Copyrigh 22 Tech Science Press CMES, vol.88, no.3, pp.229-243, 22 Haar Wavele Operaional Mari Mehod for Solving Fracional Parial Differenial Equaions Mingu Yi and Yiming Chen Absrac: In his paper, Haar wavele operaional mari mehod is proposed o solve a class of fracional parial differenial equaions. We derive he Haar wavele operaional mari of fracional order inegraion. Meanwhile, he Haar wavele operaional mari of fracional order differeniaion is obained. The operaional mari of fracional order differeniaion is uilized o reduce he iniial equaion o a Sylveser equaion. Some numerical eamples are included o demonsrae he validiy and applicabiliy of he approach. Keywords: Haar wavele, operaional mari, fracional parial differenial equaion, Sylveser equaion, numerical soluion. Inroducion Wavele analysis is a relaively new area in differen fields of science and engineering. I is a developing of Fourier analysis. Wavele analysis has been applied widely in ime-frequency analysis, signal analysis and numerical analysis. I permis he accurae represenaion of a variey of funcions and operaors, and esablishes a connecion wih fas numerical algorihms [Beylkin, Coifman, and Rokhlin (99)]. Funcions are decomposed ino summaion of basic funcions, and every basic funcion is achieved by compression and ranslaion of a moher wavele funcion wih good properies of smoohness and localiy, which makes people analyse he properies of localiy and ineger in he process of epressing funcions [Li and Luo (25); Ge and Sha (27)]. Consequenly, wavele analysis can describe he properies of funcions more accurae han Fourier analysis. Fracional differenial equaions are generalized from classical ineger order ones, which are obained by replacing ineger order derivaives by fracional ones. Fracional calculus is an old mahemaical opic wih hisory as long as ha of ineger order calculus. Several forms of fracional differenial equaions have been proposed in sandard models, and here has been significan ineres in developing nu- College of Sciences, Yanshan Universiy, Qinhuangdao, Hebei, China
23 Copyrigh 22 Tech Science Press CMES, vol.88, no.3, pp.229-243, 22 merical schemes for heir soluion. Fracional calculus and many fracional differenial equaions have been found applicaions in several differen disciplines, boh physiciss and mahemaicians have also engaged in sudying he numerical mehods for solving fracional differenial equaions in recen years. These mehods include variaional ieraion mehod (VIM) [Odiba (2)], Adomian decomposiion mehod (ADM) [EI-Sayed (998); EI-Kalla (2)], generalized differenial ransform mehod (GDTM) [Odiba and Momani (28); Momani and Odiba (27)], generalized block pulse operaional mari mehod [Li and Sun (2)] and wavele mehod [Chen and Wu e al. (2)]. The operaional mari of fracional order inegraion for he Legendre wavele and he Chebyshev wavele [Jafari and Yousefi (2); Wang and Fan (22)] have been derived o he fracional differenial equaions. In [Saeedi and Moghadam e al. (2); Saeedi and Moghadam (2)], a CAS wavele operaional mari of fracional order inegraion has been used o solve inegro- differenial equaions of fracional order. In his paper, our sudy focuses on a class of fracional parial differenial equaions: α u α + β u β = f (,) subjec o he iniial condiions u(,) = u(,) = (2) where α and β α β are fracional derivaive of Capuo sense, f (,) is he known coninuous funcion, is he unknown funcion, < α,β. There have been several mehods for solving he fracional parial differenial equaions. Podlubny [Podlubny (999)] used he Laplace Transform mehod o solve he fracional parial differenial equaions wih consan coefficiens. Zaid Odiba and Shaher Momani [Odiba and Momani (28)] applied generalized differenial ransform mehod o solve he numerical soluion of linear parial differenial equaions of fracional order. Our purpose is o proposed Haar wavele operaional mari mehod o solve a class of fracional parial differenial equaions. We inroduce Haar wavele operaional mari of fracional order inegraion wihou using he block pulse funcions. Here, we adop he orhogonal Haar wavele mari which is differen from he Haar wavele mari in he Ref. [Ray (22)]. We need no calculae he inverse of Haar wavele mari in his way. 2 Definiions of fracional derivaives and inegrals In his secion, we give some necessary definiions and preliminaries of he fracional calculus heory which will be used in his aricle [Podlubny (999) and Odi- ()
Haar Wavele Operaional Mari Mehod 23 ba (26)]. Definiion. The Riemann-Liouville fracional inegral operaor J α of order α is given by J α u() = ( T ) α u(t )dt, α > (3) Γ(α) J u() = u() (4) Is properies as following: (i) J α J β u() = J α+β u(), (ii) J α J β u() = J β J α u(), (iii) J α γ = Γ(γ+) Γ(α+γ+) α+γ Definiion 2. The Capuo fracional differenial operaor D α is given by D α u() = { d r u() d r, α = r N; Γ(r α) u (r) (T ) ( T ) α r+ dt, r < α < r. (5) The Capuo fracional derivaives of order α is also defined as D α u() = J r α D r u(), where D r is he usual ineger differenial operaor of order r. The relaion beween he Riemann- Liouville operaor and Capuo operaor is given by he following epressions: D α J α u() = u() (6) J α D α r u() = u() u (k) ( + ) k k!, > (7) k= 3 Haar wavele and funcion approimaion For [,], Haar wavele funcions are defined as follows [Chen and Wu e al. (2)]: h () = m h i () = 2 j/2 k, < k /2 2 j 2 j 2 m j/2 k /2, < k 2 j 2 j, oherwise where i =,,2,...,m, m = 2 M and M is a posiive ineger. j and krepresen ineger decomposiion of he inde i, i.e. i = 2 j + k.
232 Copyrigh 22 Tech Science Press CMES, vol.88, no.3, pp.229-243, 22 For arbirary funcion L 2 ([,) [,)), i can be epanded ino Haar series by = m i= m c i, j h i ()h j () (8) j= where c i, j = h i()d h j()d are wavele coefficiens, m is a power of 2. Le H m () = [h (),h (),...,h m ()] T, H m () = [h (),h (),...,h m ()] T, hen Eq.(8) will be wrien as = H T m() C H m (). In his paper, we use wavele collocaion mehod o deermine he coefficiens c i, j. These collocaion poins are shown in he following: l = l = (l /2)/m, l =,2,...,m. (9) Discreing Eq.(8) by he sep (9), we can obain he mari form of Eq.(8) U = H T C H () where C = [c i, j ] m m and U = [u( i, j )] m m. H is called Haar wavele mari of order m, i.e. h ( ) h ( ) h ( m ) h ( ) h ( ) h ( m ) H =....... h m ( ) h m ( ) h m ( m ) From he definiion of Haar wavele funcions, we may know easily ha H is a orhogonal mari, hen we have C = H U H T () 4 Haar wavele operaional mari of fracional order inegraion and differeniaion The inegraion of he H m () can be approimaed by Chen and Hsiao [Chen and Hsiao]: H m (s)ds = PH m () (2) where P is called he Haar wavele operaional mari of inegraion.
Haar Wavele Operaional Mari Mehod 233 Now, we are able o derive he Haar wavele operaional mari of fracional order inegraion. For his purpose, we may make full use of he definiion of Riemann- Liouville fracional inegral operaor J α which is given by Definiion. Haar wavele operaional mari of fracional order inegraion P α will be deduced by P α H m () =J α H m () where =[J α h (),J α h (),...,J α h m ()] T [ = ( T ) α h (T )dt, Γ(α) Γ(α) Γ(α) ( T ) α h m (T )dt =[Ph (),Ph (),...,Ph m ()] T α ] T ( T ) α h (T )dt,..., Ph () = [,) (3) m Γ(α + ), < k 2 Ph i () = j 2 j/2 k λ (), < k /2 2 j 2 j m 2 j/2 k /2 λ 2 (), < k (4) 2 j 2 j 2 j/2 k λ 3 (), < 2 j where λ () = ( k ) α Γ(α + ) 2 j ; λ 2 () = Γ(α + ) λ 3 () = Γ(α + ) ( k 2 j ( k 2 j ) α 2 Γ(α + ) ) α 2 Γ(α + ) ( k /2 2 j ( k /2 2 j ) α ; ) α + Γ(α + ) ( k 2 j ) α. The derived Haar wavele operaional mari of fracional inegraion is P α = (P α H) H T. Le D α is he Haar wavele operaional mari of fracional differeniaion. According o he propery of fracional calculus D α P α = I, we can obain he mari D α by invering he mari P α. For insance, if α =.5,m = 8, we have P /2 =
234 Copyrigh 22 Tech Science Press CMES, vol.88, no.3, pp.229-243, 22.7549.28.72.579.56.289.223.89.28.39.72.565.56.289.89.389.579.565.2337.32.73.52.229.44.72.72.2337.73.52.89.389.52.44.788.89.25.9,.223.89.73.229.788.89.25.289.289.52.788.89.56.56.73.788 D /2 =.229.4694.4589.396.6488.568.85.8.4694 2.678.4589.8783.6488.568.279.28.396.8783 2.8964.47.975.7547.783.432.4589.4589 2.8964.975.7547.8.28.7547.432 4.8424.524.67.5..85.279.975.783 4.8424.524.67.568.568.7547 4.8424.524.6488.6488.975 4.8424 The fracional order inegraion and differeniaion of he funcion was seleced o verify he correcness of mari P α and D α. The fracional order inegraion and differeniaion of he funcion u() = is obained in he following: J α u() = and D α u() = Γ(2) Γ(α + 2) α+ Γ(2) Γ(2 α) α. When α =.5,m = 32, he comparison resuls for he fracional inegraion and differeniaion are shown in Fig. and Fig. 2, respecively. 5 Numerical soluion of he fracional parial differenial equaions Consider he fracional parial differenial equaion Eq.(). If we approimae funcion by using Haar wavele, we have = H T m() C H m () (5)
Haar Wavele Operaional Mari Mehod 235.8.7 Our resul Eac soluion.6.5.4.3.2...2.3.4.5.6.7.8.9 Figure :.5-order inegraion of he funcion u() =..4.2 Our resul Eac soluion.8.6.4.2..2.3.4.5.6.7.8.9 Figure 2:.5-order differeniaion of he funcion u() =.
236 Copyrigh 22 Tech Science Press CMES, vol.88, no.3, pp.229-243, 22 Then we can ge α u α = α (Hm()CH T m ()) α = [ α ] H m () T CH m () α = [D α H m ()] T CH m () = H T m()[d α ] T CH m () (6) β u β = β (H T m()ch m ()) β The funcion f (,) of Eq.() can be also epressed as = H T m()c β (H m ()) β = H T m()cd β H m () (7) f (,) = H T m() F H m () (8) where F = [ f i, j ] m m. Subsiuing Eq.(6), Eq.(7) and Eq.(8) ino Eq.(), we have H T m()[d α ] T CH m () + H T m()cd β H m () = H T m()fh m () (9) Dispersing Eq.(9) by he poins ( i, j ), i =,2,,m and j =,2,,m, we can obain [D α ] T C +CD β = F (2) Eq.(9) is a Sylveser equaion. The Sylveser equaion can be solved easily by using Malab sofware. 6 Numerical eamples To demonsrae he efficiency and he pracicabiliy of he proposed mehod based on Haar wavele operaional mari mehod, we consider some eamples. Eample : Consider he following nonhomogeneous parial differenial equaion /4 u /4 + /4 u = f (,),,, /4 such ha u(,) = u(,) = and f (,) = 4(3/4 + 3/4 ) 3Γ(3/4). The numerical resuls for m = 8, m = 6, m = 32 are shown in Fig. 3, Fig. 4, Fig. 5. The eac soluion of he parial differenial equaion is given by which is shown in Fig. 6. From he Fig. 3-6, we can see clearly ha he numerical soluions are in very good agreemen wih he eac soluion.
Haar Wavele Operaional Mari Mehod 237.8.6.4.2.5.2.4.6.8 Figure 3: Numerical soluion of m = 8.8.6.4.2.5.2.4.6.8 Figure 4: Numerical soluion of m = 6.8.6.4.2.5.2.4.6.8 Figure 5: Numerical soluion of m = 32
238 Copyrigh 22 Tech Science Press CMES, vol.88, no.3, pp.229-243, 22.8.6.4.2.5.2.4.6.8 Figure 6: Eac soluion for Eample Eample 2: Consider he following fracional parial differenial equaion /3 u /3 + /2 u = f (,),,, /2 subjec o he iniial condiions u(,) = u(,) =, f (,) = 92 5/3 5Γ(2/3) + 83/2 2 3Γ(/2). Fig. 7- show he numerical soluions for various m and he eac soluion 2 2. The absolue error for differen m is shown in Table. From he Fig. 7- and Table, we can conclude ha he numerical soluions are more and more close o he eac soluion when m increases..8.6.4.2.5.2.4.6.8 Figure 7: Numerical soluion of m = 6 Eample 3: Consider he below fracional parial differenial equaion α u α + β u = sin( +),, β
Haar Wavele Operaional Mari Mehod 239.8.6.4.2.5.2.4.6.8 Figure 8: Numerical soluion of m = 32.8.6.4.2.5.2.4.6.8 Figure 9: Numerical soluion of m = 64.8.6.4.2.5.2.4.6.8 Figure : Eac soluion for Eample 2
24 Copyrigh 22 Tech Science Press CMES, vol.88, no.3, pp.229-243, 22 Table : The absolue error of differen m for Eample 2 (,) Eac m = 8 m = 6 m = 32 m = 64 soluion (,) 2.465e-6 5.933e-7 3.748e-8 4.398e-9 (/8,/8).2 8.3295e-5.8652e-5.4567e-6 2.3435e-7 (2/8,2/8).39 4.4467e-4 4.953e-5 2.253e-5.247e-5 (3/8,3/8).98.33e-3.7432e-5 9.6749e-6 6.5843e-6 (4/8,4/8).625 2.957e-3.2828e-4 8.7434e-5 7.27e-5 (5/8,5/8).526 5.474e-3 4.833e-4 3.627e-4.3465e-4 (6/8,6/8).364 8.973e-3.364e-3 8.943e-4 6.643e-4 (7/8,7/8).5862.3454e-2 2.938e-3.468e-3 8.342e-4 such ha u(,) = u(,) =. When α = β =, he eac soluion of his parial differenial equaion is sin sin. We can achieve is numerical soluion which is shown in Fig., and he eac soluion is shown in Fig. 2. Fig.3 and Fig.4 show he numerical soluions for differen values of α,β. Here, we may ake m = 32. They demonsrae he simpliciy, and powerfulness of he proposed mehod. Compared wih he generalized differenial ransform mehod in he Ref. [6], aking advanage of above mehod can grealy reduce he compuaion. Moreover, he mehod in his paper is easy implemenaion..8.6.4.2.5.2.4.6.8 Figure : Numerical soluion of α = β =
Haar Wavele Operaional Mari Mehod 24.8.6.4.2.5.2.4.6.8 Figure 2: Eac soluion of α = β =.8.6.4.2.5.2.4.6.8 Figure 3: Numerical soluion of α = /2, β = /3.8.6.4.2.5.2.4.6.8 Figure 4: Numerical soluion of α = 3/7, β = 3/5
242 Copyrigh 22 Tech Science Press CMES, vol.88, no.3, pp.229-243, 22 7 Conclusion Anoher operaional mari for he Haar wavele operaional mari of fracional differeniaion has been derived. The fracional derivaives are described in he Capuo sense. This mari is used o solve he numerical soluions of a class of fracional parial differenial equaions effecively. We ranslae he fracional parial differenial equaion ino a Sylveser equaion which is easily o solve. Numerical eamples illusrae he powerful of he proposed mehod. The soluions obained using he suggesed mehod show ha numerical soluions are in very good coincidence wih he eac soluion. The mehod can be applied by developing for he oher fracional problem. Acknowledgemen: This work is suppored by he Naural Foundaion of Hebei Province (A222347). References Beylkin, G.; Coifman, R.; Rokhlin, V. (99): Fas wavele ransform and numerical algorihms I. Commun. Pur. Appl. Mah., vol. 44, pp.4-83. Chen Y. M.; Wu Y. B. e al. (2): Wavele mehod for a class of fracional convecion-diffusion equaion wih variable coefficiens. Journal of Compuaional Science, vol., pp.46-49. Chen, C. F.; Hsiao, C. H. (997): Haar wavele mehod for solving lumped and disribued- parameer sysems. IEE Proc.-Conrol Theory Appl., vol.44, pp.87-94. EI-Sayed, A. M. A. (998): Nonlinear funcional differenial equaions of arbirary orders. Nonliear Analysis, vol.33, pp.8-86. EI-Kalla, I. L. (2): Error esimae of he series soluion o a class of nonlinear fracional differenial equaions. Commun. Nonlinear Sci. Numer. Simula., vol.6, pp.48-43. Ge, Z. X.; Sha, W. (27): Wavele analysis heorem and MATLAB applicaion, Elecronic Indusrial Publicaion, Beijing. Jafari H.; Yousefi S. A. (2): Applicaion of Legendre waveles for solving fracional differenial equaions Compuers and Mahemaics wih Applicaion, vol.62 pp.38-45. Li, Z. C.; Luo, J. S. (25): Wavele analysis and is applicaion. Elecronic Indusrial Publicaion, Beijing. Li, Y. L.; Sun, N. (2): Numerical soluion of fracional differenial equaions using he generalized block pulse operaional mari. Compuers and Mahemaics
Haar Wavele Operaional Mari Mehod 243 wih Applicaion, vol.62 pp.46-54. Momani, S.; Odiba, Z. (27): Generalized differenial ransform mehod for solving a space and ime-fracional diffusion-wave equaion. Physics Leers A, vol.37, pp.379-387. Odiba, Z. M. (2): A sudy on he convergence of variaional ieraion mehod. Mahemaical and Compuer Modelling, vol. 5, pp.8-92. Odiba, Z.; Momani, S. (28): Generalized differenial ransform mehod: Applicaion o differenial equaions of fracional order. Applied Mahemaics and Compuaion, vol.97, pp. 467-477. Odiba, Z.; Momani, S. (28): A generalized differenial ransform mehod for linear parial differenial equaions of fracional order. Applied Mahemaics Leers, vol.2, pp.94-99. Odiba, Z. (26): Approimaions of fracional inegrals and Capuo fracional derivaives. Applied Mahemaics and Compuaion, vol.78 pp.527-533. Podlubny, I. (999): Fracional Differenial Equaions. Academic press. Ray S. S (22): On Haar wavele operaional mari of general order and is applicaion for he numerical soluion of fracional Bagley Torvik equaion Applied Mahemaics and Compuaion, vol.28, pp.5239-5248. Saeedi, H.; Moghadam, M. M. e al. (2): A CAS wavele mehod for solving nonlinear Fredholm inegro-differenial equaions of fracional order. Commun. Nonlinear Sci. Numer. Simula., vol.6, pp.54-63. Saeedi, H.; Moghadam, M. M. (2): Numerical soluion of nonlinear Volerra inegro- differenial equaions of arbirary order by CAS waveles. Applied Mahemaics and Compuaion, vol.6, pp.26-226. Wang, Y. X.; Fan, Q.B. (22): The second kind Chebyshev wavele mehod for solving fracional differenial equaions. Applied Mahemaics and Compuaion, vol.28, pp.8592-86.