Numerical Solutions of Fractional Partial Differential Equations by Using Legendre Wavelets

Transcription

1 ngneerng Letters, 4:3, L_4_3_7 Numercal Solutons of Fractonal Partal Dfferental quatons by Usng Legendre Wavelets Hao Song, Mngxu Y *, Jun Huang, and Yaln Pan Abstract A numercal method based on Legendre wavelets s proposed for fractonal partal dfferental equatons Legendre wavelets operatonal matrces of fractonal order ntegraton and fractonal order dfferentaton are derved By usng these matrces, each term of the problem was converted nto matrx form Lastly, the equaton was transformed nto a Sylvester equaton he error estmaton of the Legendre wavelets method s gven n heorem 5 hree numercal examples are shown to demonstrate the valdty and applcablty of the method Index erms Fractonal partal dfferental equaton, Legendre wavelets, Operatonal matrx, Sylvester equaton, rror analyss I I INRODUCION N scence and engneerng, many dynamcal systems can be descrbed by fractonal-order equatons [-3] hese dynamcal systems generally orgnates n the felds of electrode-electrolyte [4], delectrc polarzaton [5], electromagnetc waves [6], vscoelastc systems [7] etc Varous materals and processes have been found to be descrbed usng fractonal calculus Anomalous dffuson has been dscussed n varous physcal felds [8-] he features of anomalous dffuson nclude hstory dependence, long-range correlaton and heavy tal characterstcs hese features can be accommodated well by usng fractonal calculus In order to model these phenomena, fractonal dervatves and fractonal partal dfferental equatons were proposed Nowadays, fractonal partal dfferental equatons have been employed as a powerful tool n complex anomalous dffuson modellng Apart from modellng aspects of these fractonal partal dfferental equatons, the numercal soluton technques are rather more sgnfcant aspects Varous numercal methods and approaches are avalable to solve lnear and nonlnear fractonal partal dfferental equatons Some analytc Manuscrpt receved January 6, 6; revsed June, 6 Hao Song s wth the School of Aeronautc Scence and ngneerng, Behang Unversty, Beng, PRChna (e-mal: buaasonghao@63com) Mngxu Y (Correspondng author) s wth the School of Aeronautc Scence and ngneerng, Behang Unversty, Beng, PRChna (e-mal: ymngxu88@63com) Jun Huang s wth the School of Aeronautc Scence and ngneerng, Behang Unversty, Beng, PRChna Yaln Pan s wth the School of Aeronautc Scence and ngneerng, Behang Unversty, Beng, PRChna methods are proposed However, numercal methods are n demand snce t s dffcult to obtan analytc solutons for each and every fractonal partal dfferental equaton orgnatng from real lfe problems Untl now, to the best of the author s nowledge, the man approach for solvng fractonal partal equatons were the fnte dfference method [, ], Laplace transform method [3], and generalzed dfferental transform method [4] hese approxmatons are valuable for researchers and scentsts hs research consdered a class of fractonal partal equatons: u u f () x y such that u(, y) u( x,) () where u x and u y are fractonal dervatves, f s the nown contnuous functon, uxy (, ) s the unnown functon, and, II LGNDR WAVLS Legendre wavelets () x are expressed as follows [5] / m ˆ ˆ n n P ˆ m x n x ( ), ; ( x), otherwse (3) where,,, nˆ n, n,,,, m,,, M s the degree of the Legendre polynomals and M s a fxed postve nteger; Pm () x are Legendre polynomals of degree m For any functon, f ( x) L [,) may be gven by the Legendre wavelets as f ( x) c ( x) (4) n m where c f ( x), ( x), and, s the nner product of f( x) and () x If the nfnte seres n quaton (4) s truncated, then M f ( x) c ( x) C ( x) (5) n m where C and ( x) are C [ c, c,, c, c, c,, M c,, c, c,, c ] M M M column vectors (6) (Advance onlne publcaton: 7 August 6)

2 ngneerng Letters, 4:3, L_4_3_7 ( x) [,,,,,,, M,,,,, ] M M For smplcty, quaton (5) s rewrtten as f ( x) c ( x) C ( x) (8) where c c, he ndex s determned by the rel -aton M ( n ) m herefore, the vectors can also be wrtten as C [ c, c,, cm, cm,, (9) c,, c,, c ] M M ( ) M M ( ) ( x) [,,,,,, M M,,,, ] Smlarly, the functon uxy (, ) over [,) [,) expressed as follows (7) () can be u u ( x) ( y) ( x) U( y) () where U [ u ] and u ( x), u, ( y) heorem [6] Any functon f( x ), defned over [,], s wth bounded second dervatve, say f ( x) M, can be expressed as the sum of Legendre wavelets, and the seres converges unformly to the functon f( x )hat s f ( x) c ( x) n m where c f ( x), ( x), and, s the nner product of f( x) and () x heorem [7] If a contnuous functon uxy (, ) defned over [,) [,) has bounded mxed fourth partal 4 u dervatve M, then the Legendre wavelets x y expanson of uxy (, ) converges unformly to t III OPRAIONAL MARICS OF INGRAION AND DIFFRNIAION FOR LGNDR WAVLS 3 Fractonal Calculus Defnton he Remann-Louvlle fractonal ntegral operator of order of a functon s defned as [3] x J f ( x) ( x ) f ( ) d, ( ) () J f ( x) f ( x) (3) Defnton he fractonal dfferental operator n Caputo sense s defned as r d f ( x), rn r ; dx D f ( x) (4) ( r) x f ( ) d, r r r ( r) ( x) he Caputo fractonal dervatve of order s also gven by ( ) r r D f x J r D f ( x), where D s the usual nteger dfferental operator of order r he relaton between the Caputo operator and Remann-Louvlle operator s gven as follows: D J f ( x) f ( x) (5) x J D f x f x f x (6) r ( ) ( ) ( ) ( ),! 3 Fractonal Order Operatonal Matrx of Integraton and Dfferentaton for Legendre Wavelets hs secton smply presents the operatonal matrx of fractonal ntegraton of Legendre wavelets [5] Frstly, the bass set of bloc pulse functons s consdered hese functons, defned over [,), are gven as follows [8], h x ( ) h; b ( x) (7), otherwse Note:,,,, and s a postve nteger value for ˆm and h Let B( x) [ b ( x), b ( x),, b ˆ ( )] m x Accordngly, suppose J B( x) F B( x) (8) where F s the fractonal ntegraton bloc pulse operatona l matrx [8], where ˆ m F h 3 ( ) Here, ( ) ( ),,,, here s a relatonshp between the Legendre wavelets and bloc pulse functons [9] ( x) B( x) (9) where [ ( x), ( x ),, ], x,,,, Legendre wavelets fractonal ntegraton operator J satsfes J ( x) P ( x) () where P s the Legendre wavelets fractonal ntegraton operatonal matrx quaton (8) and quaton (9) result n J ( x) J B( x) J B( x) F B( x) () Usng quaton () and quaton (), P ( x) P B( x) F B( x) () hen, matrx P s as follows P F (3) he fractonal dervatve of order n the Caputo sense of the vector ( x) can be expressed as D ( x) Q ( x) (4) where Q s the Legendre wavelets operatonal matrx of fractonal dfferentaton Due to the relatonshp n fractonal calculus, Q P I, matrx Q can easly be acqured by nvertng matrx P he fractonal order ntegraton and dfferentaton of the functon t are selected to verfy the effectveness of matrx P and Q he fractonal order ntegraton and dfferentaton of u() t t are obtaned as follows: () () - J u() t t and D u() t t ( ) (- ) When 5, 3, comparatve results for the fractonal ntegraton and dfferentaton are shown n Fg and Fg, respectvely (Advance onlne publcaton: 7 August 6)

3 ngneerng Letters, 4:3, L_4_3_7 quaton (8) s a Sylvester equaton he Sylvester equaton can be solved easly usng Matlaba Fg / order ntegraton of u() t t Fg / order dfferentaton of u() t t IV SOLUION OF H FRACIONAL PARIAL DIFFRNIAL QUAION Consder the fractonal partal dfferental equaton quaton () n secton If t s assumed the functon uxy (, ) n terms of Legendre seres, t can be wrtten as quaton () hen the followng can be obtaned: u ( ( x) U( y)) ( x) U ( y) x x x [ Q ( x)] U( y) ( x)[ Q ] U( y) (5) u ( ( x) U( y)) y y (6) ( ( y)) ( x) U ( x) UQ ( y) y he functon f of quaton () may also be wrtten as f ( x) F ( y) (7) where F [ f, ] Substtutng quaton (5), quaton (6) and quaton (7) nto quaton (), then ( x)[ Q ] U( y) ( x) UQ ( y) (8) ( x) F( y) Dspersng quaton (8) by ponts ( x, y ),,,, and,,,, then [ Q ] U UQ F (9) V RROR ANALYSIS In ths part, error analyss of the method s employed Let u be the followng approxmaton x u u of, u ( ) ( ) x y x x u u hen u ( x) ( y) x x (, ) heorem 5 Let the functon u x y obtaned usng x u Legendre wavelets be the approxmaton of, and x uxy (, ) has bounded mxed fractonal partal 4 dervatve x uxy (, ) M ˆ, then y u u ˆ ˆ M N 4 x x wher / u( x,y) u ( x,y) dxdy uxy (, ) u ( x),, ( y) x /,,, and ˆN s a constant Proof he orthonormalty of the seres () x, defned on[,), mples ( x)[ ( x)] dx I, where I s the dentfy matrx, then u u x x u u dxdy x x u ( ) ( ) x y dxdy = u u ( x) ( y) dxdy ( x) ( y) dxdy u ( x) dx ( y) dy u he Legendre wavelets coeffcents of functon uxy (, ) are defned by uxy (, ) u ( ) ( ) x y dxdy x / / P ( ˆ m x n) ( y) dxdy u m I n x Let x nˆ t By changng x nˆ t and dx dt, then (Advance onlne publcaton: 7 August 6)

4 ngneerng Letters, 4:3, L_4_3_7 u / m / nˆ t ( y) u, y Pm( t) dtdy t / (m ) nˆ t ( ), ( y u y d P m ( t) Pm ( t)) dy t / 3 (m ) nˆ t ( y) u, y ( P = t 3 (m ) / m m nt m m m m ( ), / ( t) P ( t)) dtdy ˆ P ( t) P ( t) P ( t) P ( t) y u y d dy t m 3 m 5 (m ) nˆ t Pm ( t) Pm ( t) Pm ( t) Pm ( t) ( y) u, y dtdy t m3 m Now, letm( t) (m ) Pm ( t) (m ) Pm ( t) (m 3) Pm ( t), then u / 5 (m ) (m )(m 3) nˆ t ( y) u, y m( t) dtdy t By solvng ths equaton, 4 nˆ t nˆ s u A(, m) u, ( ) ( ) m t m s dtds t s where A(, m) 5 (m ) (m ) (m 3) herefore 4 nˆ t nˆ s u A(, m) u, ( ) ( ) m t m s dtds t s Furthermore, the above equaton reveals m 3 ( ) 4 m t dt m 3 hus, 4 Mˆ (m 3) u A(, m) m 3 Mˆ (m ) (m 3)(m ) ( n) (m 3) Namely, 44Mˆ u 8 ( n) (m 3) herefore, u um ˆ x x u ˆ 44M ( n) (m 3) 8 44Mˆ 44Mˆ ( ) (M 5) ( ) (M 5) 8 8 M M p p M M ˆ 44M 8 p p M M ( ) (M 5) M p ˆ ˆ 44M ˆ p M N M p 8 8 M p ( ) (M 5) where ˆN s a constant Next, ˆ u u ˆ M N 8 x x hus u u ˆ ˆ M N 4 x x From ths theorem, t s evdent u u x x / when VI NUMRICAL XAMPLS xample Consder the nonhomogeneous partal dfferental equaton /5 /5 u u + f, x, y /5 /5 x y 4/5 4/5 5( x y xy ) Such that u(, t) u( x,) and f he 4 (4 / 5) numercal results for 8, m ˆ =6, =3 are shown n Fg 3, Fg 4, Fg 5 he exact soluton s xy, shown n Fg 6 Fg 3-6 llustrate the numercal solutons are n very good concdence wth the exact soluton Fg 3 Numercal soluton of 8 (Advance onlne publcaton: 7 August 6)

5 ngneerng Letters, 4:3, L_4_3_7 Fg 4 Numercal soluton of 6 Fg 7 Numercal soluton of 6 Fg 5 Numercal soluton of 3 Fg 8 Numercal soluton of 3 Fg 6 xact soluton for xample xample Consder the followng fractonal partal dfferental equaton [] /3 / u u + f, x, y /3 / x y 5/3 3/ 9 8 subect to u(, t) u( x,), f x y x y 5 ( / 3) 3 (/ ) Fg 7- show the numercal solutons for varous m and the exact soluton xy he absolute errors obtaned by Bloc Pulse Method (BPM) and Legendre Wavelets Method (LWM) for dfferent ˆm are shown n able, respectvely From Fg 7- and able, the absolute errors between numercal solutons and the exact soluton are clearly decreasngly smaller when ˆm ncreases Compared wth the approxmatons obtaned by BPM, LWM can acheve a hgher degree of accuracy Fg 9 Numercal soluton of 64 Fg xact soluton for xample (Advance onlne publcaton: 7 August 6)

6 ngneerng Letters, 4:3, L_4_3_7 ABL H ABSOLU RROR OF DIFFRN m FOR XAMPL ( xy, ) LWM BPM LWM BPM LWM BPM (,) (/8,/8) 7493e-6 379e e e-6 376e- 665e-6 (/8,/8) 33456e-6 3e-4 646e-7 538e e e-6 (3/8,3/8) 453e-5 67e e e-5 756e-9 457e-5 (4/8,4/8) 54356e-5 485e-4 476e-6 4e-4 588e e-5 (5/8,5/8) 7573e-5 695e e-6 69e e-8 469e-5 (6/8,6/8) 9346e e e-6 33e e e-5 (7/8,7/8) 653e-4 388e-3 99e-5 353e e-6 84e-5 xample 3 Consder the fractonal partal dfferental equaton as follows u u + sn( x y), x, y x y Such that u(, t) u( x,) he exact soluton of ths equaton s sn xsn y when = = he numercal soluton s shown n Fg, and the exact soluton s dsplayed n Fg, Fg 3 and Fg 4 show the approxmatons for varous values of, hey demonstrate the smplcty and power of the proposed method Compared wth the generalzed dfferental transform method n Ref [4], usng the aforementoned method can greatly reduce computaton Fg 3 Numercal soluton of =/, =/3 Fg 4 Numercal soluton of =3/7, =3/5 Fg Numercal soluton of = = VII CONCLUSION hs artcle ntroduced Legendre wavelets and wavelets operatonal matrces of fractonal ntegraton and fractonal dfferentaton he fractonal partal dfferental equatons mproved numercally va the operatonal matrces By solvng the Sylvester system, numercal solutons were obtaned In addton, the error analyss of Legendre wavelets was proposed he soluton obtaned usng the suggested method showed numercal solutons were n very good agreement wth the exact soluton ACKNOWLDGMNS he authors than the referees for ther careful readng of the manuscrpt and nsghtful comments, whch helped to mprove the qualty of the paper We would also le to acnowledge the valuable comments and suggestons from the edtors, whch vastly contrbuted to the mprovement of the presentaton of the paper Fg xact soluton of = = RFRNCS [] YL L, WW Zhao, Haar wavelet operatonal matrx of fractonal order ntegraton and ts applcatons n solvng the fractonal order dfferental equatons, Appl Math Comput, 6 () [] Feng, Qnghua, Interval oscllaton crtera for a class of nonlnear fractonal dfferental equatons wth nonlnear dampng term, IANG Internatonal Journal of Appled Mathematcs, vol43, no3, pp54-59, 3 [3] Asgar, M, Numercal Soluton for Solvng a System of Fractonal Integro-dfferental quatons, IANG Internatonal Journal of Appled Mathematcs, vol45, no, pp85-9, 5 [4] M Ichse, Y Nagayanag, Koma, An analog smulaton of nonnteger order transfer functons for analyss of electrode process, Journal of lectroanalytcal Chemstry, 33(97) (Advance onlne publcaton: 7 August 6)

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