Integral transform method for solving non homogenous time fractional partial differential equations

Transcription

1 DERENTAL EQATONS AND CONTROL PROCESSES Elecoic Joul eэл ФС SSN 87-7 hp:mhpuuuediffjoul e-mil: jodiff@milu el fom mehod fo olvi o homoeou ime fciol pil diffeeil equio A Ahili* - A Mohhi Depme of Applied Mhemic culy of Mhemicl Sciece iveiy of Guil POBo 8 Rh - mhili@milcom mohhi@milcom Ac hi icle he uho olved cei o-homoeeou ime fciol he equio hich i eelizio o he polem of pi- up effec o he eo ophic d qui eoophic d o loly ii poduc o dop i oi fluid he l hee decde fom mehod hve ee ued fo olvi fciol diffeeil equio iul iel equio The eul evel h he fom mehod i vey coveie d effecive Mhemic Sujec Clificio : 6A33 3A8 3K37 35R Keyod: Cpuo fciol deivive; No-homoeeou ime fciol he equio; Lplce fom; Siul iel equio oducio Eieei d ohe e of ciece c e uccefully modeled y he ue of fciol deivive Th i ecue of he fc h eliic modeli of phyicl pheomeo hvi depedece o oly he ime i u lo he peviou ime hioy c e uccefully chieved y ui fciol clculu hi ok he uho impleme fom mehod fo olvi he pil fciol diffuio equio hich ie i pplicio Sevel mehod hve ee ioduced o olve fciol diffeeil equio he popul Lplce fom mehod [] [ ] [ 3 ] [] he ouie fom mehod [] he ieio mehod [7] d opeiol mehod [ ] Hoeve mo of hee mehod e uile fo pecil ype of fciol diffeeil equio mily he lie ih co coefficie Moe deiled ifomio ou ome of hee eul c e foud i uvey ppe y Kil d Tujillo [ 9] Ackovic d Skovic [][5]d Skovic [9] ued he Lplce fom i cei pce of diiuio o olve yem of pil diffeeil equio ih fciol deivive d idiced h uch yem my eve cei model fo vico elic od Oldhm d Spie [] d [ 3] epecively y educi oudy vlue polem ivolvi ick ecod lo i eleco lyic chemiy o fomulio ed o he

2 Diffeeil Equio d Cool Pocee N pil Riem Liouville fciol ih hlf deivive Oldhm d Spie [ 3] ve ohe pplicio of uch equio fo diffuio polem chiki [ ][ ] i hich he coecio eee oluio of lie equio of fciol ode d oluio of he fi ode h ee elihedwy [ 3] d Scheide [8] coideed he ime fciol diffuio d ve equio d oied he oluio i em of o fucio Defiiio d oio The lef Cpuo fciol deivive of ode i defied y C f D f d Lplce fom of fucio f i follo f L{ f } he hee Re L{ f } e f d : L { } i ive y f ci i e d ci c o oe e C k k k L{ D f } f The diffeeil equio y y y i clled modified Beel equio eel oluio i y c ck hee K e - ode modified Beel fucio of fi d ecod kid epecively O he ohe hd K i clled Mcdold fucio The imple Wih fucio i ive y he eie z W ; z! o z We hve he folloi eliohip L{ E } Re Emple : Le L{ f } d L{ u } ep q d umi q e lyic he oe h pecil ce le u ume h hich led o The e oi L f u d q d q ep L{ u } u W he oe h ; L f W ; d Elecoic Joul hp:mhpuuuediffjoul 95

3 Diffeeil Equio d Cool Pocee N Povided h he iel i cke covee oluely : Soluio o iul iel equio ih ioomeic keel Lplce fom c e ued o olve cei ype of iul iel equio Lemm : Solvi he folloi iul iel equio of he fom i f f d Soluio: Le L f L G e he Lplce fom of f epecively he oe e he folloi elio G o i elio e eplce y o oi G 3 Comiio of 3 d d clculio of led o he folloi G G upo ui comple iveio fomul elio led o he folloi c i G G f e d 5 i c i Emple : Solve he folloi iul iel equio i f ep f d 6 Soluio: Lplce-fom of he ove iel equio led o he folloi he foml oluio i c i e f d 7 i c i y e e f dy y Elecoic Joul hp:mhpuuuediffjoul 96

4 Diffeeil Equio d Cool Pocee N Mi eul hi ecio he uho coideed cei o-homoeeou ime fciol he equio hich i eelizio o he polem of pi- up effec o he eoophic d qui eoophic d o loly ii poduc o dop i oi fluid udied y M ih [] [] he fudmel oluio fo he fciol homoeou diffuio ve equio ui joi fom mehod oied hi ok oly he Lplce fomio i coideed i i eily udeood d ei popul mo eiee d cieithe ic ol of hi ok h ee o employ he Lplce fom mehod fo udyi he ove meioed polem The ol h ee chieved y fomlly deivi ec lyicl oluiothe fom - mehod ioduce iific impoveme i hi field ove eii echique No homoeou ime fciol He equio Polem : Solve he o-homoeeou ime fciol he equio c u u u c D u u ep d ih he oudy codiio: lim u u d he iiil codiio u R Le e Lplce fomle fucio Soluio Sep: By ki he Lplce fom ih epec o vile of eliohip e e G u c The y ui he fc h Moeove e coclude fom u h u i elio e oi c G 3 u ep d lo lim u led o lim 5 Sep: The elio3 i he oudy-vlue polem ih he codiio d 5 h c e e ie follo c G The e oi i eel oluio folloi 7 oduci he che of vile 8 e e 6 Elecoic Joul hp:mhpuuuediffjoul 97

5 d d Replci 89 i 7 led o he folloi equio Th i he modified Beel equio ih eel oluio i Diffeeil Equio d Cool Pocee N 9 c ck Whee i fi ode modified Beel fucio of he fi kid h i equl o i J i 3!! d K i fi ode modified Beel fucio of he ecod kid h i equl o K J i iy i i Wih he K No o h if he d i hi ce hee i ouded heefoe coefficie of K mu e equl o zeo hich i c The i eel oluio ecome c 5 d he complemey oluio of he equio 6 i equl o he um of he oluio of he folloi equio 6 c G 7 No fo olvi o - homoeou equio 6 e ume h p i he oluio o he ove equio e hve 8 heefoe he complemey oluio ecome p 9 Similly i equio 7 e ume h i he complemey oluio of he equiotheefoe oe h p Elecoic Joul hp:mhpuuuediffjoul 98

6 Diffeeil Equio d Cool Pocee N Elecoic Joul hp:mhpuuuediffjoul 99 ui he elio 9 he complemey oluio of equio 6 ed G c p p p filly he eel oluio o equio 6 ecome c G c 3 u ice e e Coequely G c c G 5 Sep3: Replci 5 i 5 eul i cg 6 No if e eplce ih e oi cg 7 Sep : We my eie folloi 8 hee G c p G c c G c 5 3 cg cg

7 Diffeeil Equio d Cool Pocee N 9 e o lyic i hei oo of deomio hee hee oo e imple pole No e oi hee pole follo e oo of deomio uhe if 3 e he zeo of J 3 moeove ujec o 3 he zeo of J i e imple he oe h i i hee i 3 e zeo of No if e e i Theefoe 3 e he zeo of i oviou h h he pole 3 Similly e he pole of d lo 3 e he pole of 5 Sep 5: Applicio of Bomich' iel d he eidue heoem led o ci f ep Re ep d i ci k k i he pole of 3 d The ujec o 8 d he lieiy popey of he iveio of Lplce fom d he Lplce fom of covoluio of o fucio oe h f f f f c f c f 3 Theefoe o oi f m m 3 5 d he f i i ufficie h e evlue he eidue of ep m m 3 5 evey pole of m m 35 illy fe pefomi ll he clculio e e J f ep ep J e J ep J c { ep } J J J J ep * ep c c 3 Elecoic Joul hp:mhpuuuediffjoul

8 Diffeeil Equio d Cool Pocee N Sep 6: ilizi Efo' heoem e hve u ep f d ep d J J ep ep d ep ep d J ep ep J d J c ep ep d d J c ep ep d d J c ep ep d d J A hi poi e clcule evey oe of he iel o he ih hd ide of 33 i iel: By he che of vile e oi ep d ep d Secod iel: By mki he che of vile 3 e e ep ep ep d d ep ep d Similly y che of vile i he hid iel e oi v 33 ep efc 35 ep d ep efc 36 lo y che of vile v i he fouh iel e e ep d ep{ } efc 37 ifh iel: By chi he ode of ieio folloed y he che of vile i he ie iel e e Elecoic Joul hp:mhpuuuediffjoul

9 Diffeeil Equio d Cool Pocee N Elecoic Joul hp:mhpuuuediffjoul ep ep d d ep ep d d ep ep ep d d ep ep ep d dv v ep ep d efc 38 lo ih he che of vile v Similly y che of vile d he v i he ih iel e oi ep ep d d ep ep d efc 39 he l iel y he me pocedue d ui he che of vile e e v o h ep ep ep ep d d efc d illy uiuio of i 33led o he folloi foml oluio ep J u efc J ep efc

10 c Diffeeil Equio d Cool Pocee N J ep J ep J c ep J uep u uep u efc efc efc u du u du J u ep ep J c u u efc du 3 CONCLSON The ppe i devoed o udy d pplicio of Lplce fom The iel fom povide poeful mehod fo lyzi lie yem The mi pupoe of hi ok i o develop mehod fo fidi foml oluio of cei iul iel equio d lyic oluio of he ime fciol he equio hich i eelizio o he polem of pi up effec o he eo ophic d qui -eoopic d o loly ii poduc o dop i oi fluid We hope h i ill lo eefi my eeche i he diciplie of pplied mhemic mhemicl phyic d eieei Ackoledeme The uho ould like o hk Pofeo Nly BAmplyov d Pofeo V A Koi fo hei vlule comme d ueio h led o v impoveme i he ppe Refeece [] AAhili BSlkhodeh Mohddm Lplce fom pi of - dimeio d ve equio e Mh Joul Vol 5 o [] AAhili BSlkhodeh Mohddm Muli-dimeiol Lplce fom d yem of pil diffeeil equio e Mh Joul Vol 6 o - 6 [3] AAhili BSlkhodeh Mohddm Lplce fom pi of -dimeio d ecod ode lie diffeeil equio ih co coefficie Ale Mhemice e ifomice358 pp3- [] TM Ackovic BSkovic Dymic of vico -elic od of ciol deivive ype Z Ae Mh Mech [5] TM Ackovic BSkovic O yem of diffeeil equio ih fciol deivive ii i od heoy Joul of Phyic A: Mhemicl d Geel 37 No -5 [6] DGDuffy Tfom mehod fo olvi pil diffeeil equio Chpm & HllCRC [7 ] RSDhiy M Viymoohy Lplce fom pi of dimeio d he coducio polem Mh Compu Modelli vol 3No pp35-5 [8] VA Diki APPudikov Opeiol clculu o vile d i pplicio Pemo Pe Ne Yok96 [9] AAKil JJ Tujillo Diffeeil equio of fciol ode: mehod eul d polem Appl Al Elecoic Joul hp:mhpuuuediffjoul 3

11 Diffeeil Equio d Cool Pocee N [] Y Luchko H Sivv The ec oluio of cei diffeeil equio of fciol Ode y ui opeiol clculu Compu Mh Appl [] SMille BRo A ioducio o fciol diffeeil equio Wiley NeYok [] KBOldhm J Spie The ciol clculu Acdemic Pe NeYok 97 [3] KBOldhm J Spie ciol clculu d i pplicio Bull Polieh i Sec [ ] Podluy The Lplce fom mehod fo lie diffeeil equio of fciol ode Slovk Acdemy of ciece Slovk Repulic 99 [5] Podluy ciol diffeeil equio Acdemic Pe S Dieo CA999 [6 ] GERoe H KufmTle of Lplce fom Phildelphi; WBSude Co 966 [7] G Smko AKil OMchiev ciol el d deivive heoy d pplicio Godo d BechAmeedm993 [8] W Scheide Wy ciol diffuio d ve equio J Mh Phy [9] BASkovic Syem of pil diffeeil equio ih fciol deivive Mh Veik [ ]VVchikiMehod of fudmel deiviveaihock lyovky Rui8 [ ]VVchiki Sudiffuio d le l Joul of epeimel d heoeicl phyic88 999P55 [] Mih Some pi up effec o he eo ophic d qui eoopic d o loly ii poduc o dop i oi fluid PhyluidSe A [3]W Wy The fciol diffuio equio J Mh Phy Elecoic Joul hp:mhpuuuediffjoul