness θ has been shown o be relevan for anomalos diffsion models ]. In addiion, his derivaive is beer sied for a generalizaion o higher order derivaives. Anoher advanage of sing Riesz- Feller derivaive lies in he fac ha he solion of he fracional reacion-diffsion eqaion wih Riesz-Feller derivaive incldes he fndamenal solion for space-ime reacional diffsion, which iself is a generalizaion of neral fracional diffsion, space-fracional diffsion, and ime-fracional diffsion 9]. For < α < and θ min{α, α}, he qanm Riesz-Feller derivaive can be represened in he form (see e.g. ], ], ]) D α θ f()= Γ(+α) π { ( sin (α+ θ) π ) ) ( +sin (α θ) π f(+ξ) f() ξ +α f( ξ) f() ξ +α dξ dξ }. () For <α < and α and θ in is range, his formla can be rewrien as (see e.g. ], ]) where he coefficiens c ± are given by c + = c + (α,θ)= D α θ f()=(c +D α ++ c D α ) f(), () sin((α θ)π/), c = c (α,θ)= sin(απ) sin((α+ θ)π/), () sin(απ)

3 Progr. Frac. Differ. Appl., No., - () / and (D α + f)()=( d d )n (I n α + f)(), (D α f)()=( d d )n (I n α f)(), () are he wo-sided Riemann-Lioville fracional derivaives wih R and α >, n <α n, n N. In epressions () he fracional operaors I n α ± are defined as he lef- and righ-side of Weyl fracional inegrals, which given by (I+ α f)()= Γ(α) f(ξ) ( ξ) α dξ, (Iα f)()= + Γ(α) For α =, he represenaion () is no valid and has o be replaced by he formla f(ξ) dξ. () ( ξ) α D θ f()=cos(θπ/)d sin(θπ/)d] f(), () where he operaor D is relaed o he Hilber ransform as firs noed by Feller in 9 in his pioneering paper ] D = d π d + f(ξ) ξ dξ, and D refers for he firs sandard derivaive. From he above relaions one can see: - The qanm Riesz-Feller derivaive is he Riesz-Feller derivaive mliplied by -. - The Riesz-Feller fracional derivaive (in space) of order α and skewness θ can be epressed by he linear combinaion of he wo-sided Riemann-Lioville differenial operaors. - When θ =, he fracional Riesz-Feller derivaive is changed o he Riesz derivaive. - For c is any consan hen D α θ (c)=. In his paper, we consider he fracional Schrödinger eqaion wih he qanm Riesz-Feller derivaive ha describes he wave fncion Ψ of a qanm paricle ha moves in a poenial field wih he poenial V in he form: i h Ψ(,). Non-Sandard Discreizaion = C α (m)d α θψ(,)+v(,)ψ(,),, R. (8) The non-sandard finie difference (NSFD) schemes were firsly proposed by Mickens ], eiher for ordinary differenial eqaions (ODEs) or parial differenial eqaions (PDEs). A scheme is called non-sandard if a leas one of he following condiions is saisfied: - Nonlocal approimaion is sed. - Discreizaion of derivaive is no radiional and se a nonnegaive fncion i.e., when we wan o approimae dy y(+ h) y() y(+ h) y() sing Eler mehod we se insead of, where φ(h) is a d φ(h) h coninos fncion of sep size h, and he fncion φ(h) saisfies he following condiions: φ(h)=h+o(h ), <φ(h)<, h. In addiion o his replacemen, if here are nonlinear erms in he differenial eqaion, hese are replaced by non-local approimaion like for eample { y n y n+, y n+ n. Discreizaion of he Cachy-Type Problem for a Free Paricle In his secion, we presen he WA-NSFDM, o obain he discreizaion of he fracional Schrödinger eqaion wih he qanm Riesz-Feller derivaive of order α, <α < for a free paricle (V = ) in he form i h Ψ(,) = C α (m)dθ α Ψ(,), >, R (9)

4 N. H. Sweilam, M. M. Abo Hasan: Nmerical sdies for he fracional Schrödinger... Eisence and niqeness heorems of he solion of Eq. (9) sbjec o an iniial condiion and he bondary condiions Ψ(,)= f(), R, Ψ(,), as ±, were inrodced in ] and he solion was given in erms of Fo H-fncion. The problem of solving nmerically eqaion (9) lies in a properly approimaion of qanm Riesz-Feller derivaive by a WA-NSFD scheme wih a weigh facor σ,]. Le s assme ha he coordinaes of he mesh poins are n = nh, n=...,,,,,,..., m = m, m=,,,... M, where h = n n, = m m. Le s define he approimaion of he fncion Ψ(,) on he grid ( n, m ) by Ψ( n, m )= Ψ m n. Eq. (9) can be wrien in he following form: i h Ψ n m+ Ψ m n ϕ( ) = C α (m) h α + (φ(h)) (σψn+k m +( σ)ψm+ n+k )w k+ T m n, () k= where σ being he weigh facor and he coefficiens w k = w k (α,θ) have he following form ]: w k = Γ( α) ( k +) α ( λ)+( k +) α (λ ) + k α ( λ)+( k ) α (λ )+( k ) α ( λ)]c + for k, ( α ( λ)+ α (λ ) λ+ )c + +( λ)c for k=, ( α ( λ)+λ )(c + + c ) for k=, ( α ( λ)+ α (λ ) λ+ )c + +( λ)c + for k=, ( k +) α ( λ)+( k +) α (λ ) + k α ( λ)+( k ) α (λ )+( k ) α ( λ)]c for k, wih λ = λ(α,θ)= (α+ θ ). The above replacemens give rise o an error, he rncaion error, denoed here by T m n. Is vale will be discssed in Secion.. This echniqe has been sed o simlae he fracional anomalos diffsion eqaion (], ]), where Dθ α (Ψ) was approimaed by he following formla: D α θ (Ψ( n, m )) c + + wih k= (φ(h)) ( λ)ψ m n k+ +(λ )Ψ m n k +( λ)ψ m n k + λψ m n k ]v k+ c + k= (φ(h)) λψ m n+k+ +( λ)ψn+k+ m m +(λ )Ψn+k +( λ)ψm n+k ]v k], () v k = n k Γ( α) n k ( n ξ) α dξ = n+k+ dξ Γ( α) n+k (ξ n ) α = h α(k+) α k α. () Γ( α) Neglecing he rncaion error on scheme (), one ges a compable difference scheme i h Ψ n m+ Ψ m n ϕ( ) = C α (m) h α + (φ(h)) (σψn+k m +( σ)ψm+ n+k )w k. () k=

5 Progr. Frac. Differ. Appl., No., - () / The proposed mehod is eplici for σ =, parially implici for <σ <, especial case when σ = / hen we have Crank-Nicholson scheme, and flly implici for σ = ]. The nmerical scheme (), which inclded he nbonded domain < < +, has no pracical implemenaions in comper simlaions ]. Here we solve his problem in he finie domain Ω: L R wih bondary condiions for > Ψ(L,)= Ψ(,)=g L (), Ψ(R,)= Ψ( N,)=g R (). () We divide he domain Ω ino N sb-domains wih he sep h =(R L)/N. Here, we can observe addiional poins in he grid locaed oside he lower and pper limis of he domain Ω. In order o inrodce he Dirichle bondary condiions, we propose a nmerical reamen which assmes he same vales of fncion Ψ oside he domain limis as he vales prediced on bondary nodes and N. { Ψ( Ψ( k,)=,) f or k<, Ψ( N,) f or k>n. Based on previos consideraions we need o modify epressions () for he discreizaion of he qanm Riesz-Feller derivaive. Ths we have ] i h Ψ n m+ Ψn m = C α (m) h α ϕ( ) (φ(h)) (σψn+k m +( σ)ψm+ n+k )w k+(ψ m s L n +ΨN m s R ), () where s Ln = s Rn = n k= k=n+ w k = c (n+) α ( λ)+(n+) α (λ )+n α ( λ)+(n ) α. Γ( α) w k = c + (n+) α ( λ)+(n+) α (λ )+n α ( λ)+(n ) α. Γ( α) Scheme () wih he bondary condiion () can be wrien afer some simplificaion in he mari form as: CΨ m+ = AΨ m + B, () where Ψ m+ is he vecor of nknown fncion vales a ime m+, and c +c c c c N c N c c +c c c N c N C= c c c +c c N c N ,.. c N+ c N+ c N+ c N+ +c c a +a a a a N a N a a +a a a N a N A= a a a +a a N a N., a N+ a N+ a N+ a N+ +a a B=, b, b,..., b N,] T, c j = i C α(m)ϕ( )h α h(φ(h)) (σ )w j, j = N+,,...,,..., N,

6 N. H. Sweilam, M. M. Abo Hasan: Nmerical sdies for he fracional Schrödinger... a j = i C α(m)ϕ( )h α h(φ(h)) σw j, j = N+,,...,,..., N, b n = i C α(m)ϕ( )h α h(φ(h)) (Ψ m s L n +ΨN m s R ), n=,,..., N. Sabiliy Analysis and Trncaing Error. Sabiliy Analysis In his secion, John von Nemann procedre is sed o sdy he sabiliy analysis of he weighed average scheme (). Le s consider β = C α(m)ϕ( )h α, hen scheme () can be wrien in he form h(φ(h)) i(ψn m+ Ψn m )=β (σψn+k m +( σ)ψm+ n+k )w k+(ψ m s L n +ΨN m s R ) Theorem. The weighed average scheme () is condiionally sable. ]. () Proof.Assming ha Ψn m = ξ m e inhq wih q is he spaial wave nmber (which we assme o be prely real) ] hen Eq. () can be wrien in he following form i(ξ m+ ξ m )e inhq = β (σξ m +( σ)ξ m+ )w k e i(n+k)hq +(s Ln + s R e inhq )ξ ], m dividing he las eqaion by ξ m e inhq where ξ m+ = η η(q) is he amplificaion facor ], we find: so we can confirm ha also ξ m ] i(η )=β (σ+( σ)η)w k e ikhq +(s Ln e inhq + s R e i()hq ), η i β( σ) η. i β( σ) w k e ikhq ]=i+β w k e ikhq = i+β where z n = s Ln e inhq + s R e i()hq and z n is he comple conjgae of z n. The scheme will be sable as long as η, for all q i.e., i+β(σ w k e ikhq + z n ) i β( σ) σ w k e ikhq + z n ], σ w k e ikhq + z n ], w k e ikhq, his ineqaliy akes he ne form depending on properies of he comple nmber norm: ] ] i+β(σ w k e ikhq + z n ) i+β(σ w k e ikhq + z n ) i β( σ) w k e ] i β( σ) ikhq w k e ]. ikhq

7 Progr. Frac. Differ. Appl., No., - () / The above ineqaliy can be wrien as: β σ ikhq w k e i( σ)( w k e ikhq w k e ikhq + β σ(z n +β z n z n iσ( w k e ikhq Le z n = r n e iθ n, hen he previos ineqaliy (8) afer some simplificaion: β σr n (e iθ n w k e ikhq + z n w k e ikhq ) w k e ikhq ) i(z n z n ) w k e ikhq )+β( σ) ikhq w k e w k e ikhq. (8) w k e ikhq + e iθ n w k e ikhq )+β r n sinθ n which eqivalen o i( w k e ikhq w k e ikhq )+β( σ) ikhq w k e w k e ikhq, βσr n. w k cos(θ n khq) ( σ)( So he scheme () is sable nder he condiion: wih sinθ n A=sinθ n w k +,v= n,k v w k sin(khq). w k w v cos(k v)hq)+r n] β B A, (9) B=σr n w k cos(θ n khq) ( σ)( w k + w k sin(khq),,v= n,k v w k w v cos(k v)hq)+r n.. Trncaing Error Theorem. The rncaing error of WA-NSFD scheme () is: T m n = O(ϕ( )+φ(h)+h α ). Proof. From he definiion of rncaing error given by Eq. (), one ges n = i h Ψ n m+ ϕ( ) T m Ψ m n depending on Taylor series epansion we find (for all n) Ψ m+ n Ψ m n ϕ( ) and Eq.() akes he form (for all m) +C α (m) h α + (φ(h)) (σψn+k m +( σ)ψm+ n+k )w k, () k= = Ψ + Ψ (ϕ( ))+ Ψ (ϕ( )) +..., ()

8 8 N. H. Sweilam, M. M. Abo Hasan: Nmerical sdies for he fracional Schrödinger... D α θ (Ψ m n ) c + + k= +c + k= Ψ + φ(h)ψ + 8 (+λ)(φ(h)) Ψ +...]v k Ψ + φ(h)ψ + 8 (+λ)(φ(h)) Ψ +...]v k ]. () We menion here (for eample) he Taylor epansion for Ψn k+ m which we have sed o wrie Eq. () in he form () Ψ n k+ = Ψ n k +Ψ φ(h)+ Ψ (φ(h)) + Ψ (φ(h)) + Ψ (φ(h)) +... Eq. () can be wrien, sing Eq. (), in he following form D α θ (Ψ n m + h α ) (φ(h)) (Ψn+k m )w k = (c + + c )Ψ + k= φ(h)ψ (+λ)(φ(h)) Ψ +...] h α(k+) α k α. () Γ( α) Insering hese epressions (, ) ino Eq. (), he local rncaion error is k= T m n = O(ϕ( )+φ(h)+h α ). Accordingly, or scheme is convergen nder he condiion (9). Nmerical Eamples In his secion we presen he resls obained by he presen nmerical approach () wih ϕ( )=sinh( ), φ(h) = sinh(h). Eample. Consider he space fracional Schrödinger eqaion wih he qanm Riesz-Feller derivaive wih he iniial condiion: he bondary condiions: Ψ(,) and he eac solion when α = 8] is: = id α θψ(,), <<, >, <α <, () Ψ(,)=+cosh(), Ψ(,)= +cosh( )e i, Ψ(,)=+cosh()e i, Ψ(,)=+cosh()e i,. Table() shows he maimm error beween he norm of he nmerical solion obained by sing he WA-NSFDM and he norm of he eac solion, is smaller han he maimm error beween he norm of he nmerical solion obained by sing he FDM and he norm of he eac solion, when σ = a =, sing N = and differen vales of M. Table() shows he maimm errors beween he norm of he nmerical solion obained by sing he WA-NSFDM and he norm of he eac solion, when σ =,.,, a =, sing N = and differen vales of M also i shows he sabiliy bond (SB) (9). The behavior of he real pars of he analyical and nmerical solions by means of he WA-NFDM (σ = ) wih differen vales of α and θ when <. are presened in Figres () and ().

9 Progr. Frac. Differ. Appl., No., - () / 9 Eac, Alpha= NFDM, Alpha=,Thea= NFDM, Alpha=.8,Thea= NFDM, Alpha=.,Thea= NFDM, Alpha=.,Thea= NFDM, Alpha=.,Thea= NFDM, Alpha=.,Thea= NFDM, Alpha=.,Thea= NFDM, Alpha=.,Thea= Fig. : Solion of he real par of eample () for differen vales of α, θ and N =, M =. Alpha=,Thea= Fig. : Unsable solion of he real par of eample () when N =, M =, α =, θ =, here SB =..

10 N. H. Sweilam, M. M. Abo Hasan: Nmerical sdies for he fracional Schrödinger... Table : The maimm error for eample () when α = and N = a = sing WA-NSFDM and FDM (σ = ). M ma-error-wa-nsfd ma-error-sfd.e-.99e-.e-.99e- 8.e-.9e-.8e-.99e- 8.9e-.9e- Table : The maimm error for eample () when α = and N = a = sing WA-NSFDM wih σ =,., and he (SB) (σ = ) (σ =.) (σ = ) M ma-error SB ma-error SB ma-error SB divergen 9.e-.e- -9.8e-.e- -9.e- divergen 9.8e-.e- -9.8e-.e- -9.8e- divergen 9.8e-.e- -9.8e-.e- -9.8e- divergen 9.8e-.e- -9.8e-.e- -9.8e- divergen 9.8e-.8e- -9.8e-.9e- -9.8e- Eample. Consider he space fracional Schrödinger eqaion wih he qanm Riesz-Feller derivaive wih he iniial condiion: he bondary condiions: Ψ(,) and he eac solion when α = is given as follows 8]: = id α θψ(,), <<, >, <α <, () Ψ(,)=e i, Ψ(,)= e i( +), Ψ(,)=e i(+), Ψ(,)=e i(+),. Table() shows he maimm errors of WA-NSFDM, when σ =,.,, beween norm of he eac solion and norm of he nmerical solions a =, sing M = and differen vales of N also i shows he (SB) (9). Table() shows he maimm errors of WA-NSFDM, when σ =,.,, beween norm of he eac solion and norm of he nmerical solions a =, sing N = and differen vales of M also i shows he (SB) (9). The behavior of he imaginary pars of he analyical and nmerical solion by means of he WA-NFDM (σ = ) wih differen vales of α and θ when <. are presened in Figres () and (). Table : The ma-error for eample () when α = and M = a = sing WA-NSFDM wih σ =,., and he (SB) wih differen vale of N. (σ = ) (σ =.) (σ = ) N ma-error SB ma-error SB ma-error SB.9e- -.e-.8e- -.e-.99e- -.e- divergen 9.8e-.e- -9.8e-.9e- -9.8e- divergen.e-.9e- -.e-.89e- -.e- divergen.e-.e- -.e-.89e- -.e-

11 Progr. Frac. Differ. Appl., No., - () / image eac image, alpha=, hea= image, alpha=.8, hea= image, alpha=., hea= image, alpha=., hea= image, alpha=., hea= image, alpha=., hea= image, alpha=., hea=. - image, alpha=., hea= Fig. : Behavior of he imaginary par of solion of eample () for differen vales of α, θ and N =, M =. image, alpha=, hea= Fig. : Unsable solion of he imaginary par of eample () when N =, M =, α =, θ =, here SB =.8e. c NSP

12 N. H. Sweilam, M. M. Abo Hasan: Nmerical sdies for he fracional Schrödinger... Table : The ma-error for eample () when α = and N = a = sing WA-NSFDM wih σ =,., and he (SB) wih differen vale of M. (σ = ) (σ =.) (σ = ) M ma-error SB ma-error SB ma-error SB divergen.e-.e- -.9e- 8.8e- -.e- divergen.8e-.99e- -.9e-.9e- -.8e- divergen.9e-.e- -.9e-.e- -.9e- divergen.9e-.e- -.9e-.e- -.9e- Eample. Consider he space fracional Schrödinger eqaion wih he qanm Riesz-Feller derivaive sch ha wih he iniial condiion: he bondary condiions: and he eac solion is: Ψ(,) = idθ α Ψ(,) iv(,)ψ(,), <<π, >, <α <, () v(,)=/+sin θπ + cosθπ, Ψ(,)=sin(), Ψ(,)=, Ψ(,)=sin()e ( i/), Ψ(,)=sin()e ( i/), π. Table() shows he maimm errors of WA-NSFDM, when σ =,.,, beween norm of he eac solion and norm of he nmerical solions, sing N =, M =, θ = and differen vales of α when =. also i shows he (SB) (9). Table() shows he maimm errors of WA-NSFDM, when σ =,.,, beween norm of he eac solion and norm of he nmerical solions, sing N =, M =, α =. and differen vales of θ when =. also i shows he (SB) (9). Figs. () show he behavior of he real par of he eac solion and he solions of eample () sing he WA-NSFDM (σ = ) for differen vales of α and θ when N =, M =. Figs. () show he behavior of he real par of he solions of eample () sing he WA-NSFDM (σ = ) for α = and θ = when N =, M =. Table : The ma-error for eample () when N =, M =, θ = and differen vales of α, sing WA-NSFD wih σ =,., and he (SB). (σ = ) (σ =.) (σ = ) α ma-error SB ma-error SB ma-error SB divergen.e-.9e- -.e-.e- -.e-. divergen.8e-.8e- -.8e-.9e- -.8e-. divergen.e- 8.8e- -.e- 8.9e- -.e-. divergen.8e-.e- -.8e-.e- -.8e- divergen.9e-.e- -.9e-.e- -.9e-

13 Progr. Frac. Differ. Appl., No., - () / Eac NFDM, Alpha=,Thea= NFDM, Alpha=.,Thea= NFDM, Alpha=.,Thea=. NFDM, Alpha=.,Thea=. NFDM, Alpha=.,Thea= NFDM, Alpha=.,Thea=-. NFDM, Alpha=.,Thea=-. NFDM, Alpha=.,Thea= Fig. : Behavior of he real par solions of eample () for differen vales of α and θ and N =, M =. NFDM, Alpha=,Thea= Fig. : Unsable solion of he real par of eample () when N =, M =, α =, θ =, here SB =.8e.

14 N. H. Sweilam, M. M. Abo Hasan: Nmerical sdies for he fracional Schrödinger... Table : The ma-error for eample () when N=, M=, α =. and differen vales of θ, sing WA-NSFD wih σ =,., and he (SB). (σ = ) (σ =.) (σ = ) θ ma-error SB ma-error SB ma-error SB divergen.8e-.e- -.8e-.9e- -.8e-. divergen.e- 8.e- -.e- 8.e- -.e-. divergen.8e-.8e- -.8e-.9e- -.8e-. divergen.8e-.e- -.8e-.e- -.8e- -. divergen.e-.9e- -.e-.9e- -.e- Conclsions In his paper, we sed WA-NSFDM o inrodce nmerically he approimae solion of a fracional Schrödinger eqaion wih he qanm Riesz-Feller derivaive. The proposed mehod is based on choosing he weigh facor σ. The main advanage of his mehod is, i can be eplici or implici wih large sabiliy regions as we see in ables (-). Special aenion is given o sdy he sabiliy and consisency of proposed mehods. To eece his aim we have resored o he kind of John Von Nemann sabiliy analysis. Some nmerical resls are sed o show he accracy of he WA-NSFDM and some figres are sed o demonsrae how he solions change when α and θ ake differen vales. All compaions in his paper are performed sing MATLAB programming. Acknowledgmen The ahors wish o hank he referees for heir consrcive commens and sggesions which improved he presen paper. References ] N. Laskin, Fracional Schrödinger eqaion, Universiy of Torono, -8,. ] R. Becerril, F.S. Gzman, A. Rendon-Romero and S. Valdez-Alvarado, Solving he ime-dependen Schrödinger eqaion sing finie difference mehods, Rev. Me. Fis. (), - (8). ] B. Al-Saqabi, L. Boyadjiev and Y. Lchko, Commens on employing he Riesz-Feller derivaive in he Schrödinger eqaion, Er. Phys. J. Spec. Topic., 9-9 (). ] R. E. Mickens, Applicaion of nonsandard finie difference schemes, World Scienific Pblishing Co. Pe. Ld.,. ] N. H. Sweilam and T. A. Assiri, Nmerical simlaions for he space-ime variable order nonlinear fracional wave eqaion, J. Appl. Mah. Aricle ID 88,, pages (). ] N. H. Sweilam and T. A. Assiri, Non-sandard Crank-Nicholson mehod for solving he variable order fracional cable eqaion, Appl. Mah. Inf. Sci., 9-9 (). ] N. H. Sweilam and T. F. Almajbri, Large Sabiliy Regions mehod for he wo-dimensional fracional diffsion eqaion, Progr. Frac. Differ. Appl. (), - (). 8] A. Bibi, A. Kamran, U. Haya and S. Mohyd-Din, New ieraive mehod for ime-fracional Schrödinger eqaions,world J. Mod. Siml. 9(), 89 9 (). 9] L. Wei, Y. He, X. Zhang and S. Wang, Analysis of an implici flly discree local disconinos Galerkin mehods for he imefracional Schrödinger eqaion, Finie Elem. Anal. Des. 9, 8- (). ] A. Mohebbi, M. Abbaszadeh and M. Dehghan, The se of a meshless echniqe based on collocaion and radial basis fncions for solving he ime fracional nonlinear Schrödinger eqaion arising in qanm mechanics, Eng. Anal. Bond. Elem., 8 (). ] A. H. Bhrawy, E. H. Doha, S. S. Ezz-Eldien and R. A. Van Gorder, A new Jacobi specral collocaion mehod for solving + fracional Schrodinger eqaions and fracional copled Schrödinger sysems, Er. Phys. J. Pls 9, (). ] A. H. Bhrawy and M. A. Abdelkawy, A flly specral collocaion approimaion for mli-dimensional fracional Schrödinger eqaions, J. Comp. Phys. 9, -8 (). ] F. Mainardi, Y. Lchko and G. Pagnini, The fndamenal solion of he space-ime fracional diffsion eqaion, Frac. Calc. Appl. Anal. (), 9 (). ] M. Ciesielski and J. Leszczynski, Nmerical solions o bondary vale problem for anomalos diffsion eqaion wih Riesz- Feller fracional operaor, J. Theor. Appl. Mech. () 9- (). ] N. H. Sweilam and T. A. Assiri, Error analysis of eplici finie difference approimaion for he space fracional wave eqaions, SQU J. Sci., - ().

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