Numerical Solution for the Variable Order Fractional Partial Differential

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1 Numercal Soluo for he Varable Order Fracoal Paral Dffereal Equao wh Berse polyomals 1 Jsheg Wag, Lqg Lu, 3 Lechu Lu, 4 Ymg Che 1, Frs Auhor Yasha Uversy, wsheg010@163.com *,Correspodg Auhor Yasha Uversy, lulqg_yaya@163.com 3,4 Yasha Uversy, YSULIULECHUN@SINA.CN, cheym.ysu.edu.c Absrac I hs paper, we develop a framework o oba umercal soluo of he varable order fracoal paral dffereal equao usg Berse polyomals. he ma characersc behd hs approach s ha we derve a fracoal order operaoal marx of Berse polyomals. Wh he operaoal marx, he equao s rasformed o he producs of several depede marxes whch ca also be regarded as he sysem of lear equaos afer dspersg he varable. By solvg he lear sysem of algebrac equaos, he umercal soluos are acqured. Oly a small umber of Berse polyomals are eeded o oba a sasfacory resul. Numercal examples are provded o show ha he mehod s compuaoally effce. Keywords: Berse polyomals, he varable order fracoal paral dffereal equao, Numercal soluo,he absolue error 1. Iroduco I rece years, fracoal calculus has araced may researchers successfully dffere dscples of scece ad egeerg. Oe of he ma advaages of he fracoal calculus s ha he fracoal dervaves provde a superor approach for he descrpo of memory ad heredary properes of varous maerals ad processes [1]. May umercal mehods usg dffere kds of fracoal dervave operaors for solvg dffere ypes of fracoal dffereal equaos have bee proposed. he mos commoly used oes are Adoma decomposo mehod (ADM) [][3], Varaoal erao mehod (VIM) [4], geeralzed dffereal rasform mehod (GDM) [5-7], geeralzed block pulse operaoal marx mehod [8] ad wavele mehod [9][10] ad oher mehods [11-13]. Recely, more ad more researchers are fdg ha umerous mpora dyamcal problems exhb fracoal order behavor whch may vary wh space ad me. hs fac llusraes ha varable order calculus provdes a effecve mahemacal framework for he descrpo of complex dyamcal problems. he cocep of a varable order operaor s a much more rece developme, whch s a ew oreao scece. Dffere auhors have proposed dffere defos of varable order dffereal operaors, each of hese wh a specfc meag o su desred goals. he varable order operaor defos recely proposed he scece clude he Rema-Louvle defo, Capuo defo, Marchaud defo, Combra defo ad Grüwald defo [14-18]. Sce he kerel of he varable order operaors s oo complex for havg a varable-expoe, he umercal soluos of varable order fracoal dffereal equaos are que dffcul o oba, ad have o araced much aeo. herefore, he developme of umercal echques o solve varable order fracoal dffereal equaos has o ake off. o he bes of he auhors kowledge, here are few refereces arse o he dscusso of varable order fracoal dffereal equao. I several emergg, mos auhors adop dfferece mehod o deduce a approxmae scheme. For example, Combra [16] employed a cosse approxmao wh frs-order accurae for he soluo of varable order dffereal equaos. Soo e al. [19] proposed a secod-order Ruge Kua mehod whch s cossg of a explc Euler predcor sep followed by a mplc Euler correcor sep o umercally egrae he varable order dffereal equao. L e al. [0] suded he sably ad he covergece of a explc fe-dfferece approxmao for he varable-order fracoal dffuso equao wh a olear source erm. Zhuag e al. [1] obaed explc ad mplc Euler approxmaos for he fracoal adveco dffuso olear equao of varable-order. Amg a varable-order aomalous subdffuso equao, Che e al. [] employed wo umercal schemes oe Ieraoal Joural of Advacemes Compug echology(ijac) Volume 6, Number 3, May 014

2 fourh order spaal accuracy ad wh frs order emporal accuracy, he oher wh fourh order spaal accuracy ad secod order emporal accuracy. However, as far as we kow, o oe had aemped o seek he umercal soluo of he varable order fracoal equaos. So hs paper, we roduce he Berse polyomals o seek he umercal soluo of he varable order fracoal paral equao. Wh he smple srucure ad perfec properes [3][4], he Berse polyomals play a mpora role varous areas of mahemacs ad egeerg. hose polyomals have bee wdely used he soluo of egral equaos ad dffereal equaos [3-9]. I hs paper, our sudy focuses o a class of fracoal paral dffereal equao as follows: x u x, u x, x x f x,, (1) Subec o he al codos,0 g x 0, u x u h () x x u x, / x ad u x, / f x, s he kow couous fuco, u x, s he ukow fuco, x where are fracoal dervave of Capuo sese, 0, 1. he remder of he paper s orgazed as follows: Secos ad 3 are preparave, he defos ad properes of he varable order fracoal order egrals ad dervaves ad Berse polyomals are gve Secos ad 3. I Seco 4, he fracoal operaoal marxes of Berse polyomals are derved ad we appled he operaoal marx o solve he equao as gve a begg. I Seco 5, we prese some umercal examples o llusrave he mehod ad o demosrae effcecy of he mehod. We ed he paper wh a few cocludg remarks Seco 6.. Basc defos ad properes of he varable order fracoal egrals ad dervaves Form he defos ad properes of fracoal order calculus, we ca derve he defos ad properes of he varable order fracoal order calculus [14-16]. ) Rema-Louvlle fracoal egral of he frs kd wh order 1 1 Ia u ud a, 0 Re 0 (3) ) Rema-Louvlle fracoal dervae of he frs kd wh order m 1 d u Da u d m1 m m a m 1 m d (4) bu Da Ia u u. ) Capuo s fracoal dervae wh order 1 u 0 u 0 (5) D u u d where 0 1. If we assume he sarg me a perfec suao, we ca ge he defo as follows: 3

3 1 D u ud 01 1 (6) 0 wh he defo above, we ca ge he followg formula D x x a 3. Berse Polyomals ad her properes 3.1. he defo of Berse Polyomals bass he Berse Polyomals of degree are defed by By usg he bomal expaso of1 x Now, we defe: where we ca have where Clearly, B, x x 1 x 0 1 : 1,, 3, Eq. (9) ca be expressed as: k k (9) k 0 k 1 1 B x x x x x B x, B x,, B x Φ 0, 1,, (10) x x (7) (8) Φ A (11) A (1) fuco approxmao x 1, xx,,, x (13) A Φ (14) 1 x x A fuco f x L 0,1 ca be expressed erms of he Berse Polyomals bass. I pracce, oly he frs 1 erms of Berse Polyomals are cosdered. Hece 4

4 c0 c1 c where,,, c. he we have, Φ (15) 0 f x cb x c x 1 f, x whereq s a 11marx, whch s called he dual marx of x 1 Q ΦxΦ xdx 0 1 A 0 x A x dx 1 A 0 x x dx A where H s a Hlber marx: c Q Φ (16) AHA Φ. (17) H 3 (18) u x, L 0,1 0,1 as followg: We ca also approxmae he fuco ux, u B xb x,,, Φ UΦ (19) 0 0 where u00 u01 u0 u u u u0 u 1 u AdU ca be obaed as followg: U Q 1 x,, u x, Q 1 (1) 3.3. Covergece aalyss U (0) Suppose ha he fuco f :0,1 s m 1 m C 1 0,1 Y Spa{ B, B, B, B }.Ifc Φ x f, ad 0, 1,,, f ou of Y, he he mea error boud s preseed as follows: mes couously dffereable, s he bes approxmao of 5

5 m m 1 where M max f x, S max 1 x, x. m3 MS f c Φ () 1! m 3 x[0,1] 0 0 Proof: We cosder he aylor polyomals xx 0 m xx0 f1 x f x0 f x0 x x0 f x0 f x0 m! whch we kow m1 x x0 m m1 f x f1 x f 0,1 1! sce c Φ x s he bes approxmao of f, so we have m1 x x0 m 1 m f c Φ f f1 f x f 0 1 x dx m1 1 f dx 0 1! M m 1! xx dx m3 M S m1! m3 Ad akg square roos we have he above boud. 4. Proposed mehod for he umercal soluo of he fracoal paral dffereal equao Now, cosder Eq. (1). If we approxmae he fuco u x, wh he Berse Polyomals, ca be wre as Eq. (19).he we have x x ux, Φ xuφ x x x x x Φ x x UΦ x m 6

6 x x x x AUΦ 1 x 1 x 0 x x AUΦ x 1 x x x 0 x 0 x x AUΦ 1 x x 0 0 x 1 x 1 x Φ A MA UΦ (3) Also we have u x, Φ Φ x x x x 1 UANA Φ 1 1 Φ xua Φ xua Φ Φ U UA UΦ Φ (4) 7

7 Now we defe x 0 x 0 x M (5) 1 x 0 0 x 1 x we defe N (6) M, N are called he fracoal order operaoal marx of Berse polyomals. Subsug Eq. (3), Eq. (4) o Eq. (1), we have 1 1 Φ x A MA UΦ Φ x UANA Φ f x, (7) Dspersg Eq. (7) by he posx, 1,,, ; 1,,,, by usg Mahemaca x sofware, we ca obau whch s ukow. A las, we ca oba he umercal soluo as, u x Φ x UΦ. 5. umercal examples o demosrae he effcecy ad he praccably of he proposed mehod based o Berse polyomals mehod, we prese some examples ad fd her soluo va he mehod descrbed he prevous seco. Example 1: sx1 /3 cos 1 /3 ux, ux, sx1 /3 cos 1 /3 x ux,010x 1x u0, 0 x, 0,10,1 where f f x, cos 5 sx x x 5 3 cos x 8 9 x+ s x coscos cos s x 8 s x 5 s x s x

8 u x, 10x 1x 1. he exac soluo of he above equao s We solved he problem by adopg of he echque descrbed seco 4 ad by makg use of Mahemaca 9.0. akg 3 k 1 k 1, we ca oba he , dspersg x, k 1,,3; k 1,,3 marxu as follows: U he umercal soluo s ux, x Φ UΦ, as he marx U s gve above, ad , Φ Φ x x x x x x x he absolue error bewee he exac soluo ad he umercal soluo s dsplayed as follows: akg 4 Fgure 1. he absolue error for Example 1 of 3 k 1 k , dspersg x, k 1,, 5; k 1,, 5 U s dsplayed as follows: U , he marx 9

9 he umercal soluo s ux, x Φ Φ UΦ, as he marxu s gve above, ad x 1 x x 3 x 6 1 x x 4 1 x x 3 x 4, Φ he absolue error bewee he exac soluo ad he umercal soluo s dsplayed as follows: Fgure. he absolue error for Example 1 of 4 Also whe s defe, he more pos we ake, he more accurae umercal soluo we would ge. he followg fgures expla he fac. ( x, s he umber of he x, ) Fgure 3. Numercal soluo of, 5 x Fgure 4. Numercal soluo of, 8 x 30

10 Example :,05xx 1 u0, 0, x, 0,10,1 1 x /3 1 /3 u x,, 1 x/3 1 /3 x u x where f x, Fgure 5. he exac soluo of Example 1 ux f x, 5 x x 1x 3 5 1x x 40 53x x 5x x x he exac soluo of he above equao su x, x x 15 3, dspersg x, k 1,, 5; k 1,, 5 Whe 3 k 1 k s dsplayed as follows: U he umercal soluo s ux, x, he marxu Φ UΦ, as he marx U s gve above, ad , Φ Φ x x x x x x x he absolue error bewee he exac soluo ad he umercal soluo s dsplayed as follows: 31

11 Whe 4 Fgure 6. he absolue error for Example of 3 k 1 k , dspersg x, 1,, 7; 1,, 7 he marxu as follows: U he umercal soluo s ux, x, we ca oba Φ UΦ, as he marxu s gve above, where Φ x x 4 x 3 x x x x x 3 x Φ he absolue error bewee he exac soluo ad he umercal soluo s dsplayed as follows: 3

12 Example 3: x u x 1s x /3 1 /3 1s x /3 1 /3,0 x 1 3 Fgure 7. he absolue error for Example of 4 ux f x, x, 0,1 0,1 u x,, u 0, 1 where f x, 1 5 s x x 18 1 x s x s x 5 s x he exac soluo of he above equao s u x, x 11 3 akg, dspersg x, 1,,3; 1,,3 marxu as follows: U he umercal soluo s ux, x k 1 k 1,we ca oba he Φ UΦ, as he marx U s gve above, where 1 1, Φ 1 1 Φ x x x x x he absolue error bewee he exac soluo ad he umercal soluo s dsplayed able1. 33

13 able1. he absolue error bewee he exac soluo ad umercal soluo whe =0.00 =0.14 =0.9 =0.43 =0.57 =0.71 =0.86 =1.00 x= E00.9E-03.33E E E E E E00 x= E E-0.45E E E E-01.75E E-01 x= E E E-0.06E-0 1.1E-0 3.4E E-01.47E-01 x= E E E E-0.34E E E-01.78E-01 x= E00 7.E E E E E E E-01 akg 3 k 1 k 1, he marx U s , dspersg x, 1,,3; 1,,3 dsplayed as follows: U he umercal soluo s ux, Φ xuφ, as he marx U s gve above, where 3 Φ , Φ x x x x x x x he absolue error bewee he exac soluo ad he umercal soluo s dsplayed able. able. he absolue error bewee he exac soluo ad umercal soluo whe 3 =0.00 =0.14 =0.9 =0.43 =0.57 =0.71 =0.86 =1.00 x= E E E E E E E E+00 x= E E E E E E E E+00 x= E00.E E E-15.E E E-16.E-15 x= E E E E E E E+00.66E-15 x= E00.E-15.66E-15.E E E E E-15 34

14 akg 4 k 1 k , dspersg x, 1,,,6; 1,,,6 U s dsplayed as follows: U he umercal soluo s ux, x, he marx Φ UΦ, as he marxu s gve above, where Φ x x 4 x 3 x x x x x 3 x Φ he absolue error bewee he exac soluo ad he umercal soluo s dsplayed as follows. able 3. he absolue error bewee he exac soluo ad umercal soluo whe 4 =0.00 =0.14 =0.9 =0.43 =0.57 =0.71 =0.86 =1.00 x= E E E E E E E E+00 x= E E-16.E E E E-16.66E E-15 x= E E E E E-15.66E E E-14 x= E E E E E-16.E E E-15 x= E E E E E E E E-14 From ables 1-3, Fgures1-, 5-7, we ca see ha he absolue error s very y ad oly a small umber of Berse polyomals are eeded whe 3. Whe, s o surprsg ha he absolue error s very bg. As s mpossble o ge sasfacory resuls wh usg he polyomals of h degree o approxmae he polyomals of 3h. he calculag resuls also show ha combg wh Berse polyomals, he mehod hs paper ca be effecvely used he umercal soluo of he fracoal paral equao. A he same me he feasbly of he mehod ca be also proved. From he above resuls, he umercal soluos are good agreeme wh he exac soluo. 5. Coclusos hs arcle uses he Berse polyomals mehod o solve a class of he varable order fracoal paral dffereal equao by combg Berse polyomals wh he properes of fracoal dffereao. Acually we derve a fracoal operaoal marx usg Berse polyomals. he marx s used o solve he umercal soluos of a class of fracoal paral dffereal equaos effecvely. We raslae he al equao o he produc of some releva marxes whch ca also 35

15 be regarded as he sysem of lear equaos afer dspersg he varable. Ad s easy o solve by he leas square mehod. Numercal examples llusrae he powerful of he proposed mehod. he soluos obaed usg he suggesed mehod show ha umercal soluos are very good cocdece wh he exac soluo. he mehod ca be appled by developg for he oher fracoal problem. s However, here are may ssues o be resolved, such as he seco 1 0,1 0,1 rasformed o X effors of all of us. 6. Ackowledgeme: 0, 0,, or he equaos are olear ad so o. hs requres he [1] hs work s suppored by he Naural Foudao of Hebe Provce (A ). [] hs work s suppored by Qhuagdao research ad developme program of scece ad echology, Adapve Boudary Eleme Mehod of precso rollg process smulao. (0101B019) [3] hs work s suppored by Qhuagdao echology Bureau 013 research ad developme proecs of scece ad echology. (0130A03) he auhors also graefully ackowledge he helpful commes ad suggesos of he revewers, whch have mproved he preseao. 7. Refereces [1] Leda Galue, S.L. Kalla, B.N. Al-Saqab, Fracoal exesos of he emperaure feld problems ol sraa, Appled Mahemacs ad Compuao, Elsever, vol. 186, o.1, pp , 007. [] I. L. EI-Kalla, Error esmae of he seres soluo o a class of olear fracoal dffereal equaos. Commu. Nolear Sc. Numer. Smulae, Elsever, vol. 16, o.3, pp , 011. [3] Ahmed. M. A. EI-Sayed, Nolear fucoal dffereal equaos of arbrary orders. Nolear Aalyss: heory, Mehods & Applcaos, Elsever, vol. 33, o., pp , [4] Zad. Odba, A sudy o he covergece of varaoal erao mehod, Mahemacal ad Compuer Modellg, Elsever, vol.51, o. 9-10, pp , 010. [5] Shaher Moma, Zad. Odba, Geeralzed dffereal rasform mehod for solvg a space ad me-fracoal dffuso-wave equao, Physcs Leers A, Elsever, vol.370, o. 5-6, pp , 007. [6] Zad. Odba, Shaher Moma, A geeralzed dffereal rasform mehod for lear paral dffereal equaos of fracoal order, Appled Mahemacs Leers, Elsever, vol. 1, o., pp , 008. [7] Zad. Odba, Shaher Moma, Geeralzed dffereal rasform mehod: Applcao o dffereal equaos of fracoal order, Appled Mahemacs ad Compuao, Elsever, vol. 197, o., pp , 008. [8] Yualu L, Ng Su, Numercal soluo of fracoal dffereal equaos usg he geeralzed block pulse operaoal marx, Compuers ad Mahemacs wh Applcao, Elsever, vol. 6, o.3, pp , 011. [9] L Zhu, Qb Fa, Solvg fracoal olear Fredholm egro-dffereal equaos by he secod kd Chebyshev wavele. Commucaos Nolear Scece ad Numercal Smulaao, Elsever, vol. 17, o.6, pp , 01. [10] Mgxu Y, Ymg Che, Haar wavele operaoal marx mehod for solvg fracoal paral dffereal equaos. Compuer Modelg Egeerg &Sceces, ech Scece, vol. 88, o.3, pp. 9-44, 01. [11] Zad. Odba, Shaher Moma, A algorhm for he umercal soluo of dffereal equaos of fracoal order, J. Appl. Mah. Iform, Elsever, vol. 6, o. 1-, pp. 15 7, 008. [1] Ka Dehelm, Nevlle J. Ford, Mul-order fracoal dffereal equaos ad her umercal soluo, Appled Mahemacs ad Compuao, Elsever, vol. 154, o.3, pp , 004. [13] M. Javd, A. Golbaba, Modfed homoopy perurbao mehod for solvg sysem of lear 36

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