Approximate Technique for Solving Class of Fractional Variational Problems

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1 Appled Matheatcs, 5, 6, Publshed Onlne May 5 n ScRes. Approxate Technque for Solvng Class of Fractonal Varatonal Probles Ead M. Soloua,, Mohaed M. Khader,3 Departent of Matheatcs and Statstcs, College of Scence, Al-Ia Mohaad Ibn Saud Islac, Unversty (IMSIU), Ryadh, Saud Araba Departent of Matheatcs, College of Scence, Ben-Suef Unversty, Ben-Suef, Egypt 3 Departent of Matheatcs, Faculty of Scence, Benha Unversty, Benha, Egypt Eal: eads74@gal.co, ohaedbd@yahoo.co, ohaed.khader@fsc.bu.edu.eg Receved Aprl 5; accepted 6 May 5; publshed 9 May 5 Copyrght 5 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Coons Attrbuton Internatonal Lcense (CC BY). Abstract Ths paper s devoted to pleentng the Legendre spectral collocaton ethod to ntroduce nuercal solutons of a certan class of fractonal varatonal probles (FVPs). The propertes of the Legendre polynoals and Raylegh-Rtz ethod are used to reduce the FVPs to the soluton of syste of algebrac equatons. Also, we study the convergence analyss. The obtaned nuercal results show the splcty and the effcency of the proposed ethod. Keywords Fractonal Varatonal Probles, Caputo Fractonal Dervatves, Legendre Spectral Collocaton Method, Raylegh-Rtz Method, Convergence Analyss. Introducton Fractonal dervatves have recently played a sgnfcant role n any areas of scences, engneerng, flud echancs, bology, physcs and econoes. Fractonal varatonal proble s a varatonal proble n whch the perforance ndex or the objectve functonal contans at least one Caputo fractonal dervatve. The fractonal Euler-Lagrange equaton has been used to forulate fractonal varatonal probles []. The calculus of varatons has a long hstory and t has been used alost n every feld where energy prncples are applcable [] [3]. In a large nuber of probles arsng n analyss, echancs, geoetry, and so forth, t s necessary to deterne the axal and nal of a certan functonal. Because of the portant role of ths subject n scence and engneerng, consderable attenton has been receved on ths knd of probles. There are soe probles that have an portant role n the developent of the calculus of varatons [4]. How to cte ths paper: Soloua, E.M. and Khader, M.M. (5) Approxate Technque for Solvng Class of Fractonal Varatonal Probles. Appled Matheatcs, 6,

2 Agrawal [5] presented a heurstc approach to obtan dfferental equatons of fractonally daped systes. Later, Agrawal cobned the calculus of varatons and the concept of fractonal dervatves to develop Euler- Lagrange equatons for FVPs []. Also, a general fnte eleent forulaton for a class of FVPs s presented n [], where the fractonal dervatve s defned n the Reann-Louvlle sense. Fnally, Lotf and Yousef [6] ntroduced a nuercal ethod for solvng a class of FVPs wth ultple dependent varables, and ult order fractonal dervatves. They reduced the gven optzaton proble to a syste of algebrac equatons usng polynoal bass functons. For ore detals about the hstorcal coents for the varatonal probles, see [7] [8]. Spectral collocaton ethods are effcent and hghly accurate technques for nuercal soluton of non-lnear fractonal dfferental equatons (FDEs). The basc dea of the spectral collocaton ethods s to assue that the unknown soluton y( x ) can be approxated by a lnear cobnaton of soe bass functons, called the tral functons, such as orthogonal polynoals. Legendre polynoals are well known faly of orthogonal poly- noals on the nterval [,] that have any applcatons [9]-[5]. They are wdely used because of ther good propertes n the approxaton of functons. We use the Legendre collocaton ethod to dscretze FDEs to get lnear or non-lnear syste of algebrac equatons, thus greatly splfyng the proposed proble. Recently, the shfted Legendre polynoals have been used as bass functons of nuercal technques for solvng types of fractonal optal control probles, see [6] [7]. Legendre expanson ethod for solvng fractonal-order delay dfferental equatons s gven n [8]. Also, any nuercal algorths for calculatng the fractonal ntegral and the Caputo dervatve are presented n [9]. In general, we know that t s dffcult to fnd the exact soluton of FDEs, so, approxate and nuercal ethods are used to obtan the nuercal soluton []-[7]. The an a of the presented paper s concerned wth extenson of the applcaton of Legendre spectral ethod to obtan the nuercal soluton of FVPs. Also, we gve and prove an error upper bound of the approxate forula of the fractonal dervatve and study the convergence analyss. Ths paper s organzed as follows. We present few prelnares for Caputo fractonal dervatves and soe facts about shfted Legendre polynoals n Secton. In Secton 3, we study the error analyss of the ntroduced approxate forula. In Secton 4, we gve the procedure of the soluton usng the proposed ethod. In Secton 5, we ntroduce soe nuercal exaples. Fnally, soe concludng rearks are gven n the last secton.. Prelnares In ths secton, we present soe necessary defntons and atheatcal prelnares of the fractonal calculus theory that wll be requred n the present paper. Defnton The Caputo fractonal dervatve operator D of order s defned n the followng for [8] where (.) Γ ( ) ( ) D f x x = x ( f ) ( ξ ) ( ξ ) + d ξ, < < ; f x, =, Γ s Gaa functon, N, x >. For the Caputo s dervatve we have D C =, C s a constant, and, for n N and n< ; n D x = Γ ( n + ) () n x, for n N and n. Γ ( n + ) We use the celng functon to denote the sallest nteger greater than or equal to v and N = {,, }. Recall that for N, the Caputo dfferental operator concdes wth the usual dfferental operator of nteger order. For ore detals on fractonal dervatves defntons and ts propertes see [8] [9]. The well known Legendre polynoals are defned on the nterval [,] and can be deterned wth the ad of the followng recurrence forula [4] 838

3 k+ k Lk ( z) = zlk ( z) Lk ( z), k =,,, + k+ k+ where L ( z ) = and L ( z) = z. In order to use these polynoals on the nterval [ ], we defne the so called shfted Legendre polynoals by ntroducng the change of varable z = t. Let the shfted Legendre L t can be obtaned as follows polynoals Lk ( t ) be denoted by Lk ( t ). Then, k ( k+ )( t ) k k k k + ( k+ ) k+ The analytc for of the shfted Legendre polynoals Lk k k+ ( k+ )! Lk ( t) = ( ) t. = ( k )!(! ) The functon u( t ), whch s square ntegrable functon n [ ] Legendre polynoals as where the coeffcents L t = L t L t, k =,,. t of degree k s gven by (),, ay be expressed n ters of shfted u t = cl t, (3) = c are gven by In practce, only the frst ( + ) c = + u t L t d t, =,,. -ters of shfted Legendre polynoals are consdered. Then we have u t = cl t. (4) = The an approxate forula of the fractonal dervatve s gven n the followng theore. Theore [3] Let u( t ) be approxated by shfted Legendre polynoals as (4) and also suppose >, then 3. Error Analyss ( ) ( + k ) k ( + k)! = k, k, = = k= ( k)!( k)! Γ( k + ) D u t cw t, w. (5) In ths secton, specal attenton s gven to study the convergence analyss and evaluate an upper bound of the error for the proposed approxate forula. Theore [3] The Caputo fractonal dervatve of order for the shfted Legendre polynoals can be expressed n ters of the shfted Legendre polynoals theselves n the followng for where ( ) k ( ) = Θ, jk, j D L t L t, (6) k= j= + k j+ r ( ) j + k!j+ j+ r! Θ,,,., jk = j = (7), k! k! Γ k + r = j r! r! k + r+ Lea [3] For any contnuous functon u( t ) defned on [, ] wth bounded second dervatve (.e., u ( t) δ for soe constant δ ), then the coeffcents of the shfted Legendre expanson (4) s bounded n the followng for c 6δ. 3 Theore 3 For a functon u( t ). Under the two assuptons: ( ) (8) 839

4 ) u( t ) s contnuous functon on [, ] ; ) u( t ) has bounded second dervatve (.e., Then ts shfted Legendre approxaton u followng accuracy estaton u t δ for soe constant δ ). t defned n (4) converges unforly. Moreover, we have the δ L [,] = + ( 3) u t u t (9) Proof. Usng the propertes of the shfted Legendre polynoals, the orthogonal condton and the proved forula (8), we can get c L d t t c = + = + 6 u( t) u( t) = d d L [,] cl t cl t t= cl t t = = = + = + ths ples and proves the requred relaton (9). Theore 4 [3] E Dut Du t δ δ ( 3) + ( 3) 6, 3 4 = + = + The error = n approxatng Dut T E c 4. Procedure Soluton for FVPs k by Du ( t) s bounded by Θ,,. () T jk = + k= j= In ths secton, we gve a nuercal algorth usng Legendre spectral collocaton ethod for obtanng the extreal values of functonals of the general for subject to boundary condtons [ ] J y = F x, y x, D y x d, x () y = c and y = c. () To develop the forulaton for the general for (), we follow the followng steps: ) Approxate the functon y( x ) usng the forula (4) and ts Caputo fractonal dervatve D y ( x) usng the presented forula (5), then the general for of FVPs () s transfored to the followng approxated for ( ) k J = F x, cl, x cw k, x d, x (3) = = k= where w k, s defned n (5). ) Use the trapezodal ntegraton technque to copute the ntegral ter n Equaton (3) as n the followng approxated forula where Ω k ( x) = F x, cl ( x), cw x, N h J( c, c,, c) ( x) ( xn ) ( xk), Ω +Ω + Ω (4) k = ( = = k= k, ) h =, for an arbtrary nteger N, N x = h, =,,, N. 3) The extreal values of functonals of the general for (4), accordng to Raylegh-Rtz ethod gves 84

5 J J J =, =,, =, c c c whch represent + of algebrac equatons, so, after usng the boundary condtons () wth respect to (4), we obtan a syste of + algebrac equatons n the unknowns c, c,, c. 4) Solve the algebrac syste to obtan c, c,, c, then the functon y( x ) whch extrees FVPs has the for (4). 5. Nuercal Exaples In ths secton, to deonstrate the perforance of the coputatonal procedure developed above, we present nuercal results of three dfferent exaples. Proble [] Consder the followng FVP: Fnd the extreu of the functonal under the followng boundary condtons [ ] J y = ( D y( x) ) d x,, < < (5) y, y. In order to use the Legendre collocaton ethod, we approxate y( x ) wth = 3 as 3 = = (6) 3 y x = cl x. (7) = To obtan the nuercal soluton of the proposed proble (5) we follow the followng steps: ) Substtute the approxaton of the functon y( x ) fro (7) and ts approxated fractonal dervatve D y3 ( x) usng the presented forula (5), then the general for of FVP (5) s transfored to the followng approxated for 3 ( ) k J = cw k, x d, x (8) = k= where w k, s defned n (5). ) Use the trapezodal ntegraton technque to copute the ntegral ter n Equaton (8) as n the followng approxated forula N h J( c, c, c, c3) ( x) ( xn ) ( xk), Ω +Ω + Ω (9) k = where Ω ( x ) = 3 k k k, cw x, h =, for an arbtrary nteger N, x,,,,. = = = h = N N 3) The extreal values of functonals of the general for (9), accordng to Raylegh-Rtz ethod gves J J =, =, () c c also, usng the boundary condtons (6) wth respect to (7) and usng ( ) n L =, L followng two equatons 3 3 n n =, we get the c c + c c =, c + c + c + c =. () So, fro Equatons () and (), we obtan a syste of 4 algebrac equatons n the unknowns c, c, c and c 3. 4) Solve the resultng algebrac syste to obtan c, c, c and c 3, then the functon y( x ) whch extrees FVP (5) has the for (7). The behavor of the nuercal solutons of ths proble wth dfferent values of N and are gven n 84

6 Fgure and Fgure. In Fgure, the nuercal results at N = 3 for dfferent values of and the exact soluton at = are plotted. In Fgure, the nuercal results at = for dfferent values of N ( N = 5,7) and the exact soluton at = are plotted. Fro these fgures, we can conclude that the nuercal results obtaned by usng the proposed ethod are n excellent agreeent wth the exact soluton and the nuercal results obtaned usng the general fnte eleent forulaton []. Proble [] Consder the followng FVP: Fnd the extreu of the functonal under the followng boundary condtons y y J[ y] = ( D y( x) ) y( x) d, x () = =. The behavor of the nuercal soluton of ths proble wth dfferent values of N and are gven n Fgure 3 and Fgure 4. In Fgure 3, the nuercal results at N = 3 for dfferent values of and the exact soluton at = are plotted. In Fgure 4, the nuercal results for dfferent values of N ( N = 5,7) and the exact soluton at = are plotted. Fro Fgure 3 and Fgure 4, we can conclude that the nuercal results obtaned by usng the proposed ethod are n excellent agreeent wth the exact soluton and the nuercal results obtaned usng the general fnte eleent forulaton []. Proble 3 [6] Consder the followng syste of FVPs: Fnd the extreu of the functonal where [ ] J y, y = D y( x) D y( x) f ( x) + d, x (3) Fgure. The behavor of y( x ) of proble at N = 3 for dfferent values of. 84

7 Fgure. The behavor of y( x ) for proble at N = 5 and N = 7 and the exact soluton at =. Fgure 3. The behavor of y( x ) of proble at N = 3 for dfferent values of. 843

8 Fgure 4. The behavor of y( x ) for proble at N = 5 and N = 7 and the exact soluton at =. under the followng boundary condtons 4Γ 4 5Γ 5 f ( x) = x + x 3Γ 3 4Γ 4 3 4, y =, y =, y =, y =. (4) 4 For the gven proble we have y ( x) x 5 = + and y ( x) = x as nzng functons wth J[ y, y ] =. The behavor of the nuercal and exact solutons of ths proble wth dfferent values of ( =.6,.7,.8,.9,.) wth N = 5 are gven n Fgure 5. Fro ths fgure, we can conclude that the nuercal results obtaned by usng the proposed ethod are n excellent agreeent wth the exact soluton and the nuercal results obtaned usng the proposed ethod n [6]. 6. Concluson and Rearks In the presented study, Legendre spectral ethod has been successfully used to obtan the nuercal solutons of fractonal varatonal probles. In these equatons, the fractonal dervatve s consdered n the Caputo for. The fractonal dervatve s approxated by eans of the sae forula derved n [8]. The propertes of the Legendre polynoals and Raylegh-Rtz ethod are used to reduce the proposed proble to the soluton of syste of algebrac equatons. Also, the addtonal contrbuton n ths paper s gven to study the convergence analyss and deduce an error upper bound of the proposed ethod. One can easly conclude fro the presented results that the proposed ethod s a hghly good one to obtan nuercal solutons of ths knd of fractonal varatonal equatons. It s evdent that the overall errors can be ade saller by addng new ters fro the seres (7). Coparson s ade between the obtaned approxate soluton and the exact soluton wth other ethods to llustrate the valdty and the great potental of the proposed technque. All coputatons n ths paper are done usng Matlab

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