A Second Kind Chebyshev Polynomial Approach for the Wave Equation Subject to an Integral Conservation Condition
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1 SSN Eglad K Joural of forao ad Copug Scece Vol 7 No 3 pp 63-7 A Secod Kd Chebyshev olyoal Approach for he Wave Equao Subec o a egral Coservao Codo Soayeh Nea ad Yadollah rdokha Depare of aheacs Alzahra versy ehra ra Receved Jue 5 acceped February Absrac he a purpose of hs arcle s o prese a approae soluo for he oe desoal wave equao subec o a egral coservao codo ers of secod kd Chebyshev polyoals he operaoal arces of egrao ad dervao are roduced ad ulzed o reduce he wave equao ad he codos o he ar equaos whch correspod o a syse of lear algebrac equaos wh ukow Chebyshev coeffces Fally soe eaples are preseed o llusrae he applcably of he ehod Keywords: Wave equao No-local codo Secod kd Chebyshev polyoals peraoal ar ar for roduco Hyperbolc paral dffereal equaos wh a egral codo serve as odels ay braches of physcs ad echology here are ay papers ha deal wh he uercal soluo of he parabolc equao wh egral codos [ ] he prese work focuses o he oe-desoal wave equao wh he o-local boudary codo hs paper we cosder he followg oe-desoal wave equao wh al codos ad u u F < u f u f 3 ad Drchle boudary codo u g < ad he o-local codo u d g < 5 where F f f g ad g are kow fucos he esece ad uqueess of he soluo of he proble --5 are dscussed [3] Dehgha [8] preseed fe dfferece schees for he uercal soluo of proble --5 [5] he shfed Legedre au echque was developed for he soluo of he suded odel Auhor of [] developed a uercal echque based o a egro- dffereal equao ad local erpolag fucos for solvg he oe-desoal wave equao subec o a o-local coservao codo ad suably prescrbed al-boudary codos Auhors of [] cobed fe dfferece ad specral ehods o solve he oe-desoal wave equao wh a egral codo [9] varaoal erao ehod was appled for Correspodg auhor el: ; fa: E-al address: ordokha@alzahraacr ublshed by World Acadec ress World Acadec o
2 6 Soayeh Nea e al: A Secod Kd Chebyshev olyoal Approach for he Wave Equao Subec o a egral Coservao Codo solvg he wave equao subec o a egral coservao codo Auhors of [] preseed a eshless ehod for uercal soluo of proble --5 Also [6] he ehod of les was developed for he soluo of he dscussed proble rhogoal fucos have bee used o solve varous probles he a characersc of hs echque s ha reduces proble o hose of solvg a syse of algebrac equaos hus grealy splfyg he proble he prese paper he uercal soluo of he proble --5 s copued by usg wo varable shfed secod kd Chebyshev orhogoal fucos he paper s orgazed as follows: Seco basc properes of he secod kd Chebyshev polyoals are preseed Seco 3 we dscuss how o approae fucos ers of secod kd Chebyshev polyoals ad roduce operaoal arces of egrao ad dervao seco we gve a approae soluo for --5 Nuercal eaples are gve Seco 5 o llusrae he accuracy of our ehod Fally cocludg rearks are gve Seco 6 roperes of he Secod Kd Chebyshev olyoals Secod kd Chebyshev polyoals are oal orhogoal bass for he ad of he followg recursve forula [] L ad ca be deered wh For he case ha he erval s o [ ] say [ a b] we ca use he lear rasforao a b b a o rasfor he doa o [ ] he orhogoaly propery s as follows: d where s he wegh fuco Soe properes of he secod kd Chebyshev polyoals are as follows: k d k 6 d [ ] 7 [ /] Fuco Approao 3 Approao of oe-varable ad wo-varable fucos A fuco y defed over [ ] ay be epaded by he shfed secod kd Chebyshev fucos as y a a JC eal for corbuo: edor@corguk
3 Joural of forao ad Copug Scece Vol 7 No 3 pp where a y d ad f he fe seres s rucaed he ca be wre as where y y A [ a a a A [ ] Slarly a fuco h defed over [ ] [ ] ay be epaded by he shfed secod kd Chebyshev fucos as h c c 3 a ] so ha 6 c dd h ad f he fe seres 3 s rucaed he ca be wre as: h h c C such ha C [ c c c c c c c c c ] [ ] 5 order o calculae he egral par of ad we rasfor he ervals [ ] ad [ ] o he erval [ ] by eas of he rasforaos ad he use he secod kd Gauss-Chebyshev quadraure forula [] 3 peraoal ar of egrao sg equao 7 he egrao of he vecor defed equao 5 dreco ca be approaed by d 6 such ha s a ar as follows: JC eal for subscrpo: publshg@waorguk
4 Soayeh Nea e al: A Secod Kd Chebyshev olyoal Approach for he Wave Equao Subec o a egral Coservao Codo JC eal for corbuo: edor@corguk 66 where s zero ar wh deso ad peraoal ar of dervao By he defo of we have l l l d d ad usg equao 8 we ge /] [ k l k k herefore we oba where s a ar as 3 3 So ha 3 ad are respecvely 3 ad for odd ad ad for eve ad are dey ad zero ar wh deso respecvely Slarly we have 7
5 Joural of forao ad Copug Scece Vol 7 No 3 pp Nuercal Soluo of he Wave Equao ar for of he wave equao We approae u ad F equao respecvely as u A 8 F F 9 such ha A [ a a a a a a a a a ] where a are ukow secod kd Chebyshev coeffces ad s as 5 ad ad are chose posve egers wce egrag equao fro o ad usg equaos we have A f f A F Suppose ha he F ad F ca be approaed as: F f F f F F F F Subsug equaos ad o equao we ge WA V 3 so ha W s a ar ad V s a vecor as follows: W V ad s dey ar of deso F F F ar for of he Drechle boudary codo he Drchle boudary codo ca be wre as A g sg equao 9 we oba herefore A A W 5 where s as ad W s a ar as W 3 ad s dey ar wh deso We approae g as g G 6 Now usg equaos --6 he ar represeao of he boudary codo ca be wre as W A 7 G 3 ar for of he o-local codo By usg equao 8 he o-local codo 5 ca be wre as JC eal for subscrpo: publshg@waorguk
6 68 Soayeh Nea e al: A Secod Kd Chebyshev olyoal Approach for he Wave Equao Subec o a egral Coservao Codo sg equao 6 we ge herefore A d g 8 d d d k k A d A W 9 such ha W s a ar as W B B B B ad B are arces as k B k We approae g as g G 3 sg equaos 8--3 we oba he ar represeao of he o-local codo as W A 3 G ehod of soluo o oba he soluo of equao uder he codos --5 we replace rows of he augeed ar [ W ; V ] wh he rows of he augeed arces [ W ; G ] ad [ W ; G ] hs way he secod kd shfed Chebyshev polyoals coeffces are deered by solvg he ew lear algebrac syse 5 llusrave Eaples hs seco wo eaples are gve o deosrae he applcably ad accuracy of our ehod order o deosrae he error of he ehod we roduce he oao: e u u where u s he eac soluo ad u s he copued resul wh ad Eaple 5 Cosder --5 wh l ad [] F ep s < < < < f s f s < < g g ep < < for whch he eac soluo of hs proble s u ep s We appled he preseed ehod hs paper for hs proble wh dffere values of ad able we gve he error u for 5 ad 68 Fro able we see ha for fed u JC eal for corbuo: edor@corguk
7 Joural of forao ad Copug Scece Vol 7 No 3 pp as s creased he error decreases Also he error fuco e for dffere values of ad s show Fg able Values of u u for 5 ad soe values of for Eaple e66 e u u e e Fg : Graph of he e wh 68 for Eaple 5 Eaple 5 hs eaple cosder 5 wh l 5 ad [9] F < < < < 5 f cos f < < g cos g < < 5 he eac soluo of hs proble s u cos cos We appled he preseed ehod hs paper wh ad show soe uercal resuls able ad Fg Fro able we see ha he approae soluo copued by dffere values of ad coverges o he eac soluo able Values of u u u u 579 wh for Eaple Cocluso hs arcle we preseed a uercal schee for solvg he secod-order wave equao subec o a egral codo roperes of he secod kd Chebyshev polyoals were eployed he arces ad have ay zeroes hece akg secod kd Chebyshev fucos copuaoally very aracve Chebyshev coeffces of he soluo are foud very easly by usg he copuer progras whou ay JC eal for subscrpo: publshg@waorguk
![A uercal ehod for he wave equao subec o a o-local coservao codo Appl Nuer ah56 6 pp5-6 [3] S A Bel Esece of soluos for oe-desoal wave equaos wh olocal codos Elecro J Dff Eq76 pp - 8 [] J R Cao Y L ad](/docs-images/81/84009557/images/8-1.jpg "S Wag A plc fe dfferece schee for he dffuso equao subec o ass specfcao J Eg Sc8 99 pp573-578 [5] J R Cao ad A L aheso A uercal procedure for dffuso subec o he specfcao of ass J Eg Sc3 993 37-355 [6]")
8 7 Soayeh Nea e al: A Secod Kd Chebyshev olyoal Approach for he Wave Equao Subec o a egral Coservao Codo copuaoal effor ad hs process s very fas he ew descrbed ehod does' eed ay collocao po ad produces very accuracy resuls Fg : Graph of he e wh 68 for Eaple 5 7 Refereces [] W Ag A boudary egral equao ehod for he wo-desoal dffuso equao subec o a olocal codo Eg Aal Boud Ele 5 pp - 6 [] W Ag A uercal ehod for he wave equao subec o a o-local coservao codo Appl Nuer ah56 6 pp5-6 [3] S A Bel Esece of soluos for oe-desoal wave equaos wh olocal codos Elecro J Dff Eq76 pp - 8 [] J R Cao Y L ad S Wag A plc fe dfferece schee for he dffuso equao subec o ass specfcao J Eg Sc8 99 pp [5] J R Cao ad A L aheso A uercal procedure for dffuso subec o he specfcao of ass J Eg Sc [6] J R Cao ad J va der Hoek A plc fe dfferece schee for he dffuso equao subec o he specfcao of ass a poro of he doa : J Noye Ed Nuercal Soluos of aral Dffereal Equaos Norh-Hollad ublshg Aserda 98 pp [7] Dehgha Nuercal soluo of a parabolc equao wh o-local boudary specfcaos Appl ah Copu5 3 pp 85-9 [8] Dehgha he soluo of a al-boudary value proble ha cobes Neua ad egral codo for he wave equao Nuer ehods aral Dff Eq 5 pp - [9] Dehgha ad A Saadaad Varaoal erao ehod for solvg he wave equao subec o a egral coservao codo Chaos Solos ad Fracals 9 pp 8-53 [] Dehgha ad A Shokr A eshless ehod for uercal soluo of he oe-desoal wave equao wh JC eal for corbuo: edor@corguk
9 Joural of forao ad Copug Scece Vol 7 No 3 pp a egral codo usg radal bass fucos Nuer Algor5 9 pp 6-77 [] A B Guel W Ag ad E H wzell Effce parallel algorh for he wo-desoal dffuso equao subec o specfcao of ass J Copu ah6 997 pp [] JC aso ad DC Hadscob Chebyshev olyoals CRC ress LLC 3 [3] B J Noye Dehgha ad J va der Hoek Eplc fe dfferece ehods for wo-desoal dffuso wh a o-local boudary codo J Eg Sc3 99 pp [] Raeza Dehgha ad Razzagh Cobed fe dfferece ad specral ehods for he uercal soluo of hyperbolc equao wh a egral codo Nuer ehods aral Dff Eq 8 pp - 8 [5] A Saadaad ad Dehgha Nuercal soluo of he oe-desoal wave equao wh a egral codo Nuer ehods aral Dff Eq3 7 pp 8-9 [6] F Shaker ad Dehgha he ehod of les for soluo of he oe-desoal wave equao subec o a egral egral coservao codo Copuers ad aheacs wh Applcaos56 8 pp JC eal for subscrpo: publshg@waorguk
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