Research Article Finite Difference and Sinc-Collocation Approximations to a Class of Fractional Diffusion-Wave Equations

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1 Joural of Applied Maheaics Volue 4, Aricle ID 5363, pages hp://dx.doi.org/.55/4/5363 Research Aricle Fiie Differece ad Sic-Collocaio Approxiaios o a Class of Fracioal Diffusio-Wave Equaios Zhi Mao,, Aiguo Xiao, Zuguo Yu, adlogshi Hua Key Laboraory for Copuaio ad Siulaio i Sciece ad Egieerig, Xiaga Uiversiy, Xiaga, Hua 45, Chia Maheaics ad Iforaio Egieerig Depare, Togre Uiversiy, Togre, Guizhou 5543, Chia Correspodece should be addressed o Zhi Mao; zxu98@63.co Received February 4; Revised 3 May 4; Acceped 9 Jue 4; Published July 4 Acadeic Edior: azi I. Mahudov Copyrigh 4 Zhi Mao e al. This is a ope access aricle disribued uder he Creaive Coos Aribuio Licese, which peris uresriced use, disribuio, ad reproducio i ay ediu, provided he origial wor is properly cied. We propose a efficie uerical ehod for a class of fracioal diffusio-wave equaios wih he Capuo fracioal derivaive of order α. This approach is based o he fiie differece i ie ad he global sic collocaio i space. By uilizig he collocaio echique ad soe properies of he sic fucios, he proble is reduced o he soluio of a syse of liear algebraic equaios a each ie sep. Sabiliy ad covergece of he proposed ehod are rigorously aalyzed. The uerical soluio is of 3 αorder accuracy i ie ad expoeial rae of covergece i space. uerical experies deosrae he validiy of he obaied ehod ad suppor he obaied heoreical resuls.. Iroducio Fracioal diffusio-wave equaios have araced cosiderable aeio as geeralizaio of he classical diffusio/wave equaio by replacig he ieger-order ie derivaive wih a fracioal derivaive of order α ( < α < )[]. I ca be derived fro he physical case of a geeral rado iedepede velociy fucio equipped wih a algebraic iecorrelaio fucio (bu wih a Gaussia space correlaio) []. Owig o he aoalous superdiffusio of paricles, ay elecroageic, acousic, echaical, ad biological resposes ca be odelled accuraely by fracioal diffusiowave equaios, for exaple, he propagaio of sress waves i viscoelasic solids [3, 4]. There exis soe aalyical ehods o fid exac soluios of fracioal diffusio-wave equaios [5 9]. However, i is usually difficul or eve ipossible o achieve he exac soluios i ay cases, ad oes have o resor o uerical ehods. Copared wih he cosiderable lieraure o uerical soluios of fracioal diffusio equaios [ 5], oly a few wors have bee carried ou i he uerical ehods of fracioal diffusio-wave equaios. The hooopy aalysis ehod [6] ad he Adoia decoposiio ehod [7] were used o cosruc aalyical approxiae soluios of fracioal diffusio-wave equaios, respecively. Su ad Wu [8] preseed a full discree differece schee for he iiial-boudary-value proble of a diffusio-wave equaio by iroducig wo ew variables ad obaied he sabiliy ad covergece properies by usig eergy ehod. I [9], uder he wea soohess codiios, wo fiie differece schees wih firs-order accuracy i eporal direcio ad secod-order accuracy i spaial direcio were proposed o solve he sae proble. Murillo ad Yuse [] developed a explici differece ehod where he L discreizaio forula is used for solvig fracioal diffusiowave equaios. Due o he olocal aure of fracioal derivaives, all previous soluios have o be saved o copue hesoluioahecurreielevel.iaeshesoragevery expesive whe low-order ehods are eployed for spaial discreizaio. Ihispaper,weproposeauericalehodixig fiie differece wih sic collocaio o solve a class of fracioal diffusio-wave equaios. The sic ehod is widely used o solve iegral, iegro-differeial, ordiary differeial, ad parial differeial equaios [ 7]. As far as we ow, here are very liied papers o solvig fracioal

2 Joural of Applied Maheaics differeial equaio by he sic ehod [8, 9]. The prese ehod has grea advaage i sorage requiree because he sic-collocaio ehod eeds fewer grid pois o produce highly accurae soluio whe i is copared wih a low-order ehod. The reaider of his paper is orgaized as follows. I Secio, we iroduce soe ecessary defiiios ad releva resuls for developig his ehod. Secio 3 is devoed o he fiie differece ad sic-collocaio discreizaio for he fracioal diffusio-wave equaios. As a resul, a syse of liear algebraic equaios is fored, ad he soluios of he cosidered probles are obaied. Secio 4 is cocered wih he sabiliy ad covergece aalysis. I Secio 5, soe uerical experies are give o deosrae he effeciveess of he proposed ehod ad he obaied heoreical resuls. A brief coclusio eds his paper i he fial secio.. oaios ad Soe Preliiary Resuls I his secio, we iroduce soe basic defiiios ad releva resuls of he fracioal calculus [3, 3] ad sic fucios. Defiiio (see [3]). Le α R +.TheoperaorJ α a defied o L [a, b] by J α a f () Γ (α) (s) α f (s) ds () a for a b is called he Riea-Liouville fracioal iegral operaor of order α.forα,oesesj a : I,ha is, he ideiy operaor. Defiiio (see [3]). Le α R + ad α.thecapuo fracioal differeial operaor C D α a for a bis defied as C D α a f () Jα a D f () Γ (α) (s) α f () (s) ds. The sic fucios ad is properies are discussed horoughly i [, 3]. Foray h>, he raslaed sic fucios wih equidisa space odes are give as a S (, h)(z) sic ( zh ),,±,±,..., (3) h where he sic fucios are defied o he whole real lie by sic (x) { si (πx), x, { πx {, x. If f is defied o R,heforayh>he series C(f, h) (z) () (4) f (h) S (, h)(z) (5) is called he Whiaercardialexpasio off wheever his series coverges. f cabeapproxiaedby rucaig (5). To cosruc our eeded approxiaios o he ierval [a, b], we choose φ (x) l ( xa bx ) (6) which aps a fiie ierval [a, b] oo R. φ is a coforal ap which aps he eye-shaped doai i he coplex zplae D E {z C : arg (za bz ) <d π } (7) oo he ifiie srip D S of he coplex z-plae D S { z C : I ( z) <d π }. (8) The basis fucios o [a, b] areaeobehecoposie raslaed sic fucios φ (x) h S φ (, h)(x) S(, h) (φ (x)) sic ( ). (9) h Thus oe ay defie he iverse iage of he equidisa space ode {ih} as x i φ (ih) a+beih +e, i,±,±,... () ih The class of fucios such ha he ow expoeial covergece rae exiss for he sic ierpolaio is deoed by B(D E ) addefiedihefollowig. Defiiio 3 (see [, 3]). Le B(D E ) be he class of fucios f which are aalyic i D E ad saisfy φ (x+l) f (z) dz, x ±, () where L {iυ : υ < d (π/)} ad f (z) dz < () D E o he boudary of D E (deoed D E ). The followig heore gives he error which resuls fro differeiaig he rucaed cardial series. Theore 4 (see [, ]). Le φ f/g B(D E ) ad le sup x [a,b],π/h π/h ( d dx ) [g (x) e iφ(x) ] C h, (3) for,,...,;hereg is a weigh fucio ad C is a cosa depedig oly o, φ, adg. Assue ha here exiss a cosa C such ha f (x) g (x) C [ + e φ(x) ] ; β (4) e βφ(x)

3 Joural of Applied Maheaics 3 here β is a posiive cosa. The aig h πd/β, oe has d f (x) dx f(x ) g(x ) C 3 (+)/ exp ( πdβ) d dx [g (x) S φ (, h)(x)] (5) for all,,...,;herec 3 is a cosa depedig oly o, φ, g, d, β,adf. The above expressios show ha he sic ierpolaio o B(D E ) coverges expoeially. We also require he followig derivaives of he coposie raslaed sic fucios evaluaed a he odes:, i,,i [S φ (, h)(x)] xxi {, i. δ () δ (),i d dφ [S φ (, h)(x)] xxi δ (),i d dφ [S φ (, h)(x)] xxi 3. The Fiie Differece ad Sic-Collocaio Discreizaio (6) {, i, { () i {(i) h, i. (7) π { 3h, i, { () i { (i) h, i. (8) We cosider he followig fracioal diffusio-wave equaios wih a ohoogeeous field [33]: α u (x, ) c α u (x, ) x + f (x, ), a<x<b, > K (9) wih he iiial codiios u (x, ) u (x, ) φ(x), ad he boudary codiios ψ(x), a<x<b () u (a, ), u(b, ), >, () where x [a,b]ad >are space ad ie variables, c ad K are cosas, ad f(x, ) deoes he field variable. Here he ie-fracioal derivaive is defied as he Capuo fracioal derivaive, ad <α<. May auhors refer o he fracioal equaio (9) as he fracioal diffusio-wave equaio whe <α<, which is expeced o ierpolae he diffusio equaio ad he wave equaio [7, 8]. 3.. Teporal Discreizaio by a Fiie Differece Schee. Firs, we derive a fiie differece schee for eporal discreizaio of his equaio. Le : τ,,,,...,where τ is he ie sepsize. I order o discreize he ie-fracioal derivaive by usig a fiie differece approxiaio [8, 34], we iroduce he followig leas. Lea 5 (see [8]). Suppose f() C [, ].The α f () d τ α ( ) α [b f( ) (b b )f( )b f( )] α [α + 3α 3α (+α )] ax f () τ3α, where <α<ad () b (+) α α,,,,... (3) I is direc o chec ha b >b > >b >, b as. (4) Le φ (x, ) α u (x, ) α Γ (α) ad defie he eporal grid fucios u (x, s) s ds (s) α, (5) U u(x, ), Φ φ(x, ),,,,... (6) By he Taylor expasio, i follows ha U (/) τ (U U ) +r τ, (7) c Φ(/) (U (/) ) xx + K f(/) +r τ, (8) where r, r are cosas. Based o Lea 5,oehas Φ τ α Γ (3α) [b U (b l b l )U l b U ] Hece, +O(τ 3α ), Φ (/) (Φ +Φ ) τ α Γ (3α) [b U (/) +r 3 τ3α, l,,3,... l (9) (b l b l )U l(/) b U ] (3)

4 4 Joural of Applied Maheaics where r 3 are cosas. Le δ U (/) (/τ)(u U ).I view of (7), oe has Φ (/) τ α Γ (3α) [b δ U (/) l (b l b l )δ U l(/) b U ] τ 3α + Γ (3α) [b r (b l b l )r l ]+r 3 τ3α. l (3) The subsiuig (3)io(8)adobservigU ψ(x),oe obais τ α cγ (3α) [b δ U (/) where l (b l b l )δ U l(/) b ψ] (U (/) ) xx + K f(/) +R T, (3) R T τ 3α cγ (3α) [b r (b l b l )r l ] c r 3 τ3α +r τ. l (33) I follows fro (3) ha oe ca cosruc easily he followig seidiscree fiie differece schee for (9): τ α cγ (3α) [b δ u (/) l (u (/) ) xx + K f(/), which is equivale o where (b l b l )δ u l(/) b ψ] cτ α Γ (3α) (Δ +Δ ) [(u ) xx +(u ) xx ] (34) + K f(/),,3,..., (35) Accordig o (33), his schee is of (3 α)-order accuracy. A rigorous aalysis of he covergece rae will be provided laer. The above schee ca be rewrie io b cτ α Γ (3α) u (u ) xx b b cτ α Γ (3α) u cτ α Γ (3α) Δ + (u ) xx + K f(/). Specially, for he case, he schee siply reads b cτ α Γ (3α) u (u ) xx b cτ α Γ (3α) (u + τψ) + (u ) xx + K f(/). So (37) ad(38) ogeher wih he iiial codiio (37) (38) u u(x, ) φ(x), a<x<b (39) ad he boudary codiios u a u(a, ), u b u(b, ), (4) for a coplee seidiscree proble. 3.. Space Discreizaio by he Sic-Collocaio Mehod. ex we cosider space discreizaio o (37) by he siccollocaio ehod. We selec he collocaio pois x i by (). The space discreizaio proceeds by approxiaig he soluio based o he coposie raslaed sic fucios (9) u u u j S φ (j, h) (x). (4) j Δ b u (b b )u, Δ l (b l b l +b l )u l +(b b )u b τψ. (36) The uow paraeers u j will be deeried by he collocaio ehod. I is oed ha he approxiaio i (4) saisfies he boudary codiios i (4) sices φ (j, h)(x), j,+,...,are zero whe x eds o a ad b.

5 Joural of Applied Maheaics 5 ow subsiuig (4)io(37)adcollocaigi + pois x i,weobai b cτ α Γ (3α) ju j S φ (j, h) (x i ) b b cτ α Γ (3α) + j [ l u j u j j S φ (j, h) (x i ) S φ (j, h) (x i) (b l b l +b l ) u l j u j S j τ α Γ (3α) φ (j, h) (x i) j S φ (j, h) (x i )+(b b )u (x i ) where i,+,...,. To obai a arix represeaio of he above equaio, we le A C () b cτ α Γ (3α) I (+) (+), B (q ji) (+) (+), b b cτ α Γ (3α) I (+) (+), C (l) b l b l +b l cτ α I Γ (3α) (+) (+), C () b b cτ α Γ (3α) I (+) (+), D b cτ α Γ (3α) I (+) (+), U [u,u +,...,u ]T, Ψ[ψ(x ),ψ(x + ),...,ψ(x )] T, τb ψ(x i )] + K f(/) (x i ), (4) F (/) K [f(/) (x ), f (/) (x + ),...,f (/) (x )] T, (45) wherei,+,...,.based o(7)ad(8), we le q ji : S φ (j, h) (x i)φ (x i )δ () j,i +(φ (x i )) δ () j,i. (43) where he arix I is he ideiy arix ad l, 3,...,. Therefore, a each ie sep, we ge he followig syse of +liear equaios wih +uow paraeers u j, ad his syse ca be expressed i a arix for QU P, (46) Hece, (4)is reduced iediaely by (6)ad(43)o b cτ α Γ (3α) u i j q ji u j b b cτ α Γ (3α) u i cτ α Γ (3α) [ + l (b l b l +b l )u l i +(b b )u (x i )τb ψ(x i )] j q ji u j + K f(/) (x i ), (44) where PBU + l QAB, For he iiial codiio (39), we have C (l) U l +DΨ+F (/). (47) U [φ(x ),φ(x + ),...,φ(x )] T. (48) Cosequely (46) ca be solved easily for he uow coefficies U. Hece he approxiaio soluio give i (4)cabeobaied. 4. Sabiliy ad Covergece Aalysis of he Derived Mehod To aalyse he sabiliy ad covergece of he derived ehod, he ier produc is defied by u, υ b uυ dx a wih he correspodig or u (u, u) (/),whichwillbe used hereafer. We firs give he followig leas.

6 6 Joural of Applied Maheaics Lea 6 (see [8]). For ay QQ,Q,Q 3,...ad υ, here Q i,υ L (a, b), i ca be verified ha Applyig Lea 6 o he er i lef side of (54), we obai i i b Q i l (b l b l )Q il b i υ, Q i (α) α Q i i α υ,,,3,..., (49) τ α cγ (3α) b δ u (/) cγ (α) α l b υ, τδ u (/) τ δ u (/) (b l b l )δ u l(/) (55) where b l is defied i (3). cγ (3α) α υ. Lea 7. Suppose ha u,,, 3,...,saisfy τ α cγ (3α) [b δ u (/) l (u (/) ) xx +f (/), u (b l b l )δ u l(/) b υ] (5) φ, <x<, (5) u (a), u (b). (5) O he oher side, based o he fac ha δ u (/) (a) δ u (/) (b), a sraighforward calculaio of he righ ers i (54)gives (u (/) ) xx,τδ u (/) τ (u (/) ) x,δ (u (/) ) x (u ) x +(u ) x,(u ) x (u ) x The oe has he esiae (u ) x (u ) α x + cγ (3α) υ +cγ(α) α Proof. Basedo(5), we have l τ (53) f(/). [ (u ) x (u ) x ]. f (/),τδ u (/) τ [cγ (α) α f(/) + cγ (α) α δ u (/) ]. Subsiuig (55)ad(56)io(54)yieldsiequaliy(53). (56) τ α cγ (3α) b δ u (/) l b υ, τδ u (/) (b l b l )δ u l(/) Suppose ha u,,,,..., saisfy he codiios of Theore 4. Subsiuig he approxiaio soluio u give i (4)io(34)leadso + ((u (/) ) xx,τδ u (/) ) (f (/),τδ u (/) ). (54) τ α cγ (3α) [b δ u (/) l (b l b l )δ u l(/) b ψ] (u (/) ) xx + K f(/) +R,,,3,... (57)

7 Joural of Applied Maheaics 7 Subracig (57)fro(34), we obai τ α cγ (3α) (b τ [(u u )(u u )] τ l (b l b l )[(u l u l )(ul u l )]) (u (/) ) xx (u (/) ) xx R,,,3,... (58) Proof. Subracig (57) fro(3), we obai he error equaio τ α cγ (3α) [b δ e (/) (b l b l )δ e l(/) ] l (6) (e (/) ) xx +R T R, where,, 3,....oighae ad e (a) e (b),,,..., ad applyig Lea 7,wehave (e ) x cγ(α) α τ l Rl T Rl. (63) Based otheore 4,we have R e πdβ cγ (3α) τα Based o (33)ad(59), we obai (e ) x c 3 Γ (α) T α (ba) [τ 3α +c 4 τ α (3/) e πdβ ]. (64) [b (d d ) +d + (3/) e πdβ, l where d,d,...,d + are cosas. (b l b l )(d l d l )] (59) Theore 8. Le u,,,,...,beheapproxiaio soluio give i (4) ad le u saisfy he codiios of Theore 4.Theoehas u C[ (u ) α x + cγ (3α) ψ where C is cosa. +cγ(α) α τ l K fl(/) +R l(/) ], (6) Proof. Cosiderig (57) ad oig ha u φad u (a) u (b),,,,..., we ca easily obai his resul fro Lea7 ad he Poicaréiequaliy. Theore 9. Le he proble (9) () have he exac soluio U ad le u be he approxiaio soluio give i (4), where,,,....ifu saisfy he codiios of Theore 4, he for τ T,oehas e Γ (α) T α (ba)(c τ 3α +c τ α (3/) e πdβ ), (6) where e U u. Applyig he Poicaré iequaliy yields he eeded resul. Rear. I Theore 9, heerroresiaeforulacoais c τ α (3/) e πdβ (he secod er i he righ-had side), where he error coribuio fro he spaial approxiaio is affeced by he iverse of he ie sep. I is worhwhile oig ha siilar resuls are also foud for he classical diffusio/wave equaio. However for large, (3/) e πdβ cabeuchsallerhaτ; herefore, his affecio geerally would o reduce he global accuracy. 5. uerical Exaples To validae he effeciveess of he proposed ehod for he proble (9) (), we cosider he exaple give i [8]. Cosider he followig: α u (x, ) α u (x, ) x + si (πx), <x<, <, u (x, ), u (x, ), <x<, u (, ), u(, ), <. The exac soluio of he above proble is [33] (65) u (x, ) π [ E α (π α )] si (πx), (66) where E α (z) z /Γ(α+)which is oe-paraeer Miag-Leffler fucio. For solvig he above proble (wih α.3 ad.7, resp.) by usig he ehod described i Secio 3,wechoose β ad d π/, ad his leads o h π/. We will repor he efficiecy ad accuracy of he give ehod based o he L -errors ad L -errors. Figure gives he 3D

8 8 Joural of Applied Maheaics...5 u (x, ) x.6.8 u (x, ) x.6.8 (a) (b) Figure : The uerical soluios wih α.3 (a) ad α.7 (b)...5. u (x, ) ad u(x, ) u (x, ) ad u(x, ) x x u (x,.) u(x,.) u (x,.5) u(x,.5) u (x, ) u(x, ) u (x,.) u(x,.) u (x,.5) u(x,.5) u (x, ) u(x, ) (a) (b) Figure : Copariso of he uerical ad exac soluios for several fixed ie isas wih α.3 (a) ad α.7 (b). diagras of he uerical soluios u (x, ) o he whole copuaioal doai [, ] [, ] wih τ.,. I Figure, we plo he curves of he uerical soluios u (x, ) ad exac soluios u(x, ) for several fixed ie isas. A good agreee of he uerical soluio wih he exac oe is achieved. Furherore, Figure 3 shows he absolue error u (x, ) u(x, ) obaied by he prese ehod wih τ.,. To chec he covergece behavior of he uerical soluio wih respec o he ie sep τ ad he paraeer abou he uber of collocaio pois, we represe he errors U u i wo discree ors: L ad L.Allhe uerical resuls repored i Figures 4 ad 5, adtables ad are evaluaed a.8. Firsly, he copuaioal ivesigaio is cocered wih heeporalcovergecerae.wechooseheparaeer 6 adiisavaluelargeeoughsuchhaheerrorscoig fro he spaial approxiaio are egligible [35]. I Tables (α.3)ad (α.7), we lis he eporal errors whe τ decreases fro / o /3 ad he covergece order which is very close o he expecaio order 3α.Wealsoplo he errors i he L ad L ors as a fucio of he ie sepsize τ for several α i Figure 4,wherealogarihicscale has bee used for boh τ-axis ad error axis. Fro Figure 4,i is clear ha, for α.3 ad.7,heslopesofheerrorcurves i hese log-log plos approach.7 ad.3, respecively. So he proposed ehod yields a eporal approxiaio order which is close o 3α as forecased by he heoreical esiae.

9 Joural of Applied Maheaics 9 u (x, ) u(x, ) x.6.8 u (x, ) u(x, ) x.6.8 (a) (b) Figure 3: Plo of he absolue errors wih α.3 (a) ad α.7 (b). Teporal errors α.3 Teporal errors α Tie sepsize Tie sepsize τ.7 L -error L -error τ.3 L -error L -error (a) (b) Figure 4: Teporal errors i he L ad L ors versus τ for several α. ow we chec he spaial accuracy wih respec o he paraeer abou he uber of collocaio pois. For he siilar reaso eioed above, we fix he ie sep τ sufficiely sall o avoid coaiaio of he eporal error. I Figure 5, we prese he spaial errors as a fucio of wih τ., where a logarihic scale is ow used for he spaial errors axis. I is clearly observed ha he spaial errors appear i a expoeial decay. I his seilog represeaios, we observe ha he error variaios are esseially liear versus [36]. 6. Coclusio I his paper, we have developed ad aalyzed a efficie uerical ehod for a class of he iiial-boudary-value probles of fracioal diffusio-wave equaios. Based o a fiie differece schee i ie ad a global sic-collocaio ehod i space, he proble is reduced o he soluio of a syse of liear algebraic equaios a each ie sep. The proposed ehod leads o 3αorder accuracy i ie ad expoeial covergece i space. The heoreical resuls are perfecly cofired by he uerical experies.

10 Joural of Applied Maheaics α.3 α Spaial errors 4 Spaial errors L -error L -error L -error L -error (a) (b) Figure 5: Spaial errors i he L ad L ors versus for several α. Table : Teporal errors ad covergece orders wih α.3. τ L -error Order L -error Order / E E 4 /.3547E E /4 7.79E E /8.54E E / E E / E E / E E Table : Teporal errors ad covergece orders wih α.7. τ L -error Order L -error Order / E E 3 /.65563E E /4.4866E E / E E /6.6883E E / E E / E E 5.98 Coflic of Ieress The auhors declare ha here is o coflic of ieress regardig he publicaio of his paper. Acowledges The prese research is suppored by he aioal aural Sciece Foudaio of Chia (Gra os. 73 ad 376), he Chiese Progra for Chagjiag Scholars ad Iovaive Research Tea i Uiversiy (PCSIRT) (Gra o. IRT79), adhehuaproviceiovaiofoudaioforposgraduae (Gra o. CX3B5). The auhors are graeful o he aoyous reviewers for heir careful coes ad valuable suggesios o his paper. Refereces [] F. Maiardi, Fracioal diffusive waves i viscoelasic solids, i oliear Waves i Solids, J.L.Weger ad F.R.orwood, Eds.,pp.93 97,ASME/AMR,Fairfield,J,USA,995. [] R. Balescu, V-Lagevi equaios, coiuous ie rado wals ad fracioal diffusio, Chaos, Solios & Fracals, vol. 34,o.,pp.6 8,7. [3] F. Maiardi, Fracioal relaxaio-oscillaio ad fracioal diffusio-wave pheoea, Chaos, Solios & Fracals, vol. 7, o.9,pp ,996. [4] A. A. Kilbas, H. M. Srivasava, ad J. J. Trujillo, Theory ad Applicaios of Fracioal Differeial Equaios, Elsevier, Aserda, The eherlad, 6. [5] F. Maiardi, The fudaeal soluios for he fracioal diffusio-wave equaio, Applied Maheaics Leers, vol.9, o. 6, pp. 3 8, 996. [6] S.Moai,Z.Odiba,adV.S.Erur, Geeralizeddiffereial rasfor ehod for solvig a space- ad ie-fracioal diffusio-wave equaio, Physics Leers A,vol.37,o.5-6,pp , 7. [7] O. P. Agrawal, Soluio for a fracioal diffusio-wave equaio defied i a bouded doai, oliear Dyaics,vol.9,o. 4, pp ,. [8] Y. Lucho, Fracioal wave equaio ad daped waves, Joural of Maheaical Physics,vol.54,o.3,AricleID355, 6 pages, 3. [9] Y. Povseo, Axisyeric soluios o fracioal diffusiowave equaio i a cylider uder Robi boudary codiio, The Europea Physical Joural Special Topics, vol., o. 8, pp , 3.

11 Joural of Applied Maheaics [] I. Podluby, A. Chechi, T. Sovrae, Y. Che, ad B. M. Viagre Jara, Marix approach o discree fracioal calculus II: parial fracioal differeial equaios, Joural of Copuaioal Physics,vol.8,o.8,pp ,9. [] S. B. Yuse, Weighed average fiie differece ehods for fracioal diffusio equaios, Joural of Copuaioal Physics, vol. 6, o., pp , 6. []P.Zhuag,F.Liu,V.Ah,adI.Turer, ewsoluioad aalyical echiques of he iplici uerical ehod for he aoalous subdiffusio equaio, SIAM Joural o uerical Aalysis,vol.46,o.,pp.79 95,8. [3] M. M. Khader, O he uerical soluios for he fracioal diffusio equaio, Couicaios i oliear Sciece ad uerical Siulaio,vol.6,o.6,pp ,. [4] C. Pire ad E. Haer, A radial basis fucios ehod for fracioal diffusio equaios, Joural of Copuaioal Physics,vol.38,pp.7 8,3. [5] A. Mohebbi, M. Abbaszadeh, ad M. Dehgha, A high-order ad ucodiioally sable schee for he odified aoalous fracioal sub-diffusio equaio wih a oliear source er, Joural of Copuaioal Physics,vol.4,pp.36 48,3. [6] H. Jafari, A. Golbabai, S. Seifi, ad K. Sayevad, Hooopy aalysis ehod for solvig uli-er liear ad oliear diffusio-wave equaios of fracioal order, Copuers ad Maheaics wih Applicaios, vol.59,o.3,pp ,. [7] Z. M. Odiba ad S. Moai, Approxiae soluios for boudary value probles of ie-fracioal wave equaio, Applied Maheaics ad Copuaio, vol.8,o.,pp , 6. [8] Z. Su ad X. Wu, A fully discree differece schee for a diffusio-wave syse, Applied uerical Maheaics, vol. 56, o., pp. 93 9, 6. [9] J. Huag, Y. Tag, L. Vázquez, ad J. Yag, Two fiie differece schees for ie fracioal diffusio-wave equaio, uerical Algorihs,vol.64,o.4,pp.77 7,3. [] J. Q. Murillo ad S. B. Yuse, A explici differece ehod for solvig fracioal diffusio ad diffusio-wave equaios i he Capuo for, Joural of Copuaioal ad oliear Dyaics,vol.6,o.,AricleID4,6pages,. [] F. Seger, uerical Mehods Based o Sic ad Aalyic Fucios, Spriger, ew Yor, Y, USA, 993. [] F. Seger, Hadboo of Sic uerical Mehods, CRCPress, ew Yor, Y, USA,. [3]J.LudadK.L.Bowers,Sic Mehods f or Quadraure ad Differeial Equaios, SIAM, Philadelphia, Pa, USA, 99. [4] K. Parad, M. Dehgha, ad A. Pirhedri, The sic-collocaio ehod for solvig he Thoas-Feri equaio, Joural of Copuaioal ad Applied Maheaics, vol.37,o.,pp. 44 5, 3. [5] K. Al-Khaled, uerical sudy of Fisher's reacio-diffusio equaiobyhesiccollocaioehod, Joural of Copuaioal ad Applied Maheaics, vol.37,o.,pp.45 55,. [6] A. Secer, uerical soluio ad siulaio of secod-order parabolic PDEs wih Sic-Galeri ehod usig Maple, Absrac ad Applied Aalysis, vol. 3, Aricle ID , pages, 3. [7] M. Dehgha ad F. Eai-aeii, The Sic-collocaio ad Sic-Galeri ehods for solvig he wo-diesioal Schrödiger equaio wih ohoogeeous boudary codiios, Applied Maheaical Modellig,vol.37,o.,pp , 3. [8] A. Saadaadi, M. Dehgha, ad M. Azizi, The Sic- Legedre collocaio ehod for a class of fracioal covecio-diffusio equaios wih variable coefficies, Couicaios i oliear Sciece ad uerical Siulaio,vol.7,o.,pp ,. [9] G. Baua ad F. Seger, Fracioal calculus ad Sic ehods, Fracioal Calculus ad Applied Aalysis,vol.4,o. 4, pp ,. [3] I. Podluby, Fracioal Differeial Equaios, Acadeic Press, ew Yor, Y, USA, 999. [3] D. Baleau, K. Diehel, E. S, ad J. J. Trujillo, Fracioal Calculus: Models ad uerical Mehods, World Scieific, Sigapore,. [3] M. Weilbeer, Efficie uerical Mehods for Fracioal Differeial Equaios ad heir Aalyical Bacgroud, Papierflieger, Claushal-Zellerfeld, Geray, 6. [33] O. P. Agrawal, Respose of a diffusio-wave syse subjeced o deeriisic ad sochasic fields, Zeischrif für Agewade Maheai ud Mechai, vol.83,o.4,pp.65 74, 3. [34] K. Diehel, The Aalysis of Fracioal Differeial Equaios, Spriger, Berli, Geray,. [35] Y. Li ad C. Xu, Fiie differece/specral approxiaios for he ie-fracioal diffusio equaio, JouralofCopuaioal Physics,vol.5,o.,pp ,7. [36] X. Li ad C. Xu, A space-ie specral ehod for he ie fracioal diffusio equaio, SIAM Joural o uerical Aalysis,vol.47,o.3,pp.8 3,9.

12 Advaces i Operaios Research Volue 4 Advaces i Decisio Scieces Volue 4 Joural of Applied Maheaics Algebra Volue 4 Joural of Probabiliy ad Saisics Volue 4 The Scieific World Joural Volue 4 Ieraioal Joural of Differeial Equaios Volue 4 Volue 4 Subi your auscrips a Ieraioal Joural of Advaces i Cobiaorics Maheaical Physics Volue 4 Joural of Coplex Aalysis Volue 4 Ieraioal Joural of Maheaics ad Maheaical Scieces Maheaical Probles i Egieerig Joural of Maheaics Volue 4 Volue 4 Volue 4 Volue 4 Discree Maheaics Joural of Volue 4 Discree Dyaics i aure ad Sociey Joural of Fucio Spaces Absrac ad Applied Aalysis Volue 4 Volue 4 Volue 4 Ieraioal Joural of Joural of Sochasic Aalysis Opiizaio Volue 4 Volue 4