Some Numerical Methods For Solving Fractional Parabolic Partial Differential Equations

Transcription

1 Eng.& Tech. Journal,Vol.28, No.2, 2 Some Numercal Mehods For Solvng Fraconal Parabolc Paral Dfferenal Dr. Osama H.Mohammed*, Ibsam K.Hanan* & Akram A. Al-Sabbagh* Receved on: 7//29 Acceped on:/4/2 Absrac The am of hs paper so approxmae he offraconal parabolcparal dfferenal equaons usng wo numercal mehods whch are Bellman's mehod (make use of Lagrange nerpolaon formula) and he mehod of lnes. Fraconal Parabolc paral dfferenal equaons are ransformed o a sysem of frs order ordnary dfferenal equaons ha are solved usng a Runge- Kua mehod. An llusrave example usng hesemehodsare also presened and compared wh he exac. Keywords: Bellman's Mehod, Mehod of lnes, Runge-Kua Mehods. بعض الطراي ق العددية لحل المعادلات التفاضلية الجزي ية الكسورية المكافي ة الخلاصة الهدف الري يس من هذا البحث هو تقريب حل المعادلات التفاضلية الجزي ية الكسورية المكافة( equaons (fraconal Parabolc paral dfferenal باستخدام طريقتان عدديتان هما طريقة بيلمان (وبالاعتماد على صيغة لاكرانج للاندراج) وطريقة الخطوط. المعادلات التفاضلية الجزي ية الكسوريةستتحول الى منظومة معادلات تفاضلية ا عتيادية من الرتبة الاولى والتي سوف تحل بوساطة ا ستخدام طريقة رنج-كتا. تم ا عطاء مثال توضيحي لتلك الطراي ق وتم مقارنة النتاي ج مع الحل المضبوط.. Inroducon: Fraconal order paral dfferenal equaons are generalzaons of classcal paral dfferenal equaons.increasngly hese models are used n applcaons such as flud flow, fnanceand ohers [Ghareeb,27]. Fraconal calculus s a feld ofmahemacal sudy ha grows ou of he radonal deenons of he calculus negral and dervave operaors n he same way fraconal exponen s an ougrowh of exponen wh neger value,[loverro,24]. Many found, usng her own noaon and mehodology, defnons ha f he concep of a non-neger order negral or dervave. The mos famous of hese defnons ha have been popularzed n he word of fraconal calculus are he Remann-Louvlle andgrünwald-lenkov defnon.also capuo,[podlubny,999] reformulaed he more "classc" defnon of he Remann Louvllefraconal dervave n order o use neger order nal condons o solve hs fraconal order dfferenal equaons. Recenly [Kolowankar, 996] reformulaed agan, he Remann- Louvlle fraconal dervave, n * Scences College, Al-Nahran Unversy/ Baghdad 248

2 Eng.& Tech. Journal,Vol. 28, No.2, 2 Some Numercal Mehods For Solvng order o dfferenae no-where dfferenable fracal funcons. Fraconal paral dfferenal equaons have been suded and explc s have been acheved by [Manard, 23],[Manard, 25], [Yu, 25],[Langlands, 26], [Manard, 26]and several oher research works can be found n he leraure. In hs paper, we shall use some numercal mehods whch arebellman's mehod and mehod of lnes o solve he fraconal Parabolc Paral dfferenal of he form: u( x, ) u( x, ) = c( x, ) + s( x, ) () On a fne doman, L < x < R, T.Here we consder he case 2, where he parameer s he fraconal order of he specal dervave and The funcon s( s source/snk erm, he funcon c(may be nerpreed as ranspor relaed coeffcen. We wll also assume ha c( over he regon L < x < R, T. We assume an nal condon u()=f(x) for L < x < R and zero Drchle boundary condons. Ths paper consss of foursecons, In secon wo Bellman s mehodwll be consdered o solve equaon (), whle n secon hree he mehod of lnes wll be presened o solve equaon () hs mehod was proposed by Rchard Bellman for solvng orgnally Paral dfferenal equaons, whch has he general dea of evaluang he a ceran lnes of he ndependen varable of he Paral dfferenal. An llusrave example was gven n secon four n order o compare hese wo mehods wh he exac. 2.Bellman's Mehod for Solvng Fraconal Parabolc Paral Dfferenal : Consder he fraconal order parabolc paral dfferenal equaons of he form: u( u( = c( + s( L < x < R, T...(2.a) Togeher wh he nal and zero Drchle boundary condons: u( ) = f( x), L x R ul (,) =, T ur (,) =, T...(2.b) ux (,) where denoe he lef handed paral fraconal dervave of order of he funcon u wh respec ox and 2Now we solve problem (2.a, 2.b) usng Bellman s mehod[abdmech, 99];le u( x, = u ( and suppose ha = I l = u( ( x) u (.. (3) where l (x) s he Lagrange nerpolaon funcons sasfyng, = j l (x j ) =, j Afer subsung (3) no (2.a) we have I I lxu ()() = cx (,) lxu ()() + sx (,) = = x.(4)

3 Eng.& Tech. Journal,Vol. 28, No.2, 2 Some Numercal Mehods For Solvng And subsung x= x n equaon (4),j=,2,,n n order o ge he number of unknowns equals o he number of equaons herefore we have a sysem of frs order dfferenal equaons as follows : I u j() = cx ( j,) l( xj) u () + sx ( j,) x = j=,2,,n (5) Afer usng he Runge-Kua mehod [Burden, 98]he sysem (5) wll be solved and hen we ge he values of u (),j=,2,, n.whch j represen he approxmae values of u( a he ponsx j and T. 3. The Mehod of Lnes for Solvng Fraconal Parabolc Paral Dfferenal : Consder he fraconal order parabolc paral dfferenal equaon of he form: ux (,) ux (,) = cx (,) + sx (,), L x R, T (6.a) ogeher wh he nal and zero Drchle boundary condons: ux (,) = f( x), L x R ul (,) =, T ur (,) =, T (6.b) ux (,) where denoe he lef-handed paral fraconal dervave of order of he funcon u wh respec o x and 2. In hs secon, we shall use explc fne dfference ux (,) approxmaon for o solve j hs nal-boundary value problem (6.a, 6.b) by he mehod of lnes[ames 977]. To do hs, suppose ha u(x,=u (and subsung x=x no equaon (6.a) we shall ge: du() ux (,) = cx (,) + sx (,). d.. (7) Where x =Δ =,,,n, and n s he number of subnervals of he nerval [L,R]. The lef-handed shfed Grünwald esmae o he lef-handed dervaveas llusraed n [Ghareeb, 27]s: n d f( x) = gk f( x ( k ) x) dx ( x) k= Where n s he number of subnervals of he nerval [L,R] and s he fraconal number. Therefore: + ux (,) = gux k ( ( k ) x,) ( x) k= + ux (,) = gk u k () + ( x) k=...(8) Whereg = and k ( )...( k+ ) gk = (), k=,2,... k! he dervaon of equaon (8) s gven by deals n [Ghareeb, 27], by subsung equaon(8) no equaon(7) and seek he values of from o n-n order o ge he number of unknowns equals o he numbers of equaons as ge n secon wo, one can have: + du () cx (,) = gu k k () sx (,),,2,..., n d ( x) + + =. k=...(9) 2

4 Eng.& Tech. Journal,Vol. 28, No.2, 2 Some Numercal Mehods For Solvng hen he sysem(9) s of frs order dfferenal equaon and can be solved usng Runge-Kua mehod n order o ge an approxmae value of u (, =,2,,n a he pon x and Tof problem (6.a, 6.b). 4. Illusrave Example: In he presen secon, he resul of he Bellman's mehod and he mehod of lnes whch were dscussed n secon wo and hree respecvely wll be gven and mplemened on he same example. The exac s also gven for comparson purpose. The nex example appeared n [Ghareeb, 27] whch s solved by usng fne dfference mehod. Example: Consder he nal-boundary value problem:.8 u 4 u 5 = x + x( x ) (x ),.8 Γ(.2) x, ux (,) = u(, = x, u(,) = Followng able () and able (2) prescrbed he resul of he Bellman's mehod and he mehod of lnes respecvely for h=.2wh he exac of he above example whch s u(=x(x ). 5. Conclusons: From he resuls of able () and able (2) respecvely seems ha he mehod of lnes s more accurae han Bellman s mehod. Takng h large for he Bellman s mehod wll reduces he number of bass l (x) and hence reduce he l( x) calculaon of whch also gves reasonable. 6. References []-Ghareeb, L. S., Some Fne Dfference Mehods For Solvng Fraconal Dfferenal, M.Sc. Thess, Deparmen Mahemacs, Al-NahranUnversy, 27. [2]-Loverro, A., Fraconal Calculus: Hsory, Defnon and Applcaons for he Engneer, Deparmen of Aerospace and Mechancal Engneerng, Unversy of Nore Dame, Nore Dame, IN 46556, USA, May 8, 24. [3]-Podlubny I. Fraconal Dfferenal, Academc Press, New York, U.S.A, 999. [4]-Kolowankar K.M. and Gangal Fraconal Dfferenably of nowhere dfferenable funcons and dmensons, CHAOS, Amercan Insue of Physcs, V.6, No 4, 996. [5]-Manard F. andpagnng., The Wrgh funcons as s of he me-fraconal dffuson equaons. Appl. Mah. Compu., V. 4, 5-62., 23. [6]-Manard F., PagnnG. and Saxena R. K., Fox H-funcons n fraconaldffuson. J. Compu. Appl. Mah.,V. 78, , 25. [7]-Yu. R. and ZhangH., New funcon of Mag-Leffler ype and s applcaon he fraconal dffuson-wave equaon. Chaos, SolonsFrac,J. Chaos., V. 5,No. 8,25. [8]-Langlands T. A. M., Soluon of a modfed fraconal dffuson equaon. Physca A, V. 367, [9]-Mnard F.,Pagnn G. and GorenfloR., Some aspecs of 3

5 Eng.& Tech. Journal,Vol. 28, No.2, 2 Some Numercal Mehods For Solvng fraconal dffuson equaons of sngle and dsrbued order. Appl. Mah. Compu,J., V. 26, No.8, 26. []-AbdMech M. S., Numercal and Approxmae Mehods n Soluons of Paral Dfferenal, M.Sc. Thess, Deparmen Mahemacs, Baghdad Unversy, 99. []-Burden R. L. and Fares J. D., Numercal Analyss (Thrd Edon), Brndle Weber and SclumdBoson, U.S.A, 98. [2]-Ames W. F., Numercal Mehods for Paral Dfferenal, Academc Press, New York, U.S.A, 977. Table () The approxmae of Example () usng Bellman's mehod = =.2 =.4 =.6 =.8 Approxmae x = Exac x = Approxmae x = Exac x = Approxmae x = Exac x = Approxmae x = Exac x =

6 Eng.& Tech. Journal,Vol. 28, No.2, 2 Some Numercal Mehods For Solvng Table (2) The approxmae of Example () usng he mehod of lnes = =.2 =.4 =.6 =.8 Approxmae x= Exac x= Approxmae x=.4 Exac x=.4 Approxmae x=.6 Exac x=.6 Approxmae x=.8 Exac x=