The Modified Bi-quintic B-spline Base Functions: An Application to Diffusion Equation

Transcription

1 Iteratoal Joural of Partal Dfferetal Equatos ad Applcatos 017 Vol. No Avalable ole at Scece ad Educato Publshg DOI: /jpdea--1-4 The Modfed B-qutc B-sple Base Fuctos: A Applcato to Dffuso Equato S. Kutlua N. M. Yagurlu * Departet of Matheatcs Facult of Arts ad Sceces İöü Uverst Malata Turke *Correspodg author: urat.agurlu@ou.edu.tr Abstract I ths paper the b-qutc B-sple base fuctos are odfed o a geeral -desoal proble ad the the are appled to two-desoal Dffuso proble order to obta ts uercal solutos. The coputed results are copared wth the results gve the lterature. Kewords: Galerk Fte Eleet Method B-qutc B-sples Two-desoal B-sples Modfed b-qutc B-sples Two-desoal Dffuso Equato Cte Ths Artcle: S. Kutlua ad N. M. Yagurlu The Modfed B-qutc B-sple Base Fuctos: A Applcato to Dffuso Equato. Iteratoal Joural of Partal Dfferetal Equatos ad Applcatos vol. o. 1 (017): 6-3. do: /jpdea Itroducto I ths paper we wll cosder two desoal Dffuso equato of the for u u 1 u D t 0 K t wth the tal codto 0 s s u u ad boudar codtos (1) 0 () u t g t t 0 (3) u t g t t 0 (4) 0 u t h t t 0 () u t h t t 0 (6) 0 o the rego D The eact soluto of ths proble s [1] u t e Kt s s (7) K s a dffuso coeffcet. The fte eleet ethod has bee wdel appled to phscs sold ad flud echacs egeerg edce ad so o [-1]. Partcularl Kas ad Koeru [13] have used the Galerk fte eleet ethod for oedesoal ad two-desoal te depedet probles b odfg bcubc B-sple base fuctos. Moo et al. [1] have obtaed a o-separable soluto of the dffuso equato based o the Galerk s ethod usg cubc sples. Tavakol ad Dava [14] have developed a parallel ucodtoall stable full eplct fte dfferece schee for soluto of the dffuso equato. Ag [1] has proposed a boudar tegral equato ethod for the uercal soluto of the two-desoal dffuso equato subject to a o-local codto. Velvell ad Brde [16] have eaed cache optzato for the Lattce Boltza ethod both seral ad parallel pleetatos b utlzg the two-desoal dffuso equato. Aboaber ad Haada [17] have developed a geeralzed Ruge-Kutta ethod for the uercal tegrato of the stff space-te dffuso equatos. Bhaskar et al. [18] have developed Heatlets the fudaetal solutos of heat equato usg wavelets for uercall solvg hoogeeous ad hoogeeous tal value probles of dffuso equato o ubouded doas. Sterk ad Trobec [19] have gve the dervato ad pleetato of a uercal soluto of a tedepedet dffuso equato detal based o the eshless local Petrov-Galerk ethod. I ths paper we have frst odfed b-qutc b-sple fuctos o the boudar of a geeral two desoal desoal proble ad used the to obta uercal solutos of the Dffuso proble b the Galerk fte eleet ethod. The odfed b-qutc B-sples are used as bass fuctos ad rectagles as eleet shapes. We tr a b-qutc B-sple fucto of the for ( ) u t t B B j j j as a approato soluto to two desoal dffuso proble. I order to fd a approate soluto the above for to the proble b the Galerk ethod frst of all we have to redefe the B-sple bass fuctos to a ew set of fuctos ael odfed b-qutc B-sple fuctos. Ths redefg process was successfull appled to cubc B-sple fuctos to obta odfed b-cubc B-sple fuctos [13]. The ewl obtaed set of odfed fuctos are detcall zero o the boudar of the gve proble. It should be poted out that ths process s ecessar sce B-sple bass

2 Iteratoal Joural of Partal Dfferetal Equatos ad Applcatos 7 fuctos B ()( = (1)( + ) the -drecto ad the B-sple bass fuctos B j ()(j = (1)( + ) the -drecto are ot zero o the boudar of the proble. After the redefg process of the bass fuctos we ca ow tr the odfed b-qutc B-sple fuctos of the for 1 1 ( ) u t t t B B j j 1 j1 as ts approate soluto B ad Bj are the qutc B-sples to be odfed ol o the boudar the ad - drectos respectvel. Sce all of the odfed qutc B-sples are zero o the boudar of the proble o-hoogeous boudar codtos are t. satsfed b the ter [0]. B j () ca easl be foud b replacg wth j ad wth. Fgure 1 depcts a rego h = h = 1 so that t s dvded to fte eleets b the teger kots ( j) ad a sgle b-qutc B-sple B 33 whch peaks o the pot (3 3) ad also covers a total of 36 square eleets. Whe the etre set of b-qutc sples B j each of whch peaks o a kot ( j) j 6 are added to ths fgure a total of 36 sples cover each fte eleet [1].. Dervato of the Modfed B-qutc B-sples.1. B-qutc B-sple Eleet The rectagular rego D of the proble s subdvded to a uber of ufor rectagular fte eleets of sdes h ad h b the kots 0 0 j. A approato B-sple fuctos to u t s take of the for u t wth qutc j j (8) j u t t B j t's are the apltudes of b-qutc B-sples Bj gve b ad B () s defed as B B B j j (9) B 6 3 h otherwse. Fgure 1. The b-qutc B-sple B 33 cetered o (33) coverg 36 fte eleets of sde 1.. A Modfed B-qutc B-sple Eleet To show how to odf b-qutc sple fuctos o the boudar we cosder the two-desoal geeral lear equato of the for u u u u a t b t c t t t u d t e t u f t subject to the tal codto ad boudar codtos 0 u u (10) 0 () u g (1) u g 0 (13) u h (14) u h 0 (1) 0 0 ad g1 g h h 1 are gve fuctos D s a rectagular rego R wth boudar D. Now t s supposed that both the -space varable doa ad -space varable doa of the sste (10)-(1) are dvded to ad subtervals respectvel b the set of + 1 dstct grd pots 01 ad + 1 dstct grd pots j 01 such that j

3 8 Iteratoal Joural of Partal Dfferetal Equatos ad Applcatos ad Sce a qutc B-sple fucto covers s cosecutve eleets we add te addtoal grd pots the -drecto ad te addtoal grd pots the -drecto such that h h h h To fd a approate soluto the for of Eq. (8) to the proble gve b Eqs. (10)-(1) wth the Galerk ethod we do eed to redefe the bass fuctos to a ew set of bass fuctos whch all vash o D. The redefg process of the bass fuctos s doe the followg three steps. Step 1. The approate soluto u t gve b Eq. (8) ca also be wrtte as [13] u t B (16) j t Bj. (17) j t gve b Eq. (16) to satsf the boudar codtos (14) ad (1) ad elatg ad fro the resultg equatos we obta Allowg the approate soluto u u t B 1 B (18) h1 B h B 0 1 B B 0 B B B for 101 (19) B 0 B B for (0) B B B B B for 1 1. (1) Step. B evaluatg the epresso gve b Eq. (17) at 0 ad ad elatg ad fro the resultg equatos we obta B 1 B 0 j B j B 0 j1 B () Bj 0 B j B j B for j 101 (3) B 0 B j B j for j (4) B B B B j j j B for j 1 1. () Step 3. Fall substtutg Eq. () to Eq. (18) ad allowg the resultg equato to satsf the boudar codtos (1) ad (13) we obta j j (6) 1 j1 u t t t B B B B 1 t h1 h B 0 B B B g1 g B 0 B B B B h1 0 h 0 B 0 B 0 B B B B h1 h B B 0 B (7) Applg eactl the sae three steps but ow wrtg the approate soluto (8) as [13] we obta j j (8) j u t B j j t B (9) j 1 1 j j (30) 1 j1 u t t t B B B B t h1 h B 0 B B B g1 g B 0 B B B B g1 0 g1 B 0 B 0 B B B B g 0 g B B 0 B (31)

4 Iteratoal Joural of Partal Dfferetal Equatos ad Applcatos 9 (33) the weak for of the odel proble the global coordate sste s obtaed as follows 1 u u u dd 0. K t (34) D To chage fro the global coordate sste to the local oe we use the trasforatos h ad h. Thus the weak for (34) trasfors to the for h u u u d d d d 0. K t (3) Fgure. Modfed B-sple fuctos B 1 1 B takg the average of Eq. (7) ad Eq. (31) u t to we obta the geeral approato u t of the for u t t t j j 1 j1 1 1 t B B t j t B B j 1 j1 (3) the ew set of bass fuctos are B B j for 1 j 1 D ad t whch all vash o gve Eq. (3) satsfes the boudar codtos gve b Eqs. (1)-(1). The profles of the odfed qutc B-sples are show Fgure for 4 eleets. 3. Nuercal Eaple ad Results I ths secto we wll tr to obta the uercal solutos of the dffuso proble gve b Eqs. (1)-(6) usg the Galerk fte eleet ethod wth the odfed b-qutc B-sple base fuctos. The dffuso equato s tegrated space varables ad. To appl the ethod to the proble frst of all we eed to costruct the weak for of the proble Weak For of the Model Proble For ths purpose all ters Eq. (1) are take to the rght had sde of the equato ad the ultpled b the weght fucto Ψ( ). Fall b tegratg the resultg equato over the rego D ad settg t to zero we get u u 1 u dd 0 K t D (33) B B k l for k = 0(1) ad l = 0(1). B applg the Gree Theore (see e.g. Redd []) to Eq. 3.. Galerk Fte Eleet Solutos of the Model Proble I ths subsecto the uercal solutos of the odel proble are obtaed b the Galerk fte eleet ethod usg the odfed b-qutc Bsple bass fuctos. The uercal schee s pleeted b dvdg the rego D to 16h h 1/ 4 ad 1 h h 1/ 10 eleets. The obtaed solutos are copared wth those estg the lterature ad tabulated wth error ors L ad L. I all uercal 6 calculatos the coeffcet K Eq. (1) s take as 10. For ths the approate soluto u t for each eleet s wrtte the weak for (3) ad the a coeffcet atr s obtaed for each eleet. B cobg the coeffcet atr for each eleet we obta a algebrac equato the for ZA0 ad a teratve equato the for da t G dt g (36) PA t (37) 1 1 G tp A tr1 G tp A tr G g l jl g j h B B j Bk Bl d d k l P p l jl B Bk j l l j l B j Bl k (38) (39) p B B dd (40a) B B dd A t s t A 1 t A t... A 33 t ; (41) j k l = 0(1) l = j( + 1) + ad j l = l( + 1) + k

5 30 Iteratoal Joural of Partal Dfferetal Equatos ad Applcatos Z z j j j k l l l z l l B B B B d d (4) j k l = 0(1) l = j( + 1) + 1 ad j l = l( + 1) + k g g g u B B d d. l l 0 j (43) j = 0(1)1 l = j( + 1) + 1. The dvso of the rego to 4 4=16 eleets If the soluto doa D of the proble s equall dvded to 4 4 = 16 eleets 16 squares havg the sdes of h = h = h = 1/4 are obtaed. If the approate soluto u t s costructed for each eleet a total uber of 7 7 = 49 global eleet paraeters over the rego D are obtaed depedg o the local eleet paraeters α (t) ad β j (t) ( j = 1(1)). It s obvous that we eed to fd the global eleet paraeters A (t) ( = 1(1)49) order to obta the approate soluto u (ξ η t) for each eleet. B solvg these algebrac equatos eleet paraeters A (t) (=1(1)49) are obtaed at tes t = 0.0 ad t = 1.0.The obtaed eleet paraeters are put ther places the eleet equatos ad the the approate soluto u (ξ η t) for each eleet at tes t = 0.0 ad t = 1.0 s foud. The dvso of the rego to 10 10=1 eleets Now the soluto doa D of the proble s equall dvded to = 1 eleets 1 squares havg the sdes of h = h = h = 1/10 are obtaed. As 4 4 = 16 eleets f the approate soluto u t s costructed for each eleet a total uber of = 169 global eleet paraeters over the rego D are obtaed depedg o the local eleet paraeters t ad t ( = 1(1)). It s obvous that we eed to fd the global eleet paraeters A t ( = 1(1)49) order to obta the approate soluto u t for each eleet. For ths the approate soluto u t for each eleet s wrtte the weak for (3) ad the a coeffcet atr s obtaed for each eleet. B cobg the coeffcet atrces for each eleet frst we obta a algebrac equato the for Eq. (36) the a teratve equato the for of Eq. (38) b applg forward dfferece ad Crak- Ncolso fte dfferece forula to Eq. (37). B solvg these equatos wth the help of a coputer progra we obta the global eleet paraeters A t ( = 1(1)169) at tes t = 0.0 ad t = 1.0 respectvel. These obtaed eleet paraeters are put ther places the eleet equatos ad the the approate solutos u t for each eleet at tes t = 0.0 ad t = 1.0 are foud. The obtaed uercal results b the preset ethod usg the odfed b-qutc B-sples have bee dsplaed ad also copared wth ts eact oes Table 3 Table 6. It s obvousl see fro the tables that the uercal results are good agreeet wth the eact oes. Fgure 3 ad Fgure 4 show graphcall how closel the approate soluto u t atches wth the eact soluto u t at tes t = 0.0 ad t = 0.0 respectvel. Sce the uercal soluto s ver close to the eact soluto ther graphs are dscratel slar to each other. I Table ad Table 6 the uercal solutos obtaed b the Galerk fte eleet ethod wth odfed bqutc B-sple bass fuctos are copared wth the eact oes at tes t = 0.0 ad t = 1.0 respectvel. Table 1. Nuercal ad eact solutos of the odel proble for 4 4 = 16 eleets at t = 0.0 Nuercal Soluto (u ) Eact Soluto (u) ( ) 4 4 (0.0.) (0.0.0) (0.0.7) (0..) (0..0) (0..7) (0.70.) (0.70.0) (0.70.7) Table. Nuercal ad eact solutos of the odel proble for 4 4 = 16 eleets ad Δt = 0.0 at t = 1.0 Nuercal Soluto (u ) ( ) 4 4 Eact Soluto (u) (0.0.) (0.0.0) (0.0.7) (0..) (0..0) (0..7) (0.70.) (0.70.0) (0.70.7) Table 3. Nuercal ad eact solutos of the proble for = 1 eleets at t = 0.0 Nuercal Soluto (u ) Eact Soluto (u) ( ) 4 4 (0.0.) (0.0.0) (0.0.7) (0..) (0..0) (0..7) (0.70.) (0.70.0) (0.70.7) Table 4. Nuercal ad eact solutos of the proble for = 1 eleets at t = 1.0 wth Δt = 0.0 Nuercal Soluto (u ) ( ) 4 4 Eact Soluto (u) (0.0.) (0.0.0) (0.0.7) (0..) (0..0) (0..7) (0.70.) (0.70.0) (0.70.7)

6 Iteratoal Joural of Partal Dfferetal Equatos ad Applcatos 31 Fgure 3. (a) Eact soluto (b) Nuercal soluto for 4 4 = 16 eleets ad (c) Nuercal soluto for = 1 eleets at t = 0 Fgure 4. (a) Eact soluto (b) Nuercal soluto for 4 4 = 16 eleets ad (c) Nuercal soluto for = 1 eleets at t = 1.0 wth Δt = 0.0 Table. Nuercal ad eact solutos of the proble for 4 4 = 16 ad = 1 eleets at t = 0.0 Table 6. Nuercal ad eact solutos of the proble for 4 4 = 16 ad = 1 eleets at t = 1.0 wth Δt = 0.0 Nuercal Soluto (u ) Eact Soluto (u) Nuercal Soluto (u ) Eact Soluto (u) ( ) (1..0) (0.90.1) (0.80.) (0.70.3) (0.60.4) (0.0.) (0.40.6) (0.30.7) (0.0.8) (0.10.9) (0.01.0) (0..0) (0.10.1) (0.0.) (0.30.3) (0.40.4) (0.0.) (0.60.6) (0.70.7) (0.80.8) (0.90.9) (1.01.0) ( ) (1..0) (0.90.1) (0.80.) (0.70.3) (0.60.4) (0.0.) (0.40.6) (0.30.7) (0.0.8) (0.10.9) (0.01.0) (0..0) (0.10.1) (0.0.) (0.30.3) (0.40.4) (0.0.) (0.60.6) (0.70.7) (0.80.8) (0.90.9) (1.01.0)

7 3 Iteratoal Joural of Partal Dfferetal Equatos ad Applcatos Table 7. Coparso of the error ors L ad L of the odel proble wth results fro [3] Cubc [3] Qutc t=0.0 t=1.0 Nuber of Eleets L L L L I order to easure how good the uercal solutos obtaed b the Galerk fte eleet ethod wth the b-qutc B-sple bass fuctos the error ors L ad L defed as L u u p u 1 p 1 u u L u u a u u 0 p are coputed ad tabulated Table 7. I L ad L p s the uber of er pots u ad u are the eact ad approate solutos at the pot. respectvel. The error ors L ad L are coputed b takg the values u ad u at = 1681(= p ) pots obtaed b dvdg the rego D = [0 1] [0 1] to 40 equal eleets the drectos ad. As see fro the table the approate solutos becoe better as the uber of eleets crease. 4. Cocluso I ths paper a odfed b-qutc B-sple fte eleet ethod s proposed ad successfull appled to two desoal Dffuso proble to obta ts uercal solutos. The agreeet betwee our uercal results ad the eact soluto s satsfactorl good. The obtaed uercal results showed that the preset ethod s a rearkabl successful uercal techque ad ca also be appled to a large uber of phscall portat two desoal o-lear probles. Refereces [1] B. S. Moo D. S. Yoo Y.H. Lee I.S. Oh J.W. Lee D.Y. Lee ad K.C. Kwo A o-separable soluto of the dffuso equato based o the Galerk s ethod usg cubc sples Appl. Math. ad Coput. 17 (010) [] D. L. Loga A Frst Course the Fte Eleet Method Thoso Learg part of the Thoso Corporato New York 7. [3] M. A. Bhatt Fudaetal Fte Eleet Aalss ad Applcatos: wth Matheatca ad Matlab Coputatos Joh Wle & Sos Ic. New York. [4] J.N. Redd A troducto to Nolear Fte Eleet Aalss Oford Uverst Press New York 8. [] E. G. Thopso Itroducto To The Fte Eleet Method Theor Prograg ad Applcatos Joh Wle Sos Ic. USA. [6] J. Wu X. Zhag Fte Eleet Method b Usg Quartc B- Sples Nuercal Methods for Partal Dfferetal Equatos 10 (0) [7] O.C. Zekewcz The fte eleet ethod McGraw-Hll Lodo [8] S.Kutlua A. Ese I.Dağ Nuercal solutos of the Burgers equato b the least-squares quadratc B-sple fte eleet ethod J Coput Appl Math 167 (4) [9] H.Batea Soe recet researcher o the oto of fluds Mo. Weather Rev. 43 (191) [10] J.M. Burger A ateatcal odel llustratg the theor of turbulece Advaces Appled Mechacs I Acadec Press New York (1948) [] J.D. Cole O A Quas Lear Parabolc Equao Occurg Aerodacs Quart. Appl. Math. 9 (191) -36. [1] P. Jaet R. Booret Nuercal soluto of copressble flow b fte eleet ethod whch flows the free boudar ad the terfaces J. Coput. Phs. 18 (197) 1-4. [13] K.N.S. Kas Vswaadha S.R. Koeru Fte eleet ethod for oedesoal ad two-desoal te depedet probles wth B-sples Coput. Methods Appl. Mech. Egrg. 108 (1993) 01-. [14] R.Tavakol ad P. Dava D parallel ad stable group eplct fte dfferece ethod for soluto of dffuso equato Appl. Math. ad Coput.188 (7) 84-9 [1] W.T. Ag A boudar tegral equato ethod for the twodesoal dffuso equato subject to a o-local codto Egeerg Aalss wth Boudar Eleets (1) 1-6. [16] A. C. Velvell ad K. M. Brde A cache-effcet pleetato of the lattce Boltza ethod for the two-desoal dffuso equato Cocurrec Coputat. Pract. Eper. 16 (4) [17] A.E. Aboaber ad Y.M. Haada Geeralzed Ruge Kutta ethod for two- ad three-desoal space te dffuso equatos wth a varable te step Aals of Nuclear Eerg 3 (8) [18] T. Gaa Bhaskar S. Harhara ad Neela Nataraj Heatlet approach to dffuso equato o ubouded doas Appl. Math. ad Coput. 197 (8) [19] M. Sterk ad R.Trobec Meshless soluto of a dffuso equato wth paraeter optzato ad error aalss Egeerg Aalss wth Boudar Eleets 3 (8) [0] P.M. Preter Sples ad varatoal ethods Wles New York 197. [1] L.R.T. Garder G.A. Garder A two desoal b-cubc B- sple fte eleet: used a stud of MHD-duct flow Coput. Methods Appl. Mech. Egrg. 14 (199) [] J.N. Redd A troducto to the fte eleet ethod McGraw-Hll Iteratoal Edtos thrd ed. New York 6. [3] N.M. Yagurlu Nuercal Solutos of -Desoal Partal Dfferetal Equatos wth B-sple Fte Eleet Methods Ph.D. Thess I o u Uverst Malata (Turke) 0.