Numerical Solution of Fractional Telegraph Equation Using the Second Kind Chebyshev Wavelets Method

Transcription

1 J App Evo Bo Sc 59S etrod Pucto SSN: Jou o Apped Evoet d Boogc Scece tetodco Nuec Souto o cto eegph Equto Ug the Secod d Cheyhev Wveet ethod A Sdegh * H Heyd R Hoohd d S 4 4 eptet o thetc c Azd Uvety o Yzd Yzd 4 cuty o thetc Yzd Uvety Yzd ABSRAC Receved: ch 6 4 Accepted: y 7 5 th ppe the to-deo ecod d Cheyhev veet e pped o uec outo o the te-cto teegph equto th chet oudy codto th y e opeto t o cto devtve o the ecod veet deved d the th opeto t h ee epoyed to ot the uec outo o the ove etoed poe he chctetc ehd th ppoch tht t educe uch poe to thoe o ovg yte o gec equto hch gety pyg the poe he poe o th gee ethod utted EYWORS:Secod d Cheyhev veet eegph equto chet oudy codto cto devtve NROUCON ecet ye cto ccuu d deet equto hve oud eoou ppcto thetc phyc chety d egeeg ecue o th ct tht etc odeg o phyc pheoeo hvg depedece ot oy t the te tt ut o the pevou te htoy c e ucceuy cheved y ug cto ccuu he ppcto o the cto ccuu hve ee deotted y y utho o epe t pped to ode the oe octo o ethque [] ud-dyc tc [] equecy depedet dpg ehvo o y vcoetc te [] cotuu d tttc echc [4] cooed oe [5] od echc [6] ecooc [7] g poceg [8] d coto theoy [9] Hoeve dug the t decde cto ccuu h ttcted uch oe tteto o phyct d thetc ue to the ceg ppcto oe chee hve ee popoed to ove cto deet equto he ot equety ued ethod e Ado decopoto ethod A [-] hootopy petuto ethod [] hootopy y ethod [4] Vto teto ethod V [5 6] cto deet to ethod [7-] cto deece ethod [] poe ee ethod [4] geezed oc pue opeto t ethod [5] d Lpce to ethod [6] Ao ecety the opeto tce o cto ode tegto o the H veet [7] Legede veet [8] d the Cheyhev veet o t d [9 ] d ecod d [] hve ee deveoped to ove the cto deet equto th ppe code the te-cto teegph equto o ode < : hee t β β t u t u t u t t u t t t [] [] deote the Cputo cto devtve o ode β tht e deced the et ecto h equto cooy ued the tudy o ve popgto o eectc g ce to e d o ve pheoe h equto h ee o ued odeg the ecto-duo pocee vou che o egeeg cece d oogc cece y y eeche ee [] d eeece thee he pupoe o th ppe to ppy the ecod d Cheyhev veet o ovg te-cto teegph equto th y e t dece oe popete o the ecod d Cheyhev poyo d Cheyhev veet he e opeto t o cto devtve o the ecod d Cheyhev veet e deved d e pped to ot ppote outo o the ude tudy poe h ppe ogzed oo: Secto oe ecey deto o the cto ccuu e eveed Secto the ecod d Cheyhev poyo d the ecod d Cheyhev veet th oe ueu theoe e vetgted Secto 4 the popoed ethod deced Secto 5 oe uec epe e peeted y cocuo d Secto 6 Coepodg utho: A Sdegh - eptet o thetc c Azd Uvety o Yzd Yzd degh@uyzdc 64

2 Sdegh et 5 Bc deto the deveopet o theoe o cto devtve d teg y deto o cto devtve d teg e ppeed uch Re-Louve d Cputo [] hch e deced eo: eto - A e ucto u > d to e the pce C µ µ R thee et e ue p > µ uch tht u p u hee u C[ ] d t d to e the pce eto - he Re-Louve cto tegto opeto o ode t h the oog popete: hee β d ϑ > u u t C µ u C N µ o ucto µ u t dt > β β ϑ ϑ ϑ u u ϑ u deed []: eto - he cto devtve opeto o ode > the Cputo ee deed []: d u N u d t u t dt < < hee tege > d C u he ueu eto etee the Re-Louv opeto d Cputo opeto gve y the oog epeo: hee tege > d u C o oe det out cto ccuu ee [] u u u > <! he ecod d Cheyhev poyo d veet he e-o ecod d Cheyhev poyo U z o copete et o othogo ucto th epect to the eght ucto ou [4]: th z z o the tev ] [ hey c e deteed th the d o the oog ecuece U z zu z U z U z d U z z o pctc ue o thee poyo o the tev o teet [] t ecey to ht the deg do y e o the oog uttuto: z So the hted ecod d Cheyhev poyo U he othogoty codto o thee hted poyo : hee δ the oee det he ytc o o the hted ecod d Cheyhev poyo : C µ e oted o the tev [] U U U U d δ

3 J App Evo Bo Sc 59S U hee! 8!! he ecod d Cheyhev veet hch cotucted o t' coepodg poyo ued y potve tege the degee o the ecod d Cheyhev vove ou guet poyo d the ve deed ove [] hey e deed o the tev [] []: [ ] o U 9 We houd ote tht deg th the ecod d Cheyhev veet the eght ucto hve to e dte d tte A ucto u deed o [] y e epded y the ecod d Cheyhev veet : hee u c c u d deote the e poduct [] the te ee tucted the t c e tte : hee C d e cou vecto gve y: d hch C L 7 u ; c C [ c c c c c c c ] [ ] c c he de deteed y the eto u t deed ove [] Sy ty ucto o to ve Cheyhev veet : U [ u d u u t t hee ] [] y e epded to ecod d u t ; u y U t 4 he opeto t o cto devtve Hee e peet pocedue to deve the opeto t o cto devtve the Cputo ee o the ecod d Cheyhev veet o c e tte : Re By ug the hted ecod d Cheyhev poyo y copoet hee [ ] U d the chctetc ucto deed : 66

4 Sdegh et 5 ] [ o Net e peet ueu theoe out cto devtve o the ecod d Cheyhev veet: Le 4- Let e copoet o deed o the tev d e cto devtve o ode < < > o th epect to he o y e hve: < hee!!!! d! d d e deed 8 d ] [ d Poo o < o deto o the hted ecod d Cheyhev poyo d Cputo' devtve the tteet ce o o e d ytc o o the hted ecod d Cheyhev poyo e hve: 4 We otce tht th ucto zeo outde the tev ] [ No y ppyg o oth de o 4 e get:!! 5 hee ] [ d 67

5 J App Evo Bo Sc 59S Hee due to the ct tht zeo outde the tev ] [ the the ecod d Cheyhev veet epo o th ucto h oy copoet o Cheyhev veet tht e o-zeo o th tev hch yed: e 6 hee d d e 7 d Ao epdg y the ecod d Cheyhev veet ech o the tev yed: e 8 hee d d e 9 No y uttutg 6-9 to 5 e ot: hee d d! d! d Ate oe pcto d c e epeed the oog o:!!!! d! d heeoe c e tte : 68

6 Sdegh et 5 ] [ ] [ hee ] [ h copete the poo Re o o Cputo' devtve e hve: < heoe 4- Let e the ecod d Cheyhev veet vecto deed d < < > e potve cott he e hve: hee the opeto t o cto devtve o ode o the ecod d Cheyhev veet d deed oo: B B B B O hee B d e tce gve y: B 4 d 69

7 J App Evo Bo Sc 59S d 5 Poo t edte coequece o the e 4 Re o e t ut e oted tht o e hve 5 ecpto o the popoed ethod th ecto e ppy the opeto t o cto devtve o ecod d Cheyhev veet o ovg cto teegph equto th the oudy codto: o th pupoe e uppoe: hee u u u t g t u t g t u t U t U [ uo t hch houd e oud d u ] ug d e ot: d t t 6 7 the vecto hch deed No u t U t u t U u t t U t Ao ug the ucto t c e ppoted : hee B [ B ] coequet: he ete o vecto Hee e chooe t B t o t th ete B t t d t 8 9 ; Suttutg 7- [ U U U B] t e depedet o e hve: H U equto o : We c o ppote the ucto U U B H 4 g d t g : t 7

8 Sdegh et 5 C C hee C C C d 4 Appyg 7 d 5 the oudy codto 6 e hve: he ete o vecto g t C g t C C e o vecto o deo U C U C d t Λ 4 t t U y C 4 U y C e depedet o o 6 e ot: U C Λ t t Λ U C Λ4 U C4 By choog the equto o Λ d equto o Λ 4 e get 4 4 equto e Equeto 4 togethe 8 gve u t c e oud Λ Λ 4 U C equto hch c e oved o u So the uo ucto 6 Nuec epe th ecto e deotte the ececy o the popoed ethod o uec outo o the teegph equto the o o th the oudy codto 6 Epe Code the te-cto teegph equto th t t d the oudy codto : u u u t t u t t he ect outo o th poe o u t t Nuec outo o oe deet vue o d t [] o 6 e ho g he vue o ect outo d ppote outo o oe deet vue o d oe ode y [] [] o 6 e ho e ecopo etee the ect d uec outo o Epe?? Ect outo y

9 J App Evo Bo Sc 59S gnuec outo o Epe o oe deet vue o Epe Code te-cto teegph equto th t d the oudy codto : u e u e u t e u t e t t t he ect outo o th poe o u t e Nuec outo o oe deet vue o d t [] o 8 4 e ho g he vue o the ect outo d ppote outo o oe deet vue o d oe ode y [] [] o 8 e ho e e Copo etee the ect d uec outo o Epe Ect outo y

10 Sdegh et 5 gnuec outo o Epe o oe deet vue o Cocuo th ppe uec ethod o ppotg the outo o the te-cto teegph equto th chet oudy codto y cog ecod d Cheyhev veet ucto th the opeto t o cto devtve peeted he ethod ho tht vey coveet o ovg oudy vue poe Ao the peetto o the popoed ethod vey pe d vey ecet o outo o the teegph poe oeove the popoed ethod c e ued o uec outo o othe d o cto pt deet equto uch Poo d duo equto REERENCES J H He Noe octo th cto devtve d t ppcto teto coeece o vtg egeeg98 Ch: p J H He Soe ppcto o oe cto deet equto d the ppoto 'Bu Sc echo vo 5 p R L Bgey d P J ov A theoetc o the ppcto o cto ccuu to vcoetcty J Rheo 7 p d cto ccuu: oe c poe cotuu d tttc echc Cpte A d edto ct d cto ccuu cotuu echc Ne Yo: Spge Veg p B deot Soe oe th / pectu dge etee dect cuet d hte oe EEE o heoy p Y A Roh d V Shtov Appcto o cto ccuu to dyc poe o e d oe heedty echc o od App ech Rev 5 pp R Be Log eoy pocee d cto tegto ecooetc J Ecooetc 7 p R Pd d h cto geezed pe d g poceg Sg Poce vo 86 pp G W Boh Aog cto ode cotoe tepetue d oto coto ppcto J V Coto 4 p

11 J App Evo Bo Sc 59S Z Odt d S o Nuec ethod o oe pt deet equto o cto ode App th ode p S o d Z Odt Nuec ppoch to deet equto o cto ode J Coput App th 7 p 96-7 S A E-W A Ehy d Adou Ado decopoto ethod o ovg cto oe deet equto App th Coput 8 p -4 6 N H Se hde d R A-B Nuec tude o ut-ode cto deet equto Phyc Lette A 7 p Hh O Aduzz d S o Hootopy y ethod o cto vp Cou Noe Sc Nue Su 4 p N Se hde d R A-B Nuec tude o ut-ode cto deet equto Phy Lett A 7 p S ``Aytc outo o cto duo equto y vto teto ethod Coput th App vo 57 pp A Aogu d Ozo Souto o cto deet equto y ug deet to ethod Cho Soto ct 4 p A Aogu d Ozo Souto o cto tego-deet equto y ug cto deet to ethod Cho Soto ct 4 p P d A Ed A ethod o the uec outo o the tego-deet equto App th Coput 88 p V Etu d S o Sovg yte o cto deet equto ug deet to ethod J Coput App th 5 p V Etu d S o Sovg yte o cto deet equto ug deet to ethod J Coput App th 5 p V S Etu S o d Z Odt Appcto o geezed deet to ethod to ut-ode cto deet equto Co Noe Sc Nue Sut p eechet d C de te deece ppoto o to-ded pce-cto pt deet equto App Nue th 56 p Z Odt d N Shgeh Geezed tyo' ou App th Coput 86 p Y L L d N Su Nuec outo o cto deet equto ug the geezed oc pu opeto t Coput th App 6 p Poduy he pce to ethod o e deet equto o cto ode ept<v:uct-/975> Y L d W Zho H veet opeto t o cto ode tegto d t ppcto ovg the cto ode deet equto App th Coput 6 p Reh d R A h he egede veet ethod o ovg cto deet equto Coucto Noe Scece d Nuec Suto 6 p Y L Sovg oe cto deet equto ug cheyhev veet Coucto Noe Scece d Nuec Suto 59 p H Heyd R Hoohd Gh d ohd Wveet coocto ethod o ovg ut ode cto deet equto Jou o Apped thetc Atce pge do:55//544 Y Wg d Q he ecod d cheyhev veet ethod o ovg cto deet equto Apped thetc d Coputto 8 p Let d B N Sy Nuec outo o teegph equto ug tepotg cg ucto Coput th App 6 p Poduy cto eet Equto S ego: Acdec Pe 999 CCuto Hu A Quteo d Zg Spect ethod ud dyc