Ths arcle was downloaded by: [Jordan Unv. of Scence & Tech] On: 05 Aprl 05, A: 0:4 Publsher: Taylor & Francs Informa Ld Regsered n England and ales Regsered umber: 07954 Regsered offce: Mormer House, 37-4 Mormer Sree, London T 3JH, UK Journal of Dscree Mahemacal Scences and Crypography Publcaon deals, ncludng nsrucons for auhors and subscrpon nformaon: hp://www.andfonlne.com/lo/dmc0 An effcen algorhm for he soluon of second order PDE s usng Taylor expanson polynomals Mohammad H. Al-Towaq a & Hasan K. Al-Bzoor a a Deparmen of Mahemacs and Sascs, Jordan Unversy of scence and Technology, P.O. Box 3030, Irbd, 0, Jordan Publshed onlne: 03 Jun 03. To ce hs arcle: Mohammad H. Al-Towaq & Hasan K. Al-Bzoor (0) An effcen algorhm for he soluon of second order PDE s usng Taylor expanson polynomals, Journal of Dscree Mahemacal Scences and Crypography, 4:, 9-3, DOI: 0.080/097059.0.06983 To lnk o hs arcle: hp://dx.do.org/0.080/097059.0.06983 PLEASE SCROLL DO FOR ARTICLE Taylor & Francs makes every effor o ensure he accuracy of all he nformaon (he Conen ) conaned n he publcaons on our plaform. However, Taylor & Francs, our agens, and our lcensors make no represenaons or warranes whasoever as o he accuracy, compleeness, or suably for any purpose of he Conen. Any opnons and vews expressed n hs publcaon are he opnons and vews of he auhors, and are no he vews of or endorsed by Taylor & Francs. The accuracy of he Conen should no be reled upon and should be ndependenly verfed wh prmary sources of nformaon. Taylor and Francs shall no be lable for any losses, acons, clams, proceedngs, demands, coss, expenses, damages, and oher lables whasoever or howsoever caused arsng drecly or ndrecly n connecon wh, n relaon o or arsng ou of he use of he Conen. Ths arcle may be used for research, eachng, and prvae sudy purposes. Any subsanal or sysemac reproducon, redsrbuon, resellng, loan, sub-lcensng, sysemac supply, or dsrbuon n any form o anyone s expressly forbdden. Terms &
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An effcen algorhm for he soluon of second order PDE s usng Taylor expanson polynomals Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Mohammad H. Al-Towaq * Hasan K. Al-Bzoor Jordan Unversy of scence and Technology Deparmen of Mahemacs and Sascs P.O. Box 3030, Irbd 0 Jordan Absrac Taylor expanson mehod for he soluon of second order PDE s, s based on akng runcaed Taylor polynomal abou any pon of he coeffcens funcons has been nvesgaed by Kesan [3]. In hs paper, we develop an algorhm for he soluon of he PDE s of he form fxyu (,) xx - gxyu (,) yy = hxy (,), wh any boundary condons, based on akng runcaed Taylor polynomals abou any pon of he coeffcens funcons o oban he augmened sysems. e presen a modfed sparse echnque o solve he obaned sysems. e verfy our proposed algorhms va numercal expermens, hen we compare wh Kesan s work and he fne dfference mehod. The numercal resuls ndcae ha our algorhm s easly applcable on a wde range of PDE s and gve more accurae soluons han Kesan s and he fne dfference mehod. Also, provdes a reducon of boh compuaonal me and sorage requremens whou sacrfcng he numercal sably. keywords: Taylor Polynomal soluon, Paral Dfferenal Equaons, Algorhms, Sparse Technque, Lnear Sysems. Inroducon Our am n hs paper s usng compuer s echnology o solve some of paral dfferenal equaons by Taylor expanson mehod whch nvesgaed n [], snce he manual consruc for augmened marces akes a lo of me and effor, and any error n buldng wll be cosly. Furhermore, he obaned augmened marces are no well-consruced, alhough hey * E-mal: owaq@us.edu.o E-mal: albzoor80@yahoo.com Journal of Dscree Mahemacal Scences & Crypography Vol. 4 (0), o., pp. 9 3 Taru Publcaons
0 M. H. AL-TOAIQ AD H. K. AL-BZOOR Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 are sparse, hence he manual soluon s very cosly, ha s reasons nspre us o modfy a sraegy of sparse Gaussan elmnaon wh paral pvong o solve hs ype of sysems. The developmen of he algorhms based more upon he exenson of he mehod for hand compuaons, emprcsm, and nuon han on numercal analyss. ow, wh he developmen of echnology, useful mehods are evolvng whch come closer o full ulzaon of he nheren powers of hgh speed and large memory compung machne. Many sgnfcan mehods were dscovered o approxmae he soluon of paral dfferenal equaons, and many of mahemacans sll search o ge srong mehods and powerful echnques o solve problems n paral dfferenal equaons. Kesan [5], used Chebyshev polynomal o approxmae he soluon of second-order PDE s wh wo varables and varable coeffcens, usng he runcaed Chebyshev expansons of he funcons n he PDE s. Hence, he resul marx equaon can be solved and he unknown Chebyshev coeffcens can be found approxmaely. Kesan concluded ha he soluon of he marx equaon can be easly evaluaed for arbrary values of (x, y) a low compuaon effor, and he approxmaon soluon wll be beer f we ake more erms of he Chebyshev expanson of funcons. Kesan [4], used Taylor polynomals o approxmae he soluon of second order PDE s wh wo varables and varable coeffcens. He used runcaed Taylor polynomals of he funcons n he equaon and hen subsung her marx forms no he gven equaon, so he resul marx equaon can be solved and he unknown Taylor coeffcens can be found approxmaely. He concluded ha he approxmaon soluon wll be beer f we ake more erms of Taylor expanson. Tha s, for suffcenly large Taylor degree may ge he bes approxmaon soluon. If we choose small, we may no use he unknown Taylor coeffcens equaons of he funcons n he augmened sysem. Therefore, we wll arrve o bad approxmae soluon. In hs paper, we propose an effcen algorhm and echnques o buld and solve he augmened marx n general form. In secon (), we wll presen he proposed algorhm o ransform he paral dfferenal equaons of he form: fxyu (,) - gxyu (,) = hxy (,), (.) xx yy where 0 # x # L, and 0 # y # T under gven any boundary condons o he sysem XA= G, where X s n by n sparse marx, A s a vecor whch conans he unknown Taylor coeffcens of he soluon for he
ALGORITHM FOR THE SOLUTIO OF PDE PDE s (.), he vecor G conans he known Taylor coeffcens of he funcon hxy (,) n he PDE s (.) and he rgh-hand-sde funcons n he gven boundary condons. In secon (3), we presen a modfed sraegy algorhm of sparse Gaussan elmnaon algorhm o solve he obaned augmened sysem. The expermenal resuls and comparsons are presened n secon (3.3). Fnally, we conclude he paper n secon (4). Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05. General Taylor expanson algorhm In hs secon we propose a general algorhm whch can ransform he PDE s (.) no he sysem XA= G. The followng prelmnares gve he deas o se up our algorhm n buldng he augmened marces for he PDE s (.).. Prelmnares Theorem.. The recurrence relaon beween he Taylor coeffcens ( nm, ) ( n, m) u (,), x y and a rs, + ( n+, m) of u (,) x y, s gven by a ( n+, m) ( nm, ) rs, r+, s = ( r + ) a. ( nm, ) a rs, of Defnon.. Le A ( nm, ) be he marx of he Taylor coeffcens of u ( nm, ) (,) xy as follows : R ( nm, ) ( nm, ) ( nm, ) V Sa00, a0, g a0, S A (, ) ( nm, ) ( nm, ) ( nm, ) nm =, a Sa, 0 a g, and A = A (0,0) S h h g h S ( nm, ) ( nm, ) ( nm, ) a g, 0 a, a T, X ( nm, ) ( n, ) Corollary.. Take r = f,,, and assume ha a = a = f = ( nm, ) rs, rs, a rs, = 0 for r, hen we can ransform equaon (.7) no he sysem A ( n+, m) ( n, m) = MA, (.) where R 0 0 $ $ $ 0 V S S0 0 0 $ $ 0 S0 0 0 3 0 $ 0 S S $ $ $ $ $ $ 0 M = S$ $ $ $ $ $ S 0 0 $ $ $ $ 0 T X
M. H. AL-TOAIQ AD H. K. AL-BZOOR ( nm, ) Theorem.3. The recurrence relaon beween he Taylor coeffcens a rs, = 0 ( nm, ) ( nm, + ) ( nm, ) of u (,) xy and ars, of u (,) xy, s gven by a ( nm, + ) ( nm, ) rs, rs, + = ( s + ) a. Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Corollary.4. Take r = f,,,, and assume ha a ( rs n,, 0) = a ( rs n,, ) = f = ( nm, ) a rs, = 0 for s, hen we can ransform equaon (.0) no he sysem A ( nm, + ) ( nm, ) = MA where M as n corollary (.), and A = A., The deal proofs of he above heorems and corollares can be found n []. From he above we have R S a0, a, g a, - a, S 6a3 6 3 g 6a3 6a, 0 a,, - 3, S M A = S h h h h S ( - ) a, 0 ( - ) a, g ( - ) a, - ( - ) a S 0 g 0 S 0 g 0 T R a0 6a g ( - ) a0 00 V S, 03,, S a, 6a03, g ( - ) a, 00 AM ( ) = S h h h hh S a-, 6a-, 3 g ( - ) a-, 00 Sa, 6a, 3 g ( - ) a-, 00 T X The represenaon of Taylor expanson ( x-c0) ( y-c ) u (,) r s, ( r = 0,,, f, ; s = 0,,, f, ; = 0,,, f, ; = 0f,,,, ), n equaon (.), are obaned by means of he followng forms:, V X (, r 0 0 ) ( x- c ) u ( x, y) = XC M AY ( x- c ) u ( x, y) = XM A( C ) Y (, r 0) r (,) r s r s ( x- c0 ) u ( x, y) = XCAM ( ) Y (.4) ( y- c ) u ( x, y) = XA( C M ) Y ( 0, s) (,) r s 0 ( x-c ) ( y- c ) u ( x, y) = XCMA( C M ) Y G(,) x y = XGY, r s s
ALGORITHM FOR THE SOLUTIO OF PDE 3 Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 0, f u Then CA = ', where uv, = 0f,,,,. Therefore, C A au-, v, f u $ can be wren as: R 0 0 g 0 V S S 0 0 g 0 S h h h S CA= Sa00, a0, g a0, n (.5) Sa0, a, g a, n S S h h h San-, 0 an-, g a n-, n T X Hence, he rows of A shfng o below rows replacng hem wh zeros. R0 0 g 0 a a g a V 0, 0, 0, n- S A ( C ) = S0 0 g 0 a, a, g a, n- Sh h h h h h (.6) S T 0 0 g 0 an, an, g a n, n- X Hence, he columns of A shfng o he rgh columns replacng hem wh zeros. Lemma.5. Le C as n he above expressons, hen CAC ( ) shf all he rows of marx A below rows replacng hem wh zeros, and o he rgh columns replacng hem wh zeros. ow, we consder he followng hree dfferen cases of he funcon form f n equaon (.). Case() f = f ( x - c ), or / = 0 / Case( ) f = f ( y - c ), or = 0 / 0 0 Case( 3) f = f ( x - c ) ( y - c ). = 0 For Case (): we have he followng expresson: fxyu (,) = / f( x-c) u. or n marx forms xx = 0 0 xx = / C = 0 fxyu (,) f MA. For Case (): we have he followng expresson: fxyu (,) = / f( y-c) u. or n marx forms xx = 0 xx = / = 0 fxyu (,) fmac ( ). xx 0 xx
4 M. H. AL-TOAIQ AD H. K. AL-BZOOR For Case (3): we have he followng expresson : fxyu (,) / f( x c)( y c) uxx, or n marx forms xx = - 0 0 - = fxyu (,) = fcmac ( ). xx / = 0 Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Also, we consder he followng hree dfferen cases of he funcon g n equaon (.). / Case() g = g ( x -c ), or = 0 / 0 Case( ) g = g ( y -c ), or = 0 / Case( 3) g = g ( x -c ) ( y -c ) = 0 For Case (): we have he followng expresson: gxyu (,) yy = / g( x-c) u, or n marx forms / = 0 = 0 gxyuyy (,) = gcam ( ). 0 0 For Case (): we have he followng expresson: gxyu (,) yy = / g( y-c) u. or n marx forms = 0 yy yy = / = 0 gxyu (,) gcam ( )( C). For Case (3): we have he followng expresson: gxyu (,) yy = / g( x-c)( y-c) u, or n marx forms = 0 0 yy / = 0 gxyu (,) = gcam ( )( C). yy For more deals and o see he augmened marces whch have been used o mplemen he followng algorhm, see he submed hess n []. yy.. Taylor expanson algorhm The augmened sysem of he paral dfferenal equaons of he form (.) under he gven boundary condons akes huge me and effor n manual consrucon usng Taylor expanson mehod. In hs secon,
ALGORITHM FOR THE SOLUTIO OF PDE 5 we develop he followng algorhm o oban he augmened sysem o economze he me and effor. Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Sep. Inpu sep Inpu he Taylor degree n. For = 0f,,,, n Inpu Taylor coeffcens of he funcons f, g and h, and sore hem n he vecors f,g, and h respecvely. Inpu Taylor coeffcens of he rgh hand sde and he lef hand sde of he boundary condons and sore hem n he vecors r, r, r 3, r 4, l, l, l3, and l4 respecvely. ex Sore he unknown Taylor coeffcens of uxy (,) n he marx a as srng. Resore he elemens of a n he vecor dd( k), k = 0,,, f,( n + )( n + ). Sep. Compuaons Compue la f he boundary condons s a funcons of y or compue al f he boundary condons s a funcons of x, hen sore he obaned marces n he srng marces [ a ], = 34,,, respecvely. Sep 3. Inalzaon For = 0 o nn ( + ) and for = 0 o nn ( + ) Se aug(,) = 0, and bu(,) = 0. ex where au and bu are he marces forms of he expressons fxyuxx, (,) and gxyuyy (,) respecvely. For = 0 o nn ( + ) and for = 0 o ( n+ )( n+ ) Se aug(,) = 0. where aug s he frs augmened sysem. Sep 4. For case, seup he marx form of he expresson fxyuxx (,) Se m =, and couner = 0. For k = o n - For = ( + k- )( n+ ) o ( + k)( n+ ) - Se couner = 0 For = couner o nn ( + ) ncremen ( n + )
6 M. H. AL-TOAIQ AD H. K. AL-BZOOR Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Se au(,) = m( m -) f( couner ) Se couner = couner + ex Se m = m+ nex k. Sep 5. For case, seup he marx form of he expresson fxyuxx (,) Se m =, and couner = 0. For k = o n - For = ( + k- )( n+ ) o ( + k)( n+ ) - Se couner = 0 For = couner o k( n + ) - Se au(,) = m( m -) f( couner ) Se couner = couner + ex Se couner = couner + ex Se m = m+ nex k. Sep 6. For case 3, seup he marx form of he expresson fxyuxx (,) Se m =, and couner = 0. For k = o n - Se couner = 0 For = ( + k- )( n+ ) o ( + k)( n+ ) - Se couner = 0 Se couner 3= 0 For = couner o nn ( + ) ncremen ( n + ) If couner 3 # couner 4 Then Se au(,) = m )( m -)) f( couner ) Se couner = couner + Se couner 3 = couner 3+ End If ex Se couner = couner + Se couner 4 = couner 4- ex Se m = m+ ex k. Sep 7. For case, seup he marx form of he expresson gxyuyy (,) Se couner = 0. For k = 0 o n
ALGORITHM FOR THE SOLUTIO OF PDE 7 Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Se couner = 0 Se m = For = k( n+ ) o kn ( + ) + n Se couner 3= 0 If couner Then For = couner o nn ( + ) ncremen ( n + ) Se bu(,) = m )( m -)) g( couner 3) Se couner 3 = couner 3+ ex Se couner = couner +, and m = m+ End If Se couner = couner + ex Se couner = couner + ex k. Sep 8. For case, seup he marx form of he expresson gxyuyy (,) Se couner = 0 Se couner 4= n For k = 0 o n Se couner = 0 Se m = For = k( n+ )o k( n+ ) + n Se couner 3= 0 If couner Then For = couner o couner 4 Se bu(,) = m )( m -)) g( couner 3) Se couner 3 = couner 3+ ex Se couner = couner + Se m = m+ End If Se couner = couner + ex Se couner = couner + Se couner 4 = couner 4+ ( n + ) ex k. Sep 9. For case 3, seup he marx form of he expresson gxyuyy (,) Se couner = 0 For k = 0 o n
8 M. H. AL-TOAIQ AD H. K. AL-BZOOR Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Se couner = 0 m = For = k( n+ )o k( n+ ) + n Se couner 3= 0 If couner Then For = couner o nn ( + ) ncremen ( n + ) Se bu(,) = m )( m -)) g( couner 3) Se couner 3 = couner 3+ ex Se couner = couner +, and m = m+ End If Se couner = couner + ex Se couner = couner + ex k. Sep 0. Seup he frs augmened marx for he PDE (3..) Se couner = 0 For = 0 o nn ( + ). Se flage = False. For = 0o n( n+ ). If au(,) no equal 0 Or bu(,) no equal 0 Then se aug( couner, ) = au(, ) -bu(, ) se flage = rue End If ex If flage = rue Then Se couner = couner + End If ex For = 0o n( n+ ) Se aug (,( n+ )( n+ )) = h( ). ex Sep. Seup he second augmened marx of he boundary condons For = 0o d - Se bb() = 0, where d = n+ and bb s ( n + ) ( n + ) real vecor. For = 0o d - Se aug (,) = 0 ex ex
ALGORITHM FOR THE SOLUTIO OF PDE 9 Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Sep. Usng he frs boundary condon For = 0o d- For = 0o d- For k = 0o d - If srcop (rgh ( a (, ), ), rgh ( dd( k), ), vbexcompare) = 0 Then Comparson samen Se bb() k = val( a(,)) End If ex k ex For k = 0o d - Se aug (, k ) = bb ( k ) ex k Se aug (, d ) = r (). For k = 0o d -. Se bb() k = 0 ex k ex Sep 3. Usng he second boundary condon For = 0 o d - For = 0o d- For k = 0o d - I srcomp (rgh ( a (, ), ), rgh ( dd( k), ), vb ex compare) = 0 Then se bb() k = val ( a(,)) End If ex k. ex For k = 0o d - Se aug ( + d, k ) = bb ( k ) ex k Se aug( + d, d ) = r( ). For k = 0o d - Se bb() k = 0 ex k. ex. Sep 4. Usng he hrd boundary condon For = 0 o d - For = 0o d-
0 M. H. AL-TOAIQ AD H. K. AL-BZOOR Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 For k = 0o d - I srcomp (rgh ( a3 (, ), ), rgh ( dd( k), ), vbexcompare) = 0 Then se bb() k = val ( a3 (,)) End If ex k. ex For k = 0o d -, do sep 57. Se aug ( + d, k ) = bb ( k ) ex k Se aug ( d, d + ) = r ( ). 3 For k = 0o d -, do sep 60. Se bb() k = 0 ex k. ex. Sep 5. Usng he fourh boundary condon For = 0 o d - For = 0o d- For = 0o d- I srcomp (rgh ( a4 (, ), ), rgh ( dd( k), ), vbexcompare) = 0 Then se bb() k = val ( a4(,)) End If ex k. ex For k = 0o d - Se aug ( + 3 d, k ) = bb ( k ) ex k Se aug ( 3d, d + ) = r ( ). 4 For k = 0o d - Se bb() k = 0 ex k. ex. Sep 6. Seup he augmened sysem For = 0 o d -sd-, where s s he number of boundary condons For = 0 o d Se augmened (,) = aug(,) ex ex
ALGORITHM FOR THE SOLUTIO OF PDE Se couner = 0 For = d -sd o d - For = 0 o d Se augmened (,) = aug ( couner, ) ex Se couner = couner +. ex Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 3. umercal expermens In hs secon we wll presen a modfed sraegy of Gaussan elmnaon o solve he obaned augmened sysems n secon (.). Then comparng our resuls wh he Kesan s work and he Fne Dfference mehod. 3.. The modfed sraegy of Gaussan elmnaon From Taylor expanson algorhm, we have he sysem of lnear equaon Ax = b, where A s ( n + ) square and sparse marx, x s ( n + ) vecor whch conans he unknown Taylor coeffcens, and b s ( n + ) vecor whch conans he Taylor coeffcens of he rgh hand sde funcon h(x, y) n (.) and he Taylor coeffcens of he funcons n he boundary condons. Gaussan elmnaon and relaed sraeges seem o be he bes currenly avalable mehods. In hs secon, a smples form of Gaussan elmnaon s used. The augmened form we obaned s no suable for auomac compuaon unless essenal modfcaons are made. e nroduce a modfed algorhm; pvos are seleced among he nonzero elemens of X and brough o he man dagonal by means of row and column permuaons, he selecon beng done n such a way ha sparsy s preserved and numercal sably s ensured.to reduce he fll-n, preserve sparsy, and o ensure he numercal sably we do he followng:. Reorderng he rows and columns, we use he Duff e al. mehod [3], (mnrow n mn-column), n addon o hreshold pvong as follows: (a) search a column among columns kk, +, f, nwh leas number of nonzero elemens, and exchange wh column k even when here exss a nonzero elemen n locaon (k, k). (b) search among rows kk, +, f, na row wh leas number of nonzero elemens and havng a nonzero elemen n locaon (k, k).
M. H. AL-TOAIQ AD H. K. AL-BZOOR Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 (c) he elemen Xkk (,) should acheve he sably condon Xkk (,) $ a Xk (, ), = kk, +, f, n, where a s he sably facor and 0 a.. Usng he drop olerance x, dfferen values for he drop olerance have been used by dfferen varans of facorzaon algorhms, Al-Towaq e al. [6]. Choosng x and a randomly wll cause he followng problems: (a) a grea ncrease n compuaonal me and sorage, and (b) a converson of one one of he columns (rows) o zeros durng he marx facorzaon. The followng condons have allevaed he above dffcules: a = 0.000 x = 0 when n nz 3n a = 0.000 x = 0. 000 when 3n nz 5n where nz s he number of non-zero elemens. 3.. The fne dfference mehod (FD) The fne dfference mehod s based on local approxmaons of he paral dervaves n he Paral Dfferenal Equaon, whch are derved by low order Taylor polynomals expansons. The mehod s que smple o defne and raher easy o mplemen. Also, s parcularly appealng for smple regons, such as recangles, and when unform meshes are used. The marces ha resul from hese dscrezaons are ofen well srucured, whch means ha hey ypcally conss of a few nonzero dagonals. The doman ( 0, L) # ( 0, T) s dvded no an M# mesh wh he spaal sep sze h = L/ M n x drecon and he me sep sze k = T/. Grd pons ( x, y ) are defned by x = h, = 0,,, f, M, y = k, = 0,,, f,, n whch M and are posve negers. Usng he nal condon ux (, 0 ) = fx (), 0 x L. Equaon (3..) s solved approxmaely, commencng wh nal values uh (, 0) = fx ( ), = 0,,, f, M, and boundary values u (0, k ) = g0 ( y + ), u ( L, k ) = g ( y + ), = 0f,,,,. The dfference mehod s obaned usng he cenral-dfference for he second paral dervaves gven by: w, + w, w, () - + - uyy / k w+, w, w, () - + - uxx /, h
ALGORITHM FOR THE SOLUTIO OF PDE 3 w, w and he backward-dfference -, 0 u y /. here w, = w( x, y ), and k w s he approxmae soluon, for more deals see []., 3.3. umercal resuls and comparsons Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 In hs secon, we analyze he performance of he proposed algorhm and make comparson wh he fne dfference mehod. oe ha all he fgures bellow (-5) represen he relaons beween he proecon curves of he soluons obaned from he exac soluon, he fne dfference mehod, and he proposed algorhms. Examples. One good way o es an algorhm s o seup a problem whose soluon s known beforehand. e wll ake he followng four examples o oban he correspondng augmened marces, usng he above algorhm. The proposed algorhm s mplemened on P4 000 MHZ wh 56 MB RAM and 80 GB dsk space. Example 3.. Consder example (.), yy xx y u = u + 6; u( x, 0) = x, u ( x, 0) = 4x. Le = and ( c0, c) = ( 00, ), hen he marx equaons of he boundary condons usng expressons (.3.), (.3.), (.4.), and (.4.) are: A6 0 0@ = 60 0 @, A60 0@ = 60 4 0@. Then npu he prevous vecors n addon o he followng Taylor expansons n our program o ge he augmened marx: f = [ 0 0], snce fxy (,) =. And g = [ 0 0], snce gxy (,) =. Example 3.. Consder example (.), xx x u - y u = 0, yy wh boundary condons u( 0, y) =, u(, y) = e, u(, x 0) =, u(, x ) = e. y x Le = 4 and ( c0, c) = ( 00, ), hen he marx equaons of he boundary condons usng expressons (.3.), (.3.), (.4.), and (.4.) are:
4 M. H. AL-TOAIQ AD H. K. AL-BZOOR 60000@ A = [ 0000], 6 @ A = 8 0, B 6 A60000@ = 60000@, A 6 @ = 8 0. B 6 Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Then npu he prevous vecors n addon o he followng Taylor expansons n our program o ge he augmened marx: f = [ 0000], snce fxy (,) = x. And g = [ 0000], snce gxy (,) = y. Example 3.3. Consder he problem (.3.) n [], uyy - uxx = 0, 0 x, 0 y; wh boundary condons u( 0, y) = u(, y) = 0, ux (, 0) = sn( rx), 0 # x#, uy (, x 0) = 0, 0 # x #. Le = 4 and ( c 0, c ) =, `, hen he marx equaons of he boundary condons usng expressons (.3.), (.3.), (.4.), and (.4.) are: - - 8 B A = 600000@, 4 8 6 8 A, B = 600000@ 4 8 6 4 A - - 0 r 0 r 8 B = ; E, 4 8 6 4 A 0 3 - - 8 = 6 00000. 4 B @ Then npu he prevous vecors n addon o he followng Taylor expansons n our program o ge he augmened marx: f = [ 0000], snce fxy (,) =. And g = [ 0000], snce gxy (,) =. Example 3.4. Consder he problem (.3.3) n [5], u - u = 0, 0 # x # r, 0 y; yy xx wh boundary condons
ALGORITHM FOR THE SOLUTIO OF PDE 5 u( 0, y) = u( r, y) = 0, ux (, 0) = sn(), x 0 # x# r, uy (, x 0) = 0, 0 # x # r. Le = 5 and ( c0, c) = ^00, h, hen he marx equaons of he boundary condons usng expressons (.3.), (.3.), (.4.), and (.4.) are : Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 600000@ A = [ 000000], 3 4 5 6 r r r r r @ A = 600000@, A 00000 00 0 6 @ = 8, 6 B 0 A600 000@ = 6000000@. f = [ 000 00], snce fxy (,) =. And g = [ 00000], snce gxy (,) =. To ake a look a he obaned augmened marces usng above algorhm see he submed hess n [4]. Example ( 3.); we can no apply he fne dfference mehod, because we do no have boundary condons, whle we obaned exac soluon n 6.5E-4 seconds when we appled he proposed algorhms wh Taylor degree =, see Table. Table umercal soluon of example (.) wh =, and he duraon me n second s 6.5E-4 ( x, y ) Exac Appr. Error (0,0) 0 0 0 (0.05,0.05) 0.05 0.05 0 (0.,0.) 0.09 0.09 0 (0.5,0.5) 0.05 0.05 0 (0.,0.) 0.36 0.36 0 (0.5,0.5) 0.565 0.565 0 (0.3,0.3) 0.8 0.8 0 (0.35,0.35).05.05 0 (0.4,0.4).44.44 0 (0.45,0.45).85.85 0 (0.5,0.5).5.5 0
6 M. H. AL-TOAIQ AD H. K. AL-BZOOR Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Example (3.); also we can no apply he fne dfference mehod, because we do no have he requred nal condons uy (, x 0) = f() x and he coeffcens funcons of he dervaves mus equal, whle we obaned accurae soluon when we appled he proposed algorhms for Taylor expanson mehod wh = 4 and = 8, see Tables () and (3), respecvely. Table umercal soluon of example (.) wh = 4, and he duraon me n second s 4.65E- ( x, y ) Exac Appr. Error (0,0) 0 (0.05,0.05).005038.00483385.97E-05 (0.,0.).0005067.00990667.49E-04 (0.5,0.5).0755034.087656 4.67E-04 (0.,0.).04080774.039786667.0E-03 (0.5,0.5).064494459.066676.83E-03 (0.3,0.3).0947484.09305.87E-03 (0.35,0.35).3039.633698 4.09E-03 (0.4,0.4).735087.6806667 5.40E-03 (0.45,0.45).4460085.775469 6.73E-03 (0.5,0.5).840547.7604667 7.98E-03 Table 3 umercal soluon of example (.) wh = 8, and he duraon me n second s 0.9065 ( x, y ) Exac Appr. Error (0,0) 0 (0.05,0.05).005038.005038 0 (0.,0.).0005067.0755034.E-5 (0.5,0.5).0755034.087656.8E-3 (0.,0.).04080774.04080774 5.7E- (0.5,0.5).064494459.064494459 8.35E- (0.3,0.3).0947484.0947483 7.48E-0 (0.35,0.35).3039.30395 4.78E-09 (0.4,0.4).735087.7350847.38E-08 (0.45,0.45).4460085.4459987 9.86E-08 (0.5,0.5).840547.8405065 3.5E-07
ALGORITHM FOR THE SOLUTIO OF PDE 7 Examples (3.3) and (3.4); we appled he fne dfference mehod wh 00 grds, and he proposed algorhms for he Taylor expanson mehod wh = 7. Fgures () and (3) shows he curves obaned from he wo mehods. e can see ha he proposed algorhms s more accurae and performs beer. Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Fgure umercal soluons of example (.3), usng Taylor degree =7, and he FD wh 00 grds, versus he exac soluon Fgure umercal soluons of example (.3), usng Taylor degree =9, and he FD wh 0000 grds, versus he exac soluon
8 M. H. AL-TOAIQ AD H. K. AL-BZOOR Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Fgure 3 umercal soluons of example (.4), usng Taylor degree =7, and he FD wh 00 grds, versus he exac soluon Also, we appled he fne dfference mehod wh 0000 grds, and he proposed algorhms for he Taylor expanson mehod wh = 9. In example (3.3), fgure () shows ha he proposed algorhms and he fne dfference mehod almos he same. Bu for example (3.4), he curve of he exac soluon and he curve of he approxmae soluon usng he proposed algorhms are dencal, whle he curve of he approxmae soluon usng he fne dfference mehod s no, see Fgure (4). Also, when used 000000 grds n usng he fne dfference mehod we ge almos he exac soluon, see Fgure (5). Bu we pay he prce n he compuaonal me, ook 3.65 seconds whle he proposed algorhms ook.7 seconds o oban he exac soluon. Table (4) summarzes he oal me n seconds needed for solvng examples (3.3) and (3.4), usng he proposed algorhms and he fne dfference mehod.
ALGORITHM FOR THE SOLUTIO OF PDE 9 Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Fgure 4 umercal soluons of example (.4), usng Taylor degree =9, and he FD wh 0000 grds, versus he exac soluon Fgure 5 umercal soluons of example (.4), usng Taylor degree =9, and he FD wh 000000 grds, versus he exac soluon
30 M. H. AL-TOAIQ AD H. K. AL-BZOOR Table 4 Toal me n seconds for solvng examples (.3) and (.4). Taylor Expanson Fne Dfference Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Examples = 7 = 9 G s = 00 G s = 0000 G s = 000000 3.3 0.483875.7345 6.5E- 4.7E- g 3.4 0.47.7 7.6E- 4.7E- 3.65 4. Concluson The proposed Taylor expanson algorhm has been dscussed n deals hroughou hs paper o ge he augmened sysem of equaon (.) wh any boundary condons. Also, we used a modfed sraegy of sparse Gaussan elmnaon algorhm o solve he obaned augmened sysem from Taylor expanson algorhm. umercal examples are gven a dfferen ypes of paral dfferenal equaons of he form (.). Also, numercal expermens have been conduced. The expermenal resuls ndcae ha he proposed algorhms can be effecvely used n sofware packages for solvng PDE s of he form (.) usng Taylor expanson mehod. From he numercal examples and he numercal resuls, we conclude:. hen we appled he Taylor expanson mehod on examples (.) and (.), we obaned accurae soluons as shown n ables ( -3), bu we could no apply he fne dfference mehod. Hence, he Taylor expanson mehod s beer applcable mehod han he fne dfference mehod.. hen we appled he Taylor expanson mehod on examples (.3) and (.4) wh Taylor degree = 7, we obaned more accurae soluons han he soluons n usng he fne dfference mehod wh number of grds = 00 (fgures ( and 3)). Also, for examples (.3) and (.4), we appled he Taylor expanson mehod wh Taylor degree = 9, and he fne dfference mehod wh number of grds = 0000, we obaned accurae soluons n usng boh mehods for example (.3), see fgure (). Bu for example (.4) we ge accurae soluon n applyng Taylor expanson mehod, see fgure (4), where he me duraon s.73 seconds for Taylor expanson mehod and 4.7E- seconds for he fne dfference mehod (able (4)). hle, when we used he number of grds = 000000 n applyng he fne
ALGORITHM FOR THE SOLUTIO OF PDE 3 dfference mehod for example (.4), we obaned he same accuracy n usng Taylor expanson mehod (fgure (5)), bu he me duraon usng he fne dfference mehod s 3.65 seconds whch oo much larger han usng Taylor expanson mehod (Table (4)). Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05 Fnally, n hs paper we generalzed he Taylor expanson mehod for any degree and for more applcable problems. Also, he proposed algorhms gve very accurae soluons and offers a reducon of boh he compuer me and sorage requremens whou sacrfcng numercal sably. References [] Burden, Rchard L, Fares, and Douglas J. umercal Analyss. Brooks/ Cole; 00. [] Bzoor H., On Taylor polynomal soluons of second order lnear paral dfferenal equaons. MSc. hess, Deparmen of Mahemacs and Sascs, JUST, Jordan. [3] I. S. Duff, A. M. Ersman, and J. K. Red, Drec Mehods for Sparse Marces, Oxford Unversy Press, 986. [4] Kesan C., Taylor polynomal soluons of second order lnear paral dfferenal equaons. Appled Mahemacs and Compuaon. 5; 004: 9 4. [5] Kesan C. Chebyshev polynomal soluons of second order lnear paral dfferenal equaons. Appled Mahemacs and Compuaon. 34; 003: 09 4. [6] M. Al-Towaq, K. Day, and E. Al-Daoud, An Improved Algorhm for he Soluon of Sparse Lnear Sysems, J. Ins. Mah. and Compuer Scences, (Comp. Sc. Ser.) Vol. 4, o. (003) pp. 63 69. [7] Yousef Saad. Ierave Mehods For Sparse Lnear Sysems. second edon wh correcons. January 3rd; 000. Receved March, 00
Downloaded by [Jordan Unv. of Scence & Tech] a 0:4 05 Aprl 05