IN recent decades, fractional calculus (calculus of integrals

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1 IEEE/CAA JOURNAL OF AUTOMATICA SINICA 1 An Implemenaion of Haar Wavele Based Mehod for Numerical Treamen of Time-fracional Schrödinger and Coupled Schrödinger Sysems Najeeb Alam Khan, Tooba Hameed Absrac The objecive of his paper is o solve he imefracional Schrödinger and coupled Schrödinger differenial equaions (TFSE) wih appropriae iniial condiions by using he Haar wavele approimaion. For he mos par, his endeavor is made o enlarge he perinence of he Haar wavele mehod o solve a coupled sysem of ime-fracional parial differenial equaions. As a general rule, piecewise consan approimaion of a funcion a differen resoluions is presenaional characerisic of Haar wavele mehod hrough which i convers he differenial equaion ino he Sylveser equaion ha can be furher simplified easily. Sudy of he (TFSE) is heoreical and eperimenal research and i also helps in he developmen of auomaion science, physics, and engineering as well. Illusraively, several es problems are discussed o draw an effecive conclusion, suppored by he graphical and abulaed resuls of included eamples, o reveal he proficiency and adapabiliy of he mehod. Inde Terms Fracional calculus, haar waveles, operaional mari, waveles. I. INTRODUCTION IN recen decades, fracional calculus (calculus of inegrals and derivaives of any arbirary real order) has aained appreciable fame and imporance due o is manifes uses in apparenly diverse and ouspread fields of science. Cerainly, i provides poenially helpful ools for solving inegral and differenial equaions and many oher problems of mahemaical physics. The fracional differenial equaions have become crucial research field essenially due o heir immense range of uilizaion in engineering, fluid mechanics, physics, chemisry, biology, viscoelasiciy ec. Numerous mahemaicians and physiciss have been sudying he properies of fracional calculus [1], [2] and have esablished several mehods for accurae analyical and numerical soluions of fracional differenial equaions, such as he variaional ieraion mehod [3], differenial ransform mehod [4], homoopy analysis mehod [5], Jacobi specral au and collocaion mehod [6][8], Laplace ransform mehod [9], homoopy perurbaion mehod [1], Adomian decomposiion mehod [8], [9], high-order finie elemen mehods [13] and many ohers [14], [19]. The scope This aricle has been acceped for publicaion in a fuure issue of his journal, bu has no been fully edied. Conen may change prior o final publicaion. This aricle was recommended by Associae Edior Anonio Visioli. Najeeb Alam Khan is wih he Deparmen of Mahemaics, Universiy of Karachi, Karachi-7527, Pakisan. ( njbalam@yahoo.com; oobahameed@homail.com). Digial Objec Idenifier 1.119/JAS and disinc aspecs of fracional calculus have been wrien by many auhors in Refs [2][22]. In he pas few years, here has been an eensive aracion in employing he specral mehod (see [23][25]) for numerically solving he copious ype of differenial and inegral equaions. The specral mehods have an eponenial quoa of convergence and high level of efficiency. Specral mehods are o epress he approimae soluion of he problem in erm of a finie sum of cerain basis funcions and hen selecion of coefficiens in order o reduce he difference beween he eac and approimae soluions as much as possible. The specral collocaion mehod is a disinc ype of specral mehods, ha is more relevan and eensively used o solve mos of differenial equaions [26]. For he reason of he disincive aribues of wavele heory in represening coninuous funcions in he form of disconinuous funcions [27], is applicaions as a mahemaical ool is widely epanding nowadays. Besides image processing and signal decomposiion i is also used o assess many oher mahemaical problems, such as differenial and inegral equaions. Waveles comprise he incremenal concepion beween wo consecuive levels of resoluion, called muli-resoluion. The firs componen of muli-resoluion analysis is vecor spaces. For each vecor space, anoher vecor space of higher resoluion is found and his coninues unil he final image or signal is eecued. The basis of each of hese vecor spaces acs as he scaling funcion for he waveles. Each vecor space having an orhogonal componen and a basis funcion is said o be he wavele [28]. Up ill now, a number of wavele families have been presened by differen auhors, bu among all Haar wavele are considered o be he easies waveles family. Haar wavele was inroduced in 191 by Hungarian mahemaician Alfred Haar. These waveles are obained from Daubechies waveles of order 1, which consis of piecewise consan funcions on he real ais ha can ake only hree values, 1, and 1. Here we are using collocaion mehod, by increasing he level of resoluion, collocaion poins are also increasing and level of accuracy oo. Haar wavele collocaion mehod is eensively used due o is consrucive abiliy of being smooh, fas, convenien and being compuaionally aracive [29]. In addiion, i has he compeency o reduce he compuaions for solving differenial equaions by convering hem ino some sysem of algebraic equaions. The main advanages of

2 2 IEEE/CAA JOURNAL OF AUTOMATICA SINICA he proposed algorihm are, is simple applicaion and no residual or produc operaional mari is required. The mehod is well addressed in [22], [29][32]. The ime-fracional Schrödinger equaion (T-FSE) differs from he sandard Schrödinger equaion. The firs-order ime derivaive is replaced by a fracional derivaive, i makes he problem overall in ime. I describes, how he quanum sae (physical siuaion) of a quanum sysem changes wih ime, solion dispersion, deep waer waves, molecular orbial heory and he poenial energy of a hydrogen-like aom (fracional Bohr aom ). The aim of his work is o eplore he numerical soluions of he ime-fracional Schrödinger equaions by using Haar wavele mehod. Due o he large number of applicaions of he Schrödinger equaion in differen aspecs of quanum mechanics and engineering, many aemps have been eercised on analyical and numerical mehods o calculae he approimae soluion of (T-FSE). Some of hem are sudied [6], [1], [18], [33][36], and enumeraed here for beer percepion of he presened analysis. Also, he eisence and uniqueness of soluions of fracional Schrödinger equaion have been proved by muliple auhors [35], [37], [38]. II. PRELIMINARIES In his secion, some noaions and properies of fracional calculus, he basis of Haar funcion approimaion for parial differenial equaion and soluion of Haar by muli-resoluion analysis are given ha will help us in eploring he main heme of he paper. A. Riemann-Liouville Differenial and Inegral Operaor Assume ν >, m = ν and f(, ) C m ([, 1] [, 1]) hen he parial Capuo fracional derivaive of f(, ) wih respec o is defined as where I ν as { ν I mν m f(, ) = f(, ) m ν m f(, ) m (1) is he Riemann-Liouville fracional inegral, given I ν f() = 1 ( ϕ) ν1 f(ϕ)dϕ Γ(ν) I f() = f() (2) we use he noaion D ν in replacemen of ν for he Capuo ν fracional derivaive. The Capuo fracional derivaive of order ν > for f() = α is given as D ν f() = { Γ(α1) Γ(αν1) αν, m > α m 1, if α {, 1, 2,..., m 1} (3) In he following, some main compuaional properies and relaions of fracional inegral and differenial operaors are defined as i)i α I β f() = I αβ f() = I β I α f() ii) β β Iα f(, ) = I αβ f(, ) iii) α α f(, ) = f(, ) n1 k= k k f(, ) = k! k n1 = f(, ) ζ k () k (4) k= where,ζ k () = 1 k f(,) = k!. For more deails see [1]. k B. Haar waveles and funcion approimaion Basis of Haar waveles is obained wih a muli-resoluion of piecewise consan funcions. Le he inerval [, 1) be divided ino 2m subinervals of equal lengh, where m = 2 j and J is maimal level of resoluion. Ne, wo parameers are inroduced, j =, 1, 2,..., J and k =, 1, 2, ldos, m 1, such ha he wavele number i saisfies he relaion i = k m 1. The ih Haar wavele can be deermined as 1, [ϑ 1, ϑ 2 ) h i () = 1, [ϑ 2, ϑ 3 ) (5), elsewhere where ϑ 1 = k m, ϑ 2 = k.5 m, ϑ 3 = k1 m For he case i = 1, corresponding scaling funcion can be defined as: { 1, [ϑ 1, ϑ 3 ) h 1 = (6), elsewhere Here, we consider he wavele-collocaion mehod, herefore collocaion poins are generaed by using, l = l.5, l = 1, 2, 3, ldos, 2m (7) 2m The Haar sysem forms an orhonormal basis for he Hilber space f() L 2 (, 1). We may consider he inner produc epansion of f() L 2 ([, 1)) in Haar series [31] as: J1 2 j 1 f() f, ϕ ϕ() f, h j,k h j,k () = C T H() (8) j= i= where, C is 1 2 J coefficien vecor and H() = [h (), h 1 (), ldos, h m1 ()] T. Also, a funcion of wo variables can be epanded by Haar waveles [32] as: u(, ) m1 i= m1 j= u i,j h i ()h j () = H T ().U.H() (9) where, U is 2 J 2 J coefficien mari calculaed by he inner produc u i,j = h i (), u(, ), h j The operaional mari of fracional inegraion of Haar funcion is needed o solve PDE of fracional order. A more rigorous derivaion for

3 N. A. KHAN e al.: AN IMPLEMENTATION OF HAAR WAVELET BASED METHOD FOR NUMERICAL TREATMENT OF 3 he generalized block pulse operaional marices is proposed in [39]. The block pulse funcion forms a complee se of orhogonal funcions which is defined in inerval [a, b) as { i1 1, ψ i () = m b < i m b (1) elsewhere for i = 1, 2, ldos, m I is known ha any absoluely inegrable funcion f() on [a, b), can be epanded in block pulse funcions as f() = F T ψ (m) () (11) so ha he mean square error of approimaion is minimized. Here, F T = [f 1, f 2, f 3, ldos, f m ] and ψ T (m) () = [ψ 1 (), ψ 2 (), ψ 3 (), ldos, ψ m ()]. where, f i = m b b a f()ψ i ()d = m b (i/m)b (i1)b/m f()ψ i ()d (12) The Riemann-Liouville fracional inegral is simplified and epanded in block pulse funcions o yield he generalized block pulse operaional mari F ν as where F ν = ( b m )ν 1 Γ(ν 2) (I ν ψ m )() = F ν ψ m () (13) 1 ξ 2 ξ 3... ξ m 1 ξ 2... ξ m ξ m wih ξ 1 = 1, ξ p = p ν1 2(p 1) ν1 (p 2) ν1 (p = 2, 3, 4, ldos, m i 1) For furher deails see refs. [39]. The Haar funcions are piecewise consan, so i may be epanded ino an merm block pulse funcions (BPF) as H m () = H m m ψ m () (14) In [31] Haar waveles operaional mari of fracional order inegraion is derived by (I ν H m )() P ν H m () (15) where P ν is m m order Haar waveles operaional mari of fracional order inegraion. Subsiuing Eq.(2) in Eq.(14) we ge (I ν H m )() (I ν H m m ψ m )() = H m m (I ν ψ m )() From Eq.(15) and Eq.(16), i can be wrien as: H m m F ν ψ m () (16) P ν H m () = H m m F ν (17) Therefore, P ν can be obained as P ν = H.F ν.h 1 (18) C. Muli Resoluion Analysis (MRA) Any space V can be consruced using a basis funcion h(2 m ) as: V m = span{h(2 m n)} n,m Z h() is called scaling funcion, also known as Faher funcion. The chain of subspaces ldosv 2, V 1, V, V 1, V 2 ldos wih he following aioms is called muli-resoluion analysis (MRA) [28]. i){ V m } m Z = L 2 (R) ii){ V m } m Z = {} iii)there eiss h()such ha,v = span{h( n)} n Z iv){h( n)} n Z is an orhogonal se. v)iff() V m hen f(2 m ) V, m Z) vi)iff() V hen f( n) V, n Z) (19) Under he given aioms, here eiss a ψ(.) L 2 (R), such ha {ψ(2 m n)} m,n Z spans L 2 (R). The wavele funcion ψ(.) is also called Moher wavele. Convergence of he mehod are coninuous and bounded funcions on (, 1) (, 1), hen M 1, M 2 >,, (, 1) (, 1) Le 3 u(,) and 3 v(,) u(, ) 2 M 1and 3 v(, ) 2 M 2 (2) Le u m (, ) and v m (, ) are he following approimaions of u(, ) and v(, ), hen we have u m (, ) m1 i= v m (, ) u(, ) u m (, ) = m1 j= m1 i= u ij h i ()h j () and m1 j= i=m j=m v ij h i ()h j () (21) u ij h i ()h j () = u ij h i ()h j () and i=2 p1 j=2 p1 v(, ) v m (, ) = v ij h i ()h j () = i=m j=m i=2 p1 j=2 p1 v ij h i ()h j () (22)

4 4 IEEE/CAA JOURNAL OF AUTOMATICA SINICA Theorem 1: Le he funcions u m (, ) and v m (, ) obained by using Haar waveles are he approimaion of u(, ) and v(, ) hen we have he errors bounded as following where u(, ) u m (, ) E M 1 3m 3 and u(, ) E = ( See proof in [32]. v(, ) v m (, ) E M 2 3m v(, ) E = ( u 2 (, )dd) 1/2 and 1 1 (23) v 2 (, )dd) 1/2 (24) III. THE SCHRÖDINGER EQUATIONS A. The ime-fracional Schrödinger equaion The ime-fracional Schrödinger equaion (T-FSE) has he following form i ν φ(, ) ν λ 2 φ(, ) 2 η φ 2 φ µ()φ = q(, ), <, 1 (25) wih iniial condiions φ(, ) = f(), φ(, ) = g(), φ (, ) = h() where < ν 1, λ and η are real consans, µ() is he rapping poenial and φ(, ), f(), g(), h() and q(, ) are comple funcions. We can epress comple funcions φ(, ), f(), g(), h() and q(, ) ino heir respecive real and imaginary pars as φ(, ) = u(, ) iv(, ) µ() = µ 1 () iµ 2 f() = f 1 () if 2 () (26) g() h() = g 1 () ig 2 () = h 1 () ih 2 () q(, ) = q 1 (, ) iq 2 (, ) Subsiuing Eq. (26) ino Eq. (25) and collecing real and imaginary pars, hen Eq. (25) can be wrien as coupled imefracional nonlinear parial differenial equaions as: B. The ime-fracional coupled Schrödinger sysem The ime-fracional coupled Schrödinger sysem (T-FCSS) has he following form i ν φ(, ) ν i 2 φ(, ) 2 2 φ(, ) 2 λ 1 ( φ 2 ψ 2 ) φ(, ) α 1 ()φ(, ) β 1 ψ(, ) f 1 (, ) = i ν ψ(, ) ν i 2 ψ(, ) 2 2 ψ(, ) 2 λ 2 ( φ 2 ψ 2 ) ψ(, ) α 2 ()φ(, ) β 2 ψ(, ) f 2 (, ) = wih iniial condiions φ(, ) = f 3 (), φ(, ) = g 3 (), φ (, ) = h 3 (), (29) ψ(, ) = f 4 (), ψ(, ) = g 4 (), ψ (, ) = h 4 () (3) where < ν 1, λ 1, λ 2, α 1, α 2, β 1 andβ 2 are real consans and φ(, ), ψ(, ), f 3 (), f 4 (), g 3 (), g 4 (), h 3 (), h 4 (), q 1 (, ) and q 2 (, ) are comple funcions. We can epress comple funcions ino heir respecive real and imaginary pars as φ(, ) = u(, ) iv(, ), ψ(, ) = r(, ) is(, ) f 3 () = f 5 () if 6 (), f 4 () = f 7 () if 8 () g 3 () = g 5 () ig 6 (), g 4 () = g 7 () ig 8 () h 3 () = h 5 () ih 6 (), h 4 () = h 7 () ih 8 () q 3 (, ) = q 5 (, )iq 6 (, ), q 4 (, ) = q 7 (, )iq 8 (, ) (31) Subsiuing Eq. (31) ino Eq. (29) and equaing real and imaginary pars we ge he sysem of wo coupled imefracional nonlinear parial differenial equaions. ν v(, ) v(, ) ν 2 u(, ) 2 λ 1 (u 2 v 2 r 2 s 2 ) u(, ) α 1 u(, ) β 1 r(, ) q 3 (, ) = ν u(, ) u(, ) ν 2 v(, ) 2 λ 1 (u 2 v 2 r 2 s 2 ) v(, ) α 1 v(, ) β 1 s(, ) q 4 (, ) = ν s(, ) s(, ) ν 2 r(, ) 2 λ 2 (u 2 v 2 r 2 s 2 ) r(, ) α 2 u(, ) β 2 r(, ) q 5 (, ) = ν r(, ) r(, ) ν 2 s(, ) 2 λ 2 (u 2 v 2 r 2 s 2 ) s(, ) α 2 v(, ) β 2 s(, ) q 6 (, ) = (32) ν v(, ) ν λ 2 u(, ) 2 η(u 2 v 2 )u µ 1 () u(, ) q 1 (, ) = ν u(, ) ν λ 2 v(, ) 2 η(u 2 v 2 )v µ 2 () v(, ) q 2 (, ) = (27) wih iniial condiions u(, ) = f 1 (), v(, ) = f 2 (), u(, ) = g 1 (), v(, ) = g 2 (), u (, ) = h 1 (), v (, ) = h 2 () (28) IV. THE PROPOSED METHOD A. For ime-fracional Schrödinger equaion Any arbirary funcion u(, ) L 2 ([, 1) [, 1)) and v(, ) L 2 ([, 1) [, 1)), can be epanded ino Haar series [32] as: u (, ) = v (, ) = u ij h i ()h j () v ij h i ()h j () (33)

5 N. A. KHAN e al.: AN IMPLEMENTATION OF HAAR WAVELET BASED METHOD FOR NUMERICAL TREATMENT OF 5 where 2M 2M Haar coefficien mari of u ij and v ij in Eq. (33) can be wrien as: U = H T ().U.H() V = H T ().V.H() (34) Le dos and primes in Eq. (33) represen differeniaion wih respec o and, respecively. By inegraing Eq. (34) wih respec o from o, we ge U = H T ().U.P 1.H() u (, ) V = H T ().V.P 1.H() v (, ) (35) on inegraing Eq. (35) wice wih respec o from o, hen we ge U = H T ().[P 1 ] T.U.P 1.H() u (, ) u (, ) u (, ) V = H T ().[P 1 ] T.V.P 1.H() v (, ) v (, ) and hen U = H T ().[P 2 ] T.U.P 1.H() u(, ) u(, ) u (, ) u (, ) u(, ) V = H T ().[P 2 ] T.V.P 1.H() v(, ) v(, ) v (, ) (36) v (, ) v (, ) v(, ) (37) Applying differenial operaor D ν on boh sides of Eq. (37) and using propery (ii) of Eq. (4) D ν U = H T ().[P 2 ] T.U.P 1ν.H() D ν u (, ) D ν u(, ) D ν V = H T ().[P 2 ] T.V.P 1ν.H() D ν v (, ) D ν v(, ) (38) Subsiuion of Eqs. (35), (37) and (38) ino Eq. (27) may lead o coupled sysem of ime-fracional differenial equaions. This sysem will have some unknown funcions u (, ), u (, ), u (, ), u(, ), D ν u (, ), D ν u(, ), v (, ), v (, ), v (, ), v(, ), D ν v (, ), and D ν v(, ). Wih he help of iniial condiions all hese funcions are calculaed. We solve Eq. (27) for unknown Haar coefficiens by using collocaion mehod. Finally, for Haar soluion of T-FSE, we subsiue values of Haar coefficiens in Eq. (37). B. For ime-fracional coupled Schrödinger sysem Now, we approimae u(, ), v(, ), r(, ) and s(, ), by he Haar series [32] as: u (, ) = v (, ) = ṙ (, ) = ṡ (, ) = u ij h i ()h j () v ij h i ()h j () r ij h i ()h j () s ij h i ()h j () (39) Mari form of Eq. (39) can be wrien as: U V Ṙ = H T ().U.H() = H T ().V.H() = H T ().R.H() Ṡ = H T ().S.H() (4) Le dos and primes in Eq. (4) represen differeniaion wih respec o and, respecively. By inegraing Eq. (4) wih respec o from o, we ge U = H T ().U.P 1.H() u (, ) V = H T ().V.P 1.H() v (, ) R = H T ().R.P 1.H() r (, ) S = H T ().S.P 1.H() s (, ) (41) on inegraing Eq. (41) wice wih respec o from o, once we ge U = H T ().[P 1 ] T.U.P 1.H() u (, ) u (, ) u (, ) V = H T ().[P 1 ] T.V.P 1.H() v (, ) v (, ) v (, ) R = H T ().[P 1 ] T.R.P 1.H() r (, ) r (, ) r (, ) S = H T ().[P 1 ] T.S.P 1.H() s (, ) s (, ) and U = H T ().[P 2 ] T.U.P 1.H() u(, ) u(, ) u (, ) u (, ) u(, ) V = H T ().[P 2 ] T.V.P 1.H() v(, ) v(, ) v (, ) v (, ) v(, ) R = H T ().[P 2 ] T.R.P 1.H() r(, ) r(, ) r (, ) r (, ) r(, ) S = H T ().[P 2 ] T.S.P 1.H() s(, ) s(, ) s (, ) (42) s (, ) s (, ) s(, ) (43) Applying differenial operaor D ν on boh sides of Eq. (43) and using propery (ii) of Eq. (4) D ν U = H T ().[P 2 ] T.U.P 1ν.H() D ν u (, ) D ν u(, ) D ν V = H T ().[P 2 ] T.V.P 1ν.H() D ν v (, ) D ν v(, ) D ν R = H T ().[P 2 ] T.R.P 1ν.H() D ν r (, ) D ν r(, ) D ν S = H T ().[P 2 ] T.S.P 1ν.H() D ν s (, ) D ν s(, ) (44) Subsiuion of Eqs. (41), (42), (43) and (44) ino Eq. (32) may lead o he wo coupled sysems of ime-fracional differenial

6 6 IEEE/CAA JOURNAL OF AUTOMATICA SINICA equaions. This sysem has some unknown funcions. Wih he help of iniial condiions all hese funcions are calculaed. We solve sysem of Eq. (32) for unknown Haar coefficiens by using collocaion mehod. Finally, for Haar soluion of imefracional coupled Schro dinger sysem (T-FCSS), we subsiue values of Haar coefficiens in Eq. (43). V. N UMERICAL P ROBLEMS 2i2ν 2 )ei (45) Subjeced o iniial condiions 2 2 φ(, ) =, φ(, ) =, φ (, ) = i (46) he eac soluion for ν = 1 is φ(, ) = 2 ei (47) Subsiue Eq. (45) in Eq. (25) he deermined coupled sysem of equaions is TABLE I C OMPARISON B ETWEEN H AAR S OLUTIONS (J = 1, m = 4) AND HAM [5] OF P ROBLEM 1 In his secion, four es problems are aken o es he efficiency and accuracy of he proposed scheme. The compuaions associaed wih he problems were eecued using Mahemaica 1. Problem 1: Consider he linear T-FSE which is also found in [6] wih λ = 1, η =, µ() = and q(, ) = ( from Eq. (37). Comparison of he Haar soluions by Homoopy analysis mehod in [5] is shown in Table I wih differen values of ν. ν =.1 ν =.3 ν =.5 HWCM HAM[5] HWCM HAM[5] HWCM HAM[5] ν v(, ) 2 u(, ) sin 2 ( cos )= ν 2 ν u(, ) 2 v(, ) cos 2 ( sin ) = (48) ν 2 By using he mehod given in Secion IV A, Eq. (48) can be wrien as H T ().[P 2 ]T.V.P 1ν.H() H T ().U.P 1.H() Γ(ν 1)2ν sin (2 cos )= Γ(2 ν 1) H T ().[P 2 ]T.U.P 1ν.H() H T ().V.P 1.H() Γ(ν 1)2ν cos 2 sin ) = (49) Γ(2 ν 1) Towards he approimae soluion, we firs collocae Eq. (49) a poins i = i.5 j.5, j = 2m 2m Fig. 1. Haar soluions of Problem 1 a ν =.1,.3,.5 Problem 2: Consider a nonlinear cubic form of T-FSE [6] wih (5) Eq. (49) λ = 1, η = 1, µ() = and q(, ) = ( (4π )i)e2πi H T (i ).[P 2 ]T.V.P 1ν.H(j ) H T (i ).U.P 1.H(j ) Γ(ν 1)i j2ν sin i j (2j cos i )= Γ(2 ν 1) and iniial condiions H T (i ).[P 2 ]T.U.P 1ν.H(j ) H T (i ).V.P 1.H(j ) cos i Γ(ν 1)2ν j j φ(, ) =, φ(, ) = i2, φ (, ) = 2π2 2j sin i ) = (51) Γ(2 ν 1) Eq. (51) generaes wo sysems of 2M algebraic equaions of Haar coefficiens. The values of Haar coefficiens are obained from sysem of Eq. (51) by using Newon s ieraive mehod. Wih he help of hese coefficiens, Haar soluions are aained (52) (53) wih eac soluion for ν = 1 is φ(, ) = 2 ie2πi (54)

7 N. A. KHAN e al.: AN IMPLEMENTATION OF HAAR WAVELET BASED METHOD FOR NUMERICAL TREATMENT OF 7 Subsiue Eq. (52) in Eq. (25) he deermine coupled sysem of equaions is ν v(, ) ν 2 u(, ) 2 (u 2 v 2 )u (( 6 4π 2 2 ) sin 2π 23ν cos 2π) = (55) ν u(, ) ν 2 v(, ) 2 (u 2 v 2 )v (( 6 4π 2 2 ) cos 2π 23ν sin 2π) = (56) On following he mehod described in Secion IV A, we have H T ().[P 2 ] T.V.P 1ν.H() H T ().U.P 1.H() ((H T ().[P 2 ] T.U.P 1.H() 2π 2 ) 2 (H T ().[P 2 ] T.V.P 1.H() 2 ) 2 ).(H T ().[P 2 ] T.U.P 1.H() 2π 2 ) Γ(3) 2ν (( 6 4π 2 2 ) sin 2π 2 3ν cos 2π) =, H T ().[P 2 ] T.U.P 1ν.H() H T ().V.P 1.H() ((H T ().[P 2 ] T.U.P 1.H() 2π 2 ) 2 (H T ().[P 2 ] T.V.P 1.H() 2 ) 2 ).(H T ().[P 2 ] T.V.P 1.H() 2 ) 2πΓ(3) 2ν (( 6 4π 2 2 ) cos 2π 2 3ν sin 2π) = (57) For approimae soluion of Eq. (56), puing collocaion poins of Eq. (5) in above equaions generaes wo sysems of 2M non linear algebraic equaions of Haar coefficiens. The values of Haar coefficiens are obained by using Newon s ieraive mehod and hen wih he help of hese coefficiens Haar soluions are aained from Eq. (37). The Haar soluions comparisone wih he mehod in [5] for he differen values of ν is shown in Table II. Problem 3: Consider T-FSE wih rapping poenial [6] for λ = 1, η = 1, µ() = cos 2 and 6 3ν q(, ) = (i Γ(4 ν) cos 2 )e i 2 (58) Subjeced o iniial condiions φ(, ) =, φ(, ) =, φ (, ) = i 3 2 wih eac soluion for ν = 1 is (59) φ(, ) = 3 e i 2 (6) TABLE II COMPARISON BETWEEN HAAR SOLUTIONS (J = 1, m = 4) AND HAM [5] OF PROBLEM 2 ν =.1 ν =.5 ν =.9 HWCM HAM[5] HWCM HAM[5] HWCM HAM[5] Deermine he coupled sysem of equaions by subsiuing Eq. (57) in Eq. (25). Ne, following he mehod illusraed in Secion IV A, by following same seps of problem 1 Haar soluions are aained. The obained Haar soluions are compared wih he Homoopy analysis mehod [5] and a differen values of ν are shown in Table III. TABLE III COMPARISON BETWEEN HAAR SOLUTIONS (J = 1, m = 4) AND HAM [5] OF PROBLEM 3 ν =.1 ν =.3 ν =.5 HWCM HAM[5] HWCM HAM[5] HWCM HAM [5] Problem 4: Take ino consideraion non-linear T-FCSS [35] for

8 8 IEEE/CAA JOURNAL OF AUTOMATICA SINICA λ1 = 2, λ2 = 4, α1 = α2 = 1, β1 = 1 and β2 = 1 q1 (, ) = sin 46 cos i( cos sin 86 cos 46 sin ) andq2 (, ) = i( cos 86 sin ) (61) φ(, ) = ψ(, ) =, φ(, ) = ψ(, ) =, φ (, ) = ψ (, ) = i2 (62) and wih eac soluion for hese values of λ, α, β and ν = 1 is φ(, ) = ψ(, ) = 2 ei, Fig. 2. Γ(ν 1) 2ν Γ(2 ν 1) H T ().[P 1 ]T.U.P 1.H() H T ().V.P 1.H() V 2(U2 V2 R2 S2 )V S ( cos 46 sin ) =, Γ(ν 1) (H T ().[P 2 ]T.S.P 1ν.H() 2ν ) Γ(2 ν 1) H T ().[P 1 ]T.S.P 1.H() H T ().R.P 1.H() U 4(U2 V2 R2 S2 )R R sin 86 cos =, Γ(ν 1) 2ν ) (H T ().[P 2 ]T.R.P 1ν.H() Γ(2 ν 1) H T ().[P 1 ]T.R.P 1.H() H T ().S.P 1.H() V 4(U2 V2 R2 S2 )SS( cos 86 sin ) = (66) H T ().[P 2 ]T.U.P 1ν.H() wih iniial condiions Γ(ν 1) 2ν ) Γ(2 ν 1) H T ().[P 1 ]T.V.P 1.H() H T ().U.P 1.H() U 2(U2 V2 R2 S2 )U R sin 46 cos =, (65) (H T ().[P 2 ]T.V.P 1ν.H() (63) Haar soluions of Problem 3 a ν =.1,.3,.5 where U = H T ().[P 2 ]T.U.P 1.H() 2, V = H T ().[P 2 ]T.V.P 1.H() 2, R = H T ().[P 2 ]T.R.P 1.H() 2 and S = H T ().[P 2 ]T.S.P 1.H() 2 Deermine he coupled sysem of wo equaions by subsiuing Eq. (6) in Eq. (29) as ν v(, ) v(, ) 2 u(, ) 2(u2 v 2 r2 s2 ) ν 2 sin 46 cos =, u(, ) u(, ) r(, ) ν u(, ) u(, ) 2 v(, ) 2(u2 v 2 r2 s2 ) ν 2 v(, ) v(, ) s(, ) ( cos 46 sin ) =, ν s(, ) s(, ) 2 r(, ) 2(u2 v 2 r2 s2 ) ν 2 r(, ) u(, ) r(, ) sin 86 cos =, ν r(, ) r(, ) 2 s(, ) 2(u2 v 2 r2 s2 ) ν 2 s(, ) v(, ) s(, ) ( cos 86 sin ) = (64) Ne, following he mehod illusraed in Secion IV B, we have TABLE IV C OMPARISON B ETWEEN H AAR S OLUTIONS (J = 1, m = 4, ν =.1) AND HAM [5] OF P ROBLEM 4 φ(, ) HWCM HAM [5] ψ(, ) HWCM HAM [5] Using Eq. (5), he values of Haar coefficiens are obained and finally wih he help of hese coefficiens Haar soluions

9 N. A. KHAN e al.: AN IMPLEMENTATION OF HAAR WAVELET BASED METHOD FOR NUMERICAL TREATMENT OF 9 are aained from Eq. (43). The Haar soluions are compared wih he mehod in [5] wih ν =.1 and resuls are shown in Table IV, and for ν =.5 and ν =.9 in Table V and Table VI, respecively. TABLE V COMPARISON BETWEEN HAAR SOLUTIONS (J = 1, m = 4, ν =.5) AND HAM [5] OF PROBLEM 4 φ(, ) ψ(, ) HWCM HAM [5] HWCM HAM [5] TABLE VI COMPARISON BETWEEN HAAR SOLUTIONS (J = 1, m = 4, ν =.9) AND HAM [5] OF PROBLEM 4 φ(, ) ψ(, ) HWCM HAM [5] HWCM HAM [5] VI. CONCLUSION In his paper, we eended he capabiliy of he Haar wavele collocaion mehod (HWCM) for he soluion of imefracional coupled sysem of parial differenial equaions. The main advanage of HWCM is he abiliy o achieve a good soluion and rapid convergence wih small number of collocaion poins. The presence of maimum zeros in he Haar marices reduces he number of unknown wavele coefficiens ha is o be deermined, which as a resul diminishes he compuaion ime as well. The scheme is esed on some eamples of ime fracional Schrödinger equaions. The presened procedure may very well be eended o solve wo dimensional Schrödinger equaion and oher similar nonlinear problems of parial differenial equaions of fracional order. The problem discussed here is jus for showing he applicabiliy of he proposed compuaional echnique o handle he comple sysem of differenial equaion in fracionalorder problems in a sraigh forward way. Also, he Haar wavele mehod proves o be capable o efficienly handle he nonlineariy of parial differenial equaions of fracional order. The main advanages of he proposed algorihm are, is simple applicaion and no requiremen of residual or produc operaional mari. Numerical soluions for differen order of fracional ime derivaive by Haar wavele are shown in Tables and Figures. The increasing values of ν show ha he soluions are valuable in undersanding heir respecive eac soluions for ν = 1. Comparisons beween our approimae soluions of he problems wih heir acual soluions and wih he approimae soluions achieved by a homoopy analysis mehod [5] confirm he validiy and accuracy of our scheme. REFERENCES [1] K. Diehelm, The analysis of fracional differenial equaions, Springer- Verlag, Berlin, 21. [2] I. Podlubny, Fracional Differenial equaions. Academic Press San Diego, [3] M. G. Sakar, F. Erdogan, and A. Yldrm, Variaional ieraion mehod for he ime-fracional Fornberg-Whiham equaion, Compuers and Mahemaics wih Applicaions, 212, 63(9): [4] J. Liu and G. Hou, Numerical soluions of he space- and ime-fracional coupled Burgers equaions by generalized differenial ransform mehod, Applied Mahemaics and Compuaion, 211, 217(16): [5] N. A. Khan, M. Jamil, and A. Ara, Approimae Soluions o Time-Fracional Schrödinger Equaion via Homoopy Analysis Mehod, ISRN Mahemaical Physics, 212, Aricle ID 19768hp://d.doi.org/1.542/212/ [6] A. H. Bhrawy, E. H. Doha, S. S. Ezz-Eldien, and R. A. Van Gorder, A new Jacobi specral collocaion mehod for solving 11 fracional Schrödinger equaions and fracional coupled Schrödinger sysems, THE EUROPEAN PHYSICAL JOURNAL PLUS, :(26) DOI 1.114/epjp/i [7] A. H. Bhrawy and M. A. Zaky, A mehod based on he Jacobi au approimaion for solving muli-erm ime-space fracional parial differenial equaions, Journal of Compuaional Physics, 215, 281: [8] A. Bhrawy and M. Zaky, A fracional-order Jacobi Tau mehod for a class of ime-fracional PDEs wih variable coefficiens, Mahhemaical Mehods in he Applied Sciences, 215, DOI: 1.12/mma.36. [9] Z. J. Fu, W. Chen, and H. T. Yang, Boundary paricle mehod for Laplace ransformed ime fracional diffusion equaions, Journal of Compuaional Physics, 213, 235: [1] H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, An Efficien Mehod for Time-Fracional Coupled Schrödinger Sysem, Inernaional Journal of Parial Differenial Equaions, 214, Aricle ID 13747: hp://d.doi.org/1.1155/214/ [11] V. Dafardar-Gejji and a. Babakhani, Analysis of a sysem of fracional differenial equaions, Journal of Mahemaical Analysis and Applicaions, 293, 24, :

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Dehghan, The use of a meshless echnique based on collocaion and radial basis funcions for solving he ime fracional nonlinear Schrödinger equaion arising in quanum mechanics, Engineering Analysis wih Boundary Elemens, 213, 37(2): [34] D. Wang, A. Xiao, and W. Yang, Crank-Nicolson difference scheme for he coupled nonlinear Schrödinger equaions wih he Riesz space fracional derivaive, Journal of Compuaional Physics, 213, 242: [35] L. Wei, X. Zhang, S. Kumar, and A. Yildirim, A numerical sudy based on an implici fully discree local disconinuous Galerkin mehod for he ime-fracional coupled Schrödinger sysem, 212, Compuers and Mahemaics wih Applicaions, 64(8): [36] A. H. Bhrawy and M. A. Abdelkawy, A fully specral collocaion approimaion for muli-dimensional fracional Schrödinger equaions, Journal of Compuaional Physics, 215, 294: [37] X. Guo and M. Xu, Some physical applicaions of fracional Schrödinger equaion, Journal of Mahemaical Physics, 26, 47 Aricle ID 8214, doi: 1.163/ [38] Y. Wei, Some Soluions o he Fracional and Relaivisic Schrödinger Equaions, Inernaional Journal of Theoreical and Mahemaical Physics, 215, 5(5): [39] W. Chi-hsu, On he Generalizaion Operaional Marices and Operaional, The Franklin Insiue, 1983, 315(2): Najeeb Alam Khan obained he Ph. D. degree from Universiy of Karachi in 213. He is he assisan professor in he Deparmen of Mahemaics, Universiy of Karachi. His research ineres include he areas of Nonlinear sysem, Numerical mehods, Approimae analyical mehods, Fluid Mechanics, differenial equaions of applied mahemaics, fracional calculus, fracional differenial equaion and heoreical analysis. He has published more han 75 refereed journal papers. Tooba Hameed is a research scholar. Her research ineres include Numerical mehods, fracional calculus and differenial equaion.