A Numerical Technique-Recursive form of Bi-cubic B-spline Collocation Solution to Laplace equation

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1 Iteatoal Joual of Coutatoal ad Aled atheatc ISS Volue ube Reeach Ida Publcato htt://wwwublcatoco A uecal Techue-Recuve fo of -cubc -le Collocato Soluto to Lalace euato Y Raahekha Redd Ch Sdha Redd V Raaa uth Atat Pofeo Deatet of atheatc JT vet college of Egeeg Jagtal achuall kodagattu Kaaga Telagaa State Ida E-al: edd4@gal co Aocate Pofeo Deatet of echacal egeeg JT vet college of Egeeg Jagtal achuall Kaaga Telagaa State Ida E-al: Sdha tuhce@gal co Pofeo Det of atheatc Oaa vet Hdeabad Telagaa State Ida E-al: v 50@gal co Abtact Teo oduct of thd degee -le ba fucto ued a ba fucto collocato ethod fo aoate oluto of ecod ode atal dffeetal euato Recuve fo of -le fucto ued a ba th eet ethod Th ethod aled to fd the aoate oluto of heat tafe goveg atal dffeetal euato wth Dchlet ouda codto fo dffeet kd of doa The eult how t effcec ad cotec The eae of the ethod educe the colet ad te coug whe coaed wth othe eted ethod Ke wod: -le collocato Lalace euato Dchlet ouda value oble Itoducto The wdel ued eh baed ethod ae fte volue ethod ad fte eleet ethod Thee ethod deed o dcetzato of doa It eued that oete egula cole geoet hould be dcetzed ad total geoet eeeted a cooed of thee all eleet Thee all eleet whch ae at oble doa a ot fo eact gve geoet Th lead to

2 7 Y Raahekha Redd et al geoetcal eo I the oce of develog the aoate oluto fo the goveg dffeetal euato the eact geoet to be ataed ode to evaluate tegato ove the ecfed doa The othe oble eo whch aed odfg the goveg dffeetal euato to weak fo ubeuetl aoate oluto develoed fo th weak fo ol Addto to thee eo eh geeato oe te coug ad cotl Owg to the dffcult of above eh ethod eh geeato dffeet te of eh fee ethod ae develoed Soe of the wdel ued eh fee ethod ae dcued[ ] The focu of the eet tud cocetated o the heat coducto oble Fo above bef evew o ehle ethod alcato olvg heat coducto oble we ca ee that evou eeache have focued al o ug EFG SPH ad LPG ethod Howeve the EFG ethod eed a backgoud eh fo the tegal the weak fo hece t ot eall eh le ethod; the SPH ad DA ad LWS ethod ae bult o the collocato ot chee fo whch the electo of the collocato ot ae otat ad the uecal accuac goe dow ea the bouda A wde age of oble have bee vetgated b Atlu ad h co-autho ug LPG ethod Alot all of the evou wok lted to heat coducto oble of egula doa Howeve a oble egeeg ae egula doa ad FV ad FE ae dffcult to decbe accuatel boudae of the egula doa ule the eh ve fe o ecal gd geeato ethod adoted whch uuall te-coug eh le ethod ca ovecoe th dffcult becaue the do ot eed eh eh le ethod dtbute abtal catteg ot the oble doa o the wll have oe advatage olvg oble wth egula doa tha FV ad FE So the eet ae we al b-cubc -le collocato ethod to coute the oluto to Lalace euato g -le ba fucto a ba collocato ethod a te edato evaluato ca be avoded uch a tog fo dffeetal euato to weak fo coveo evaluato of tegal whch ae eetal fo ot of the eted ethod ad at the ae te oe addtoal foato that alo eued to leet uch ethod I th auct a ethodolog develoed fo the oluto of Lalace euato ug -cubc -le Collocato ethod efoe gog to detaled leetato ad alcato of the eet ethod defto ad oete of -le ba fucto ad t aocated teolog eeted -Sle Suface A -le uface defed a whee ae the vetce of the olgo et called cotol ot the th degee -le ba fucto whch defed at the kot ove the kot vecto ace KV X-decto ad the th degee -le ba fucto

3 A uecal Techue-Recuve fo of -cubc -le Collocato Soluto 7 whch defed at the kot ove the kot vecto ace KV Y-decto The -le uface alo defed b ectagula aa of cotol ot Th fo et local cotol of cuve hae The degee of t ba fucto ae defed deedet of the cotol ot The uface euato ca be eeed at ado ot a at fo a [ ] Q whee Q [ T [ ] Euato the lea euato * cotol ot Fo the gve cotol ot ad choe degee -le ba fucto X- decto degee -le ba fucto Y-decto decde the kot vecto each decto The elato aog cotol ot o degee of the ba fucto ad the ube of kot vecto ube of kot vecto = ube of cotol ot + degee of the ba fucto+ The euato hould be evaluated at each cotol ot Reult of th the eued uface ot ae obtaed to geeate the uface Suoe cotol ot ae gve the euato hould be evaluated at ot wth kot vecto defed baed o the degee of ba[-4] Soe otat Poete of -le uface The au degee of the uface each aaetc decto eual to the ube of defg olgo vetce that decto The cotut of the uface each aaetc decto oe le tha the degee each decto ] -Sle Collocato ethod fo D Teo oduct of -le ba fucto ued a ba fucto collocato ethod to fd the uecal oluto fo ecod ode atal dffeetal euato Recuve fo of -le fucto eloed a ba oal collocato ethod Th ethod develoed baed o the auto that the kot vecto ae aocated wth the coutatoal odal ot ad cotat aoate oluto ae teated a cotol ot Codeg the ecod ode atal dffeetal euato of the teeatue dtbuto ove the ego a b c d

4 74 Y Raahekha Redd et al f wth the bouda codto c g a b d g a b a h c d ad b h c d whee abcd ae cotat gg ae ad h h ae fucto fucto a h Let 4 whee ' ae cotol ot the th degee le ba fucto the X decto ad th degee le ba fucto the Y decto be the aoate oluto of heat dtbuto fucto fo whch the goveg atal dffeetal euato The fucto the teeatue dtbuto ove the codeed coutatoal doa Patculal the aoate oluto aued baed o the thd degee -le ba fucto that eloed the collocato ethod I ode to have the - le oet whch atto of ut thee addtoal kot hould be take both de of kot vecto ace Let the coutatoal doa be a b c d the ode X decto ae { a b} ad the ode Y decto ae { c d } Aug that the kot vecto each decto ae ode each decto eectvel ad cotol ot ae teated a cotat o ukow euato 4

5 A uecal Techue-Recuve fo of -cubc -le Collocato Soluto 75 The ft ad ecod ode atal devatve of aoate fucto wth eect to ae deved b dffeetatg the cooet fucto wth eect to e h 5 h 6 Slal the ft ad ecod ode atal devatve of aoate oluto wth eect to gve a h 7 h 8 Slal h 9 4 Collocato ethod Collocato ethod a uecal techue It ued to etablh the elato aog cotol ot whch ae ued to ee the lea cobato of the bae fucto Th ethod covet aued aoate oluto the fo of te of lea euato ad becoe a oweful tool develog vaou aoate ethod becaue of t ot baed ad dcete atue Redue the dffeece betwee the eact oluto ad the aoate oluto The edue ade to zeo at oe dcete odal value ode to get the cotat aog the cotol ot Geeal wokg ocedue of collocato ethod If -le fucto ae ued a bae aoate oluto ad the collocato ocedue followed to obta the te of lea euato cotol ot of aoato oluto the th ethod kow a -le collocato ethod Subttutg the euato 4-9 goveg dffeetal euato 4 The we have f 0 Let be the a odal ot the coutatoal doa ad eadg the euato 0 the we have

6 76 Y Raahekha Redd et al f Eeg the above foulato at fo f cd A cd A = ' ' ' ' A A A A T A T A

7 A uecal Techue-Recuve fo of -cubc -le Collocato Soluto 77 whee = = = = f cd A T A

8 78 Y Raahekha Redd et al T A Euato the at fo of euato 0 at gle coutatoal doa ode ot It the euato cotol ot whch the eult of auto that the aoate oluto 4 atfe the goveg atal dffeetal euato at the coutatoal doa ot The coutatoal doa cot * ode ot The euato hould be evaluated at each thee coutatoal doa ode ad aued that thee ode ot ae collocato ot alo The the euato becoe te of * lea euato +*+ cotol ot whch ae ukow The at fo of above te of * lea euato gve below a A f cd * The at A cd ot uae at ad et ouda codto ae ot aled to aoate oluto Alg ouda codto to aoate oluto e 4 The total ube of odal ot o bouda 4 ad the auto that the aoate oluto atfe the bouda codto the alg euato 4 fo gve bouda codto a we have Alog the bouda ot we have h Alg all the bouda codto to aoate oluto we get Takg the bouda alog =c we have

9 A uecal Techue-Recuve fo of -cubc -le Collocato Soluto 79 c h c g c c g c c g Eeg the above euato to the at fo we have A b g 4 whee Ab c c c c c c c c c c c c c c c T The at fo 4 eteded to all odal ot alog the le =c the the te of -lea euato ae obtaed +*+ cotol ot ad t at fo gve a A b [ g ] 5 whee take the -ode alog the le =c; Slal alog the eag bouda le we have Alog the above le e =d A a [ g ] 6 whee take above bouda le ode Alog the left bouda odal ot A l [ h ] 7 Alog the ght bouda odal ot we have the lea te of -euato o we have A [ h ] 8 Aeblg all the te of lea euato whch ae geeated ove the coutatoal doa bouda ot e cobg all the euato 5 6 7

10 80 Y Raahekha Redd et al ad 8 the we have Ab Aa Al g g h A h 4 whee 4 Aeblg all the atce ode to fo the global at the ae ad 9 A cd * f ad Ab Aa Al g g h A h 4 whee 4 e The ube of euato geeated the coutatoal doa ae * ad the ube of euato obtaed b ug bouda codto ae + but coe ode ae ued two te theefoe deductg thee eetto the we have +-4 bouda codto ae ol cluded cotuctg the global at The eeated evaluato at coe ode ae eglected Acd f Ab Aa Al g g h A 4 h 4 Th ca be wtte the lfed at fo we have A C 0 Acd 9 whee A Ab Aa A l A 4

11 A uecal Techue-Recuve fo of -cubc -le Collocato Soluto 8 C f g g h = h 4 Solve the euato0 fo ukow cotat cotol ot Relacg thee value wth ukow euato 4 the the aoato oluto becoe kow aoate oluto to the euato The whole aebl of atce ad t oluto obtaed b codg atlab a baed o the leetato ocedue gve below 5 uecal Eale I th ecto a two deoal uecal eeet ae codeed a the at of tetg of alcablt of b-cubc -le collocato ethod to vaou kd of bouda value oble Lalace euato wth the o zeo bouda codto codeed ude the uecal eeet ad coaed the obtaed oluto wth the well etablhed uecal techue Fte Eleet ethod b takg a 80 eleet e 8 8 atto Th oble olved b the eet ethod codeg the et of cotol ot 5 X 5 5 cotol ot 5 X 9 ad 5 X whee the ft dgt eeet the ube of cotol ot X-decto ad the ecod dgt eeet the ube of cotol ot Y-decto The eult ae eeted table Goveg dffeetal euato fo the teeatue dtbuto 0 0 wth the bouda 5 C ; 0 0 Codto 00 C; 50 C; 75 C; Gve coutatoal doa [0 ] [0 ]ad bouda codto ae how below Fgue Fgue

12 8 Y Raahekha Redd et al Cae : Patto of the doa fo 5 5 Dvdg each de 5 ufo at of the teval [0 ] gve the total 5 collocato ot fo the uae coutatoal doa A thd degee -le ba fucto eloed collocato ethod to obta the uecal oluto fo the uecal eale Kot vecto aocate the odal ot each decto So the kot vecto X-decto ae KV = { } ad the kot vecto Y- decto KV = { } Thee addtoal kot vecto ae added both de of the kot vecto ace fo both decto ode to ata the atto of ut oet of -le ba fucto Thee addtoal kot ae codeed ol to fd the weght of kot at de the kot coutatoal doa Thee kot ae ot teated a collocato ot becaue thee kot ae outde of the doa The euato 4 aoate oluto wth the auto that the cubc degee - le ued a the ba fucto collocato ethod The coutatoal doa fo 5 5 atto ha 5 ode whch ae teated a kot to fd -le bae fucto The ode the coutatoal doa ae take a the collocato ot ad the auto that the aoate oluto atfe goveg dffeetal euato at thee collocato ot Th gve the te of 5-lea euato cotol ot ukow 6 ode ae bouda ot The bouda codto ae gve fo the gve goveg dffeetal euato The aoate oluto the oluto to the goveg dffeetal euato a the auto collocato ethod Thefoe the euato 4 hould atf the bouda codto alo aed o the latato ocedue whch gve below fo th uecal eale leeted atlab Ileetato ocedue of the ethod gve below Auto of aoate oluto a the Catea oduct of -le ba fucto each decto e 4 Subtutg the aoato oluto e 4 goveg dffeetal euato e Ste of lea euato develoed e 4 Iog the bouda codto e 5 e 6 e 7 & e 8 5 Aeblg all the euato e e 5 e 6 e 7 &e 8 6 Solve e 0 fo cotol ot [] 7 Subttute thee cotol ot cotat e 4 e 4 Table : eet the oluto of -le collocato ad Fte eleet oluto ode -le collocato oluto Fte eleet ethod

13 Y-a A uecal Techue-Recuve fo of -cubc -le Collocato Soluto Abolute elatve eo Table : Coao of abolute elatve eo ode Soe of the ot of doa ae evaluated b the eet ethod ad ae how the above Table ad value at thee ode calculated b the Fte Eleet oluto ae cluded fo the uoe of coao ad to calculate elatve eo The eet ethod efoace ove the doa cheatcall how Fgue b ug the cotou fo th 5 5 atto X-a Fgue : -cubc -le collocato oluto fo the 5 5 atto The cotou the fgue how the teeatue dtbuto of the coutatoal doa whch havg the bouda teeatue a how fgue The fgue

14 Y-a 84 Y Raahekha Redd et al eflect the teeatue cleal e the cotou whch cloe to the X-a vae betwee the teeatue 90 0C to 00 0 C alog the bouda ovded teeatue 00 0 C It alo obeved fo the fgue that the teeatue chagg fo00 0 C to 75 0 C vetcall wheea hozotall vag the teeatue fo 5 0 C to 50 0 C To ove the oothe of the cotou of teeatue dtbuto the ube of atto of coutatoal doa ceaed fo 5 5 to 5 9 whch dcued the cae Cae : Patto of the coutatoal doa fo 5 9 ube of collocato ot ceaed cae to ove the accuac of the uecal oluto oe dvo ae take the cae Total 45collcato ot ae obtaed b dog the 5 9 atto Table eet eet ethod oluto at the ae ot a calculated eale table FE oluto ad abolute elatve eo alo gve at thee ot Table Peet oluto at all the collocato ot llutated b cotou gah Fgue X-a Fgue : -cubc -le collocato oluto fo the 5 9 atto The oothe of the teeatue dtbuto cotou fo the atto 5 9 oved whe coaed wth the cotou whch ae geeated b the eet

15 Y-a A uecal Techue-Recuve fo of -cubc -le Collocato Soluto 85 oluto fo the doa 5 5 Th how the fgue Th ca be obeved thoughout doa Futhe the doa ade to oe ube of atto ode to get the oe ooth cotou ad tet the covegece of the eet - le collocato ethod whch tuded cae Cae ube of collocato ot ceaed cae to tet the covegec of the uecal oluto oe dvo ae take the cae Total 65collcato ot ae obtaed b dog the 5 atto Table eet eet ethod oluto at the ae ot a calculated eale FE oluto ad abolute elatve eo alo gve at thee ot Table Peet oluto at all the collocato ot llutated b cotou gah Fgue X-a Fgue 4: -cubc -le collocato oluto fo the 5 atto 6 Reult ad Dcuo Lalace -D heat coducto oble wth the bouda codto llutated to deotate the eet ethod Teted the ethod b chagg the ube of collocato ot Itall coutatoal doa ade to 5 5 atto ad etated the teeatue at vaou ode the coutatoal doa Thee etated value ae how Table The teeatue at d-ot

16 Aveage abolute Relatve Eo 86 Y Raahekha Redd et al 96 Whe ceaed the ube of atto of coutatoal doa 5 5 to 5 9 ad the to5 t obeved that teeatue at the d 5 5 deceaed fo to ad the to 6 45 We ca ee that the aveage abolute elatve eo cotatl deceag a the ube of collocato ot ae ceaed whch gahcall how Fgue 5 Coeuece of thee eult we ca a that the eet ethod coveget Fgue Fgue ad Fgue 4 eet the efoace of eet uecal ethod thoughout the coutatoal doa Alo t obeved fo the Fgue that oothe oved a the ube of collocato ot ae ceaed ube of ode Fgue 5 Coae the Aveage Abolute Relatve Eo ad ube of ode 7 Cocluo Recuve fo of -cubc -le collocato ethod develoed ad aled fo the two deoal Poo euato wth teeatue a the feld vaable Dchlet fo of bouda codto ae codeed fo the aoate oluto The eult obtaed b ug the b-cubc -le collocato ethod ae good ageeet wth the fte eleet oluto

17 A uecal Techue-Recuve fo of -cubc -le Collocato Soluto 87 Refeece [] LuG R Lu Soothed Patcle Hdodac: A ehfee Patcle ethod Wold Scetfc Publhg Co Pte Ltd Sgaoe 00 [] Luc L A uecal aoach to tetg of the fo hothe Ato J [] oagha J J Soothed atcle hdodac Au Rev Ato Atoh [4] eltchko T Y Y Lu L Gu Eleet-fee Galek ethod It J ue ethod Eg [5] Oate E S Ideloh O Zekewcz R L Talo A fte ot ethod coutatoal echac alcato to covectve taot ad flud flow It J ue ethod Eg [6] Atlu S T Zhu A ew ehle local Petov Galek LPG aoach coutatoal echac Cout ech [7] Atlu S T Zhu A ew ehle local Petov Galek LPG aoach to olea oble coute odelg ad ulato Cout odel Sulat Eg [8] Atlu S S P She The ehle Local Petov Galek LPG ethod Tech Scece Pe Eco SA 00 [9] Sadat H S Coutue Pefoace ad accuac of a ehle ethod fo laa atual covecto ue Heat Tafe Pat [0] Schee C E E Kazkuewcz ad agav acch A ultlevel Schwaz hootg fo the oluto of the Poo euato two deoal coeble flow ulato Al ath ad Cout [] Cal De oo O Calculatg wth -le JORAL OF APPROXIATIO THEORY6 SO-6 97 [] Cu H ad Schoebeg I J O le dtbuto ad the lt: the Pola dtbuto Abt ull Ae ath Sot [] Cu H ad I J Schoebeg O Pola feuec fucto IV: The fudaetal le fucto ad the lt J Aal ath [4] CRRY H AD I J SCHOEERG O Pola feuec fucto IV: The fudaetal le fucto ad the lt J Aal ath [5] Davd F Roge ad J Ala Ada atheatcal Eleet fo Coute Gahc d ed Tata cgaw-hll Edto ew Delh

18 88 Y Raahekha Redd et al