The line method combined with spectral chebyshev for space-time fractional diffusion equation
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1 Apped and Computatona Mathematcs 014; 3(6): Pubshed onne December 31, 014 ( do: /j.acm ISS: (Prnt); ISS: (Onne) The ne method combned wth spectra chebyshev for space-tme fractona dffuson equaton I. K. Youssef 1, *, A. M. Shukur 1 Department of Mathematcs, An Shams Unversty, Caro, Egypt Department of Apped Mathematcs, Unversty of Technoogy, Baghdad, Iraq Ema address: kaoud@hotma.com (I. K. Youssef), ahmhshad@yahoo.com (A. M. Shukur) To cte ths artce: I. K. Youssef, A. M. Shukur. The Lne Method Combned wth Spectra Chebyshev for Space-Tme Fractona Dffuson Equaton. Apped and Computatona Mathematcs. Vo. 3, o. 6, 014, pp do: /j.acm Abstract: The Method of Lnes Combned wth Chebyshev Spectra Method respect to weghted resdua (Coocaton Ponts) for Space-Tme fractona dffuson equaton s consdered, the drect way w be used for approxmatng Tme fractona and the expaton of shfted frst knd of Chebyshev poynoma w be used to approxmate unknown functons, the structure of the systems and the matrces w be fund, the agorthm steps s ustrated, The tabes and fgures of the resuts of the mpementaton by usng ths method at dfferent vaues of fractona order w be shown, wth the hepng of programs of matab. Keywords: Space-Tme Fractona Dffuson Equaton, Chebyshev-Spectra Method, Fnte Dfference Method 1. Introducton Dffuson equaton provde an mportant too for modeng a numerous probems n engneerng, physcs and others scence, the generazaton of the nteger order dffuson equaton to space fractona order, tme fractona order or more genera s the space-tme fractona order dffuson equaton (S-TFPDE), s more mportant to study t and to fnd the easer ways to sove t. When tme dependent PDEs are soved numercay by spectra methods[1, ], the pattern s usuay the same: empoy spectra treatment to space dependency and use fnte dfference n tme or eave the tme dependency to obtan a system of ordnary dfferenta equatons (ODE) n tme. In ths paper, S-TFPDE has been consdered, =, +,; 0< 1<,0,0, Wth nta and boundary condtons 0,=,=0 ; 0,,0=; 0. The dea of the treatment n ths paper depends on the method of dscretzaton n tme (the method of nes) [3]. In (1) () ths method the tme nterva 0, s dvded nto subntervas,,, " ( # =$ #%, # &,' =1,,, of ength ( = ) ", and the probem s transformed to sove fractona boundary vaue probem aong each tme eve. For the fractona boundary vaue probem a spectra coocaton method s empoyed. Spectra method [1, ] nvove seekng the souton to dfferenta equatons n terms of a seres of known, smooth functons. Spectra methods may be vewed as an extreme deveopment of the cass of dscretzaton schemes for dfferenta equatons known generay as the method of weghted resduas (WRM). The key eements of the WRM are the tra functons and the test functons. The tra functons are used as the bass functons for a truncated seres expanson of the souton. The test functons are used to ensure that the dfferenta equaton s satsfed as cosey as possbe by the truncated seres expanson. The choce of test functon dstngushes whch used spectra schemes, s coocaton. In the coocaton approach the test functons are the transated Drac deta functons centered at the coocaton ponts. The treatment consdered n ths work depends on the use of the coocaton approach. Coocaton The core of the coocaton method s defnton of the resdue functon and the coocaton ponts. Consder the boundary vaue probem, [4, 5].
2 Apped and Computatona Mathematcs 014; 3(6): L + =,, -, +, = +-=0. Where L s any operator. An approxmate souton. w not, n genera, satsfy (3) exacty, and assocated wth such an approxmate souton s the resdua defned by (3) /. = L.. (4) It s cear that f. s the exact souton then /. =0. 4 In the coocaton the tra functons, 1+ # #3, are used as the bass functons for a truncated seres expanson of the souton = # #. The shfted Chebeshev poynomas 4 are used as the bass functon + #, and the Gauss-Labatto ponts # =0.5cos? #@ A+0.5, ' = 0,1,,B are used as the 4 coocaton ponts. Accordngy, /C. 4 # D= L. 4 #, ' = 0,1,,B. (5) Represents a system of B+1 agebrac equatons n B+1 unknowns 6 7. Determnaton of 6 7 enabes us to wrte the approxmate souton as a near combnaton of the bases functons.. The Caputo s Fractona Dervatves I HE G F, [] For any rea functon f(x) 6 K, the Caputo fractona dervatve of order s: L M = P M ГK% QK%% UhWW >0;Y 1< Y? R RS AK QTQ, (6) 0 ^ _ < ] L M Z 0 ^ Z `aq,a = c (7) \ Г_+1 [ Г_ +1 Z% ^ _ The Caputo fractona dervatve operator s near n the sense that for any two constants A, B and any two functons {f(x) and g(x)} 6 K, one can wrte L M {e g}=e L M g L M () 3. Chebyshev Poynomas CPs [1,, 9] Chebyshev poynomas (CPs) of the frst knd denoted by, 1 1, a = 0,1,, and defned as =1, = and = % %,a. The sgnfcance can be mmedatey apprecated from the fact that the functon cosaj s a Chebyshev poynoma (9) functon of cosj. Specfcay, for a 0 cosaj = cosj, (10) The anaytc form of the Chebyshev poynoma of degree a s gven by o p q r = a Гa m k 1 m! Гa m+1 3 where o r s the foor of =1. %, (the nteger part of ). (11) It s cear that CP s a poynoma of degree a, The CP s have n zeros n the nterva [-1, 1], gven by # = cos j=0,1,,,n-1, The CP s have n+1 extremes ponts n same nterva [-1,1], gven by # = cos #@, j= 0,1,,, n. (1) The CP are orthogona functons n the sense that P ) t M ) p M % u%m q 0 Y T = v Y = a 0. (13) x Y = a = 0 In order to use CPs on ntervas of the form,,- other than the nterva 1,1 one can use the change of the varabe =? y%z A 1, whch transforms the ponts {%z,,- to 1,1, conversey, =? {%z A+? {sz A, transforms the ponts 1,1 to,,- Accordngy, the shfted Chebyshev poynomas (SHCP), of degree n on nterva [a,b] can be wrtten as =?, A 1}, a,a 1, -, =1. Where s the frst knd CP defned n (11). 4. Fractona Dervatves of SHCP (14) The Caputo s Fractona dervatve of SHCP defned n (14), of order >0, can be wrtten wth the hepng of propertes n (6-) and the defnton of the SHCP formua (1). Lemma: The Caputo s fractona dervatves of order Y 1< <Y,Y =, of shfted Chebyshev poynoma of degree a,0 a,a 1, on nterva [a,b] s zl M = a o p q r K -,, K% k 1 Гa m % ;a,m,y, Гm+1 3 (15)
3 c c 33 I. K. Youssef and A. M. Shukur: The Lne Method Combned wth Spectra Chebyshev for Space-Tme Fractona Dffuson Equaton ;a,m,y 0, Y>a m ] 1 ƒ ГY +1, Y=a m = 17s% %K % %K \ k Г+Y +1Гa m Y +1 ƒ [?, Y<a m -, A7, Proof Consder the Caputo s fractona dervatves of power functons and the nearty property ^ _=0 0, L M e Z = c e Г_+1 UhWW e ^Q `aq,a Г_ +1 Z%, ^ _. Wrte formua (11) of SHCP and appyng the Caputo s dervatve on t wth usng the nearty property, yeds zl M = zl M?, -, A 1} o p q r = a k 1Гa m% m! Гa m+1 3?, -, A 1} % zl M (16) The Caputo s dervatve part n ths formua can be determned as foows, Use the defnton of CFD, zl M? M%z {%z A 1}% = then the foowng three cases are obtaned K% z M L K M? M%z {%z A 1}%, -If Y >a m,l M K? M%z {%z A 1}% =0, -If Y =a m, L K M? M%z {%z A 1}% % =a m! -,, Whch s the constant term and Y 0, usng R-L FI to opten, z K% M L K M?, % -, A 1} = a m! * M%zt ГK%s, Thus zl M?, % -, A 1} Гa m+1, K% -, K ГY +1, -If Y a m, L M K? M%z {%z A 1}% % -, Г% s =? Г% %Ks {%z AK? M%z %K {%z A 1}%. Snce a m Y Z s,at a m 0, by appyng the bnoma expanton to the ast term and consequenty usng the R-L FI term by term, there s z K% M?, % %K -, A 1} =? A % %K K% {%z z M,? {%z A} % %K, = % %K -, *? {%z A % %K%7 z M % %K K% k a m Y 7 % %K = 1 % %K Гa m Y+1 k Г+1Гa m Y ? A 7 K% {%z z M, 7, = Гa m Y+1, K% % %K % Šp q t? Г% %K%7sГ7sK%s {%z A7? M%z Thus obtaned zl M?, % -, A 1} = =Гa m+1 K -,, K% {%z A 1}7, 0, Y >a m, ] 1 ƒ ГY +1, Y =a m, 17s% %K % %K \ k Г+Y +1Гa m Y +1 ƒ [?, Y <a m. -, A7, Whch competes the prove 5. Descrpton of the Method The consdered method has two steps, the frst step s usng the drect way to approxmate the tme fractona part, the dffuson equaton w be system of space fractona ordnary dfferenta equatons formua (17), the second step s usng the shfted Chebyshev poynomas to approxmate the unknown functons then, furmua(17) yed the system of agebrac equaton formua () wth unknown coeffcents. Usng the spectra method wth the weghted resdua (coocaton method) to fnd the unknown coeffcents. Consder the space-tme fractona dffuson equaton Œ ŽM,y = Œ ŽM,y +, ; 0 < 1<, Œy ŒM 0, 0, respect to nta and boundary condton
4 Apped and Computatona Mathematcs 014; 3(6): I.C,0=; 0, B.C 0,=,=0 ; 0, To sove S-TFPDE, and t s ceary to show that =,,m =0,1,, 1,1<, s the numerca functon at tme-step = m, = ). " The drect way (fnte dfference method for Caputo s fractona dervatves) s L y, s k - # %#s %# + τ, L y s % #3 0< 1, = k - #s - # %# + s - #3 - # = '+1 % ' %, = Г %, for a 0,, =,, so that one can rewrte the S-TFPDE eq(1) by usng the approxmaton of tme fractona as s =C L M +1 - D % + k- # - #s %# # (17) In genera the dea of the Spectra Method(SM), [6, 7], s fndng the tra approxmaton for the unknown functons, by usng known bass and assumng ths unknown functons equa fnte sum of ths bass,(n ths paper, shfted Chebyshev poynomas have been used), as 4 = 6 7 7, UhWW a,a 1, (1) Where m = 0,1,, 1,1<, t s we known formua (17) can be wrtten as 4 =,UhWW, L, and M denoted an operator whch maps set of the functons U nto set of G, s.t U, G and D s the prescrbed doman, snce the approxmate soutons gven n eq(1) w be not (n genera) satsfy (1) exacty, then we defne the resdua /,6 as /,6 = 4, (19) The resdua depended on the coeffcents 6. ow the probem s, how one can fnd (n+1) coeffcents 6 7 n equaton (19), to fnd the soutons n equaton(1)? The answer of ths queston s, by usng the weghted resdua method. Weghted Resduas Methods (WRM)[4,5], n ths method one has many ways to defne weghted functon on D to make the weghted ntegra on the D equa zero, n ths work the coocaton method s used, 1, hw ` `,^a _^aq Defned =, now 0 hwu^qw one can wrte š /,6 = 0, (0) œ UhWW,^ =0,1,,B 1, are the coocaton ponts of Chebyshve whch are gven n eq(1), from the defnton of weghted functon one gets /,6 = 4, or 4 =. (1) ow from equatons (17), (1), and eq(1) one can rewrtes the formuas (17) as k6 s 7 7 =k6 7 C L M +1 - D 7 % + $ 6 %# #3 7 C- # - #s D 7 +k6 7-7 & + s () From defnton of resdua eq(1) and the coocaton ponts eq(1), one can obtan a near system of agebrac equatons n the unknowns {6 s 7 } n the form e 6 7 s =-, where e6 7 =. (3) Where the rght hand sde - s defned expcty n terms of known data n the prevous tme eves. - =e6 % 7 + ž # 6 %# #3 7 +Ÿ6 7 + s, (4) 6 7 s = s = s 6 6,m = 0,1,,, 6 s,m =0,1,,. - # - #s - # - #s ž # = - # - #s - # - #s - # - #s - # - #s e = 6 7 = 6 6, = 6, - # - #s - # - #s - # - #s
5 334 I. K. Youssef and A. M. Shukur: The Lne Method Combned wth Spectra Chebyshev for Space-Tme Fractona Dffuson Equaton e= L M 0- L M 0- L M 0- L M 0- L M 0- L M 0- we can cacuate the startng vectors 6 7, from the nta condton as Snce 0,,, 0,1, then e umerca Exampes (5) Exampe wth known exact souton s consdered. The exampe s chosen such that the behavor of the souton has dfferent characterzatons wth space and tme rangng from poynoma, snusoda and exponentay decay. In the cacuatons, the expct fnte dfference methods s used to approxmate the fractona tme dervatve and obtaned a fractona boundary vaue probem aong every tme eve. The souton of the fractona boundary vaue probem s appro xmated by a truncated shfted Chebyshev poynomas 6 7 7, as a tra souton n the coocaton method wth the coocaton ponts # cos #@, ' 0,1,,a. Aso, three graphs have been ntroduced, n the frst and dfferent vaues of ( 0.6, 0.7, 0.9 and 1.0), n the second 1.5 and dfferent vaues of ( 0.6, 0.7, 0.9 and 1.0), and n the thrd group 1 and dfferent vaues of ( 1.6, 1.7, 1.9 and.0). Moreover, tabes ntroduced whch contan the vaues of the coeffcents of the shfted Chebyshev Poynoma 6, (m s the tme eve, s the order of tme dervatve and s the order of the space dervatve). The coeffcents are determned aong each tme step by sovng a near system of equatons n the form defned n (), Exampe Consder the S-TFPDE L y,0l M,,, 0 < α 1 < β ; 0t1 and, 0 x 1, where, snxew %y 0W %y?x sn?x x AA; e 01, Subject to the boundary condtons 0, 1, 0, and the nta condton,,0 sn x, 0 x 1,whose exact souton s, W %y sn x. Soutons Consder a tra souton of the form ± 4 k ² ² 6, ow for the frst nta souton one can cacuate t from the nta condton by,0 snx, ^ 0,1,,,7; UVWW 0.5 A0.5, sove ±, UVWW,,^ 0,1,,7. The resuts of the souton of ths exampe, By hepng of Matab are gven n the foowng fgures(1-3) and tabes (1-3). Fgure 1. the exact souton and the numerca souton at dfferent orders of α wth fxed β=, ,a 7, h=0.015, Tabe 1. ( ,a 7. I H.,ª I H.,ª I H.,ª I,ª H I µ Co-pont ¹.² ¹ ¹.º E E ¹ E-0.60E-0.64E-0 3E E E E E E E E E E E E E Ģ
6 Apped and Computatona Mathematcs 014; 3(6): Fgure. the exact souton and the numerca souton at dfferent orders of α wth fxed β=1.5, ( , a 7, h=0.015, 0.01 Tabe. ths resuts at( ,a 7. I,. I,. I,. I,ª H I µ Co-pont ¹.² ¹ ¹.º E E E E ¹ E-0.64E-0.64E-0 3E E E E E E E E E E E E E Ģ Fgure 3. the exact souton and the numerca souton at dfferent orders of β wth fxed α =1,( ,a 7, h=0.015, 0.01 Tabe 3. ( ,a 7. H I µ Co-pont ² ¹ ¼ E E E ² ¹.¼ ² ¹.¼ E E E E ² ¹.¼ E-0.79E-0..7E-0. 3E E-05.94E E E E E E E E E E E Ģ 7. Dscusson and Concusons It s known that many dfferenta equatons can be soved by usng the WRM. The coocaton method s the smpest method among the WRM. The effcency of usng ths method depends on the seecton of the coocaton ponts. The treatment depends on transformng the tme dependent paraboc FPDE nto a famy of fractona boundary vaue
7 336 I. K. Youssef and A. M. Shukur: The Lne Method Combned wth Spectra Chebyshev for Space-Tme Fractona Dffuson Equaton probems aong tme eves wth the hep of the fnte dfference method. the shfted Chebyshev poynomas of the frst knd s used as the bases functon of the tra souton, and the Gauss Lobatto ponts are used as the coocaton ponts., thus the probem s transformed to one of sovng an agebrac near systems n the coeffcents of the bases functons ncuded n the tra souton. Thus an expct approxmaton s obtaned for the souton aong each tme eve whch enabes to approxmate the souton at any pont aong ths tme eve not ony at some grd ponts as n the fnte dfference. Athough the agebrac system s dense n ths treatment n comparson wth those obtaned wth the fnte dfference method the number of equatons s reatvey very sma. From the graphs one can see that ths treatment performs more effcenty n the nteger case (as the fractona orders of the dervatves approaches the cassca nteger case as the approxmate souton approach the exact souton). References [1] G. Caporae, and M. M. Cerrato, Usng Chebyshev Poynomas to Approxmate Parta Dfferenta Equatons, Comput Econ, 010. [] B. Costa, Spectra Methods for Parta Dfferenta Equatons, A Mathem-Journ, Vo. 6, December, 004. [3] Z. C. Kuruo gu, Weghted-resdua methods for the souton of two-partce Lppmann-Schwnger equaton wthout parta-wave decomposton, rxv, physcs.comp-ph, 17 Ju, 013. [4]. e, J. Huang, W. Wang, and Y. Tang, Sovng spata-fractona parta dfferenta dffuson equatons by spectra method, Journa of Statstca Computaton and Smuaton, 013 [5] S. A. Odejde, and Y. A. S. Aregbesoa, appcatons of method of weghted resduas to probems wth sem-nfnte doman, Rom. Journ. Phys., Vo. 56, os. 1-, P. 14 4, Bucharest, 009. [6] R. K. Rektorys, The method of dscretzaton n tme and parta dfferenta equatons, Sprnger etherands, Dec 31, 19. [7] L. Xanjuan, and Xu. Chuanju, A Space-Tme Spectra Method for the Tme Fractona Dffuson Equaton, Ths work supported by atona SF of Chna, Hgh Performance Scentfc-Computaton Research Program CB31703, 005. [] J. P. Boyd, Chebyshev and Fourer Spectra Methods, Second Edton, Unversty of Mchgan, Ann Arbor, Mchgan , 000. [9] M. Dar, and M. Bashour, Appcatons of Fractona Cacuus, App- Math- Sce- Vo. 4, , 010.
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