Tenth Order Compact Finite Difference Method for Solving Singularly Perturbed 1D Reaction - Diffusion Equations

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1 Internatonal Journal of Engneerng & Appled Senes (IJEAS) Vol.8, Issue (0)5- Tent Order Compat Fnte Dfferene Metod for Solvng Sngularl Perturbed D Reaton - Dffuson Equatons Faska Wondmu Galu, Gemes Fle Duressa and Tesfae Aga Bullo Department of Matemats, Dlla Unverst, Dlla, P. O. Box 9, ETHIOPIA Department of Matemats, Jmma Unverst, Jmma, P. O. Box 78, ETHIOPIA e-mal: gammeef@aoo.om Reeved date: Jul 0 Aepted date: September 0 Abstrat In ts paper, tent order ompat fnte dfferene metod ave been presented for solvng sngularl perturbed two-pont boundar value problems of D reaton-dffuson equatons. Te dervatves n te gven dfferental equaton ave been replaed b fnte dfferene approxmatons and transformed to tr-dagonal sstem w an easl be solved b Dsrete Invarant Imbeddng algortm. Te teoretal error bounds ave been establsed for te metod. Tree model examples ave been onsdered to ek te applablt of te proposed metod. Te numeral results presented n tables sow tat te present metod approxmates te exat soluton ver well. Kewords: Sngular perturbaton, ompat dfferental dfferene metod, reaton-dffuson equatons.. Introduton An dfferental equaton n w te gest order dervatve s multpled b a small postve 0 s alled sngular perturbaton problem and te parameter s known as te parameter perturbaton parameter. Tese tpes of problems arse frequentl n man felds of appled matemats and engneerng, lke quantum matemats, flud dnams, emal reatons, eletral network, nulear pss, elastt, dro-dnams, modelng of semondutor deves, dffraton teor and reaton-dffuson proesses and man oter alled areas. Classal omputatonal approaes to sngularl perturbed problems are known to be nadequate as te requre extremel large numbers of mes ponts to produe satsfator omputed solutons Farrell et al. [] and Roos et al. []. Detaled dsussons on te teor of asmptotal and numeral solutons of sngular perturbaton problems ave been publsed (see [ - 9]). So, te treatment of sngularl perturbed problems presents severe dffultes tat ave to be addressed to ensure aurate numeral solutons (see [0 - ]). It s well-known fat tat te soluton of sngular perturbaton problem exbts a mult-sale arater tat s; tere are tn laer(s) were te soluton vares rapdl, wle awa from te laer(s) te soluton beaves regularl and vares slowl. However, most of te exstng lassal fnte dfferene metods w ave been used n solvng sngular perturbaton problems gve good result onl wen te mes sze s mu less tan te perturbaton parameter w s ver ostl and tme onsumng. 5

2 In ts paper, tent order ompat fnte dfferene metod s presented for solvng seond-order self-adont sngularl perturbed D reaton-dffuson problems. Compat fnte dfferene metod s a fnte dfferene metod w emplos a lnear ombnaton of tree onseutve ponts of dervatves to approxmate a lnear ombnaton of te same tree onseutve values of a funton ( x ),,,. To valdate te effen of te metod, tree modal examples are solved for dfferent values of te parameter, mes lengt and ompare te maxmum absolute error wt te more urrentl publsed papers.. Desrpton of te Metod Consder te followng sngularl perturbed D reaton-dffuson equaton of te form: ( x) a( x) ( x) f ( x); 0 x, () wt te Drlet boundar ondtons ( 0), () () were s a small postve parameter (dffuson oeffent) su tat 0 and, are gven onstants and a (x); f (x) are assumed to be suffentl ontnuousl dfferentable funtons su tat a ( x) 0 for ever x [0, ], were s some postve onstant. To desrbe te seme, we dvde te nterval [0, ] nto N equal subntervals of mes lengt Let x, x,..., x be te mes ponts. Ten, we ave x 0, 0,,..., N. 0 N ( n) ( n) For onvenene, let a( x ) a, f ( x ) f, ( x ), ( x ), ( x ), ( x ). Assume tat (x) as ontnuous ger order dervatves on [ 0, ]. Usng Talor seres expanson, we ave: (5) () (7) (8)!!! 5!! 7! 8! 9 0 (9) (0) () () () O( ) 9! 0!!! (5) () (7) (8)!!! 5!! 7! 8! 9 0 (9) (0) () () O( ) 9! 0!!! Subtratng Eq. from Eq. (), we obtan te seond order fnte dfferene approxmaton ( ) for te frst dervatve of s: (5) were Addng Eqs. () and, we obtan te seond order fnte dfferene approxmaton ( ) for te seond dervatve of s: x

3 () were Substtutng Eqs. () and nto Eqs. (5) and () elds 8 (5) (7) (9) 0 5,00,880 (7) 0 () were 9,9,800 8 () (8) (0) 0 0,0,8,00 (8) 0 () were 9,500,800 Wrtng Eq. () at dsretzed mes, we obtan a f (9) Dfferentatng Eq. (9) twe, four and sx tmes respetvel, we ave a f (0) ( ) a a f f () () ( 8) a a a f f f () B applng to n Eq. (), we obtan: (8) (0) (8) (8) T () Substtutng Eq. () n Eq. () elds: ( ) 0 () 8 0,0,8, (8) ( ) ( ) were 5,8,00 9,958,00 Substtutng Eqs. (0), () and () nto Eq. (), gves: 8 a a a a f 0 0,0,8, ,0 8 8 a () f f 5,8, 00 0,0,8, 00 8 a a a a 0 0,0,8, 00 (5) (8) T 5 () 7

4 B substtutng Eq. (5) for te value nto Eq. () and rearrangng, we obtan: a a a a, 8, ,0 8 a 8 a a a a f 0 0,0,8, 00 f 8 a a a f 0 0,0, 8, a () () f f f 5, 8, 00 0,0, 8, 00 a f f 0 0,0 () Substtutng Eq. () nto Eq. () for ( ) togeter wt f f f f f f f, f, () () () f f f () f f f f f and rearrangng, we obtan te equvalent tree-term reurrene relaton gven b: E F G H,,,..., N (7) were F E a G,8, 00 a a a 0 a,,800 a a a a a H f f f,8,00 0,800,8,00 a a a a f f f,8, 00 0,800,8, 00 a f,8, 00 a a f f 0,800,8, 00 () () () f f f,8, 00,800,8, 00 Eq. (7) gves us te tr-dagonal sstem w an easl be solved b applng Tomas Algortm.. Convergene Analss Wrtng te tr-dagonal sstem Eq. (7) above n matrx vetor form, we obtan 8

5 AY C (8) were A m ),, N s a tr-dagonal matrx of order N, wt m m ( a,8,00 a a a 0 a,800 a m,8,00 and C H For,,,..., N and wt te loal trunaton error 0 0 a () ( ) (9),77,800 9,958,00 We also ave AY ( ) C (0) t werey,,,..., ) denotes te exat soluton and ( 0 N t ( ) ( ( ), ( ),..., ( )) 0 N N denotes te loal trunaton error. Makng use of Eq. (9) and Eq. (0), we obtan an error equaton: AE () t were E Y Y e, e,..., ). ( 0 e N Let S be te sum of elements of te N S m, for. t row of A, ten a a a a a 0,800,8,00 a a 89a Terefore, S a, for 0,8,00 N m S, for,,..., N a a a a a,8, 00 0,800 a a a Terefore, S a A 0, were A0 for a mn S 0 8,0 N N S, for N. N m N, () 9

6 a a a a a,8,00 0,800 a a 89a Terefore, S N a 0,8,00 for N Sne 0, we an oose suffentl small so tat te matrx A s rreduble and monotone [7]; Ten t follows tat A exsts and ts elements are non-negatve. Hene, from Eq. (), we get E A. T( ) () and E A. T( ) () Let m, be te k t ( k, ) elements of A. Sne mk, 0, matres wt ts nverses we ave N b te defnton of multplaton of mk,. S, k,,..., N () Terefore, t follows tat N m k, mn S a (5) N We defne A max N mk, and T( ) max T ( ). N N From Eqs. (0), () and () and (), we obtan: e e N mk,. T ( ),,..., N. T ( ) a 0 k Terefore, e,,,..., N a a () Were, k w s a onstant and ndependent of.,77,800 9,958,00 Terefore, E o( 0 ). Ts mples tat te metod gves a tent order onvergene.. Numeral Examples To demonstrate te applablt of te metods, two model sngularl perturbed problems ave been onsdered. Tese examples ave been osen beause te ave been wdel dsussed n te lterature and ter exat solutons are avalable for omparson. 0

7 Example : Consder te followng sngular perturbaton problem wt onstant oeffents: x, 0 x wt boundar ondtons ( 0), () exp. Te exat (analtal) soluton s gven b: x ( x) x exp Te numeral solutons n terms of maxmum absolute errors and ts omparson wt oter autors are tabulated n Table for dfferent values of and N. Table Maxmum Absolute Error for Example. N = N = N = N = 8 N = 5 Our metod / 7.E-5.07E-.E-5.08E-5.99E-5 /.99E-.07E- 5.55E-.E E-5 / 7.7E- 7.75E E E-.07E-5 /8.7E-0.05E-.775E E-.5E-5 /5.7755E E- 7.70E E-.5075E- Faska et al. [] / 8.07E-09.8E-0.970E-.75E-.575E- /.7E-08.0E E-.77E-.595E- / 5.0E E-09.77E-0.988E-.07E- /8.7E-0.0E-08.05E E-.70E- /5.989E E E-09.77e-0.99e- Arsad and Pooa [0] /.5E-08.08E-0.5E-.9E-.5E- /.9E-07.59E-09.0E-.8E-.E- /.8E-0.50E-08.08E-0.0E-.59E- /8.59E-05.9E-07.59E-09.0E-.907E- /5.9E-0.8E-0.50E-08.08E-0.599E- Example : Consder te followng sngular perturbaton problem wt onstant oeffents: os ( x) os( x), 0 x wt boundar ondtons (0) 0 () Te exat (analtal) soluton s gven b: x exp( ( x) / ) exp( x / ) ( ) os ( x) exp( / ) Te numeral solutons n terms of maxmum absolute errors and ts omparson wt oter autors are tabulated n Table for dfferent values of and N.

8 Table : Maxmum Absolute Errors for Example N N N N 8 N 5 Our metod /.5E-.708E-5.78E- 5.E-.708E-5 /.79E-.90E-5.09E-.0E-.77E-5 / 7.9E- 7.09E-5.0E- 8.7E-.09E-5 /8.E-0.05E-.07E-.07E-.E-5 /5.7755E E- 7.05E E-.E- Faska et al [] /.E-07.87E-09 7.E-.8E-.5059E- /.00E-07.89E-09.59E-.0078E- 8.E- /.7E E E-0.8E-.807E- /8.957E-0.8E E E-.7E- /5.97E E E-09.77E-0.99E- Arsad and Pooa [0] /.707E E E-.089E- 5.00E- /.8E E E- 9.0E-.559E- /.0E-0.908E E-.50E-.98E- /8.50E-05.07E-07.58E-09.9E-.05E- /5.90E-0.80E-0.9E E-0.579E- Example : Consder te followng sngular perturbaton problem wt onstant oeffents: x x ( x) ( x) {exp exp } wt boundar ondtons (0) 0 () Te exat (analtal) soluton of te above problem s: x x ( x) ( x)exp x exp Te numeral solutons n terms of maxmum absolute errors and ts omparson wt oter metod are tabulated n Table for dfferent values of and N.

9 Table : Maxmum Absolute Errors for Example N N N N 8 N 5 Our metod /.75E- 5.55E-.887E E E-5 / 7.0E E-.09E-.77E-5.09E-5 /.70E-.79E E-.998E-5.09E-5 /8.9E-0.55E-.09E E-.5E-5 /5.98E-08.E-.E E- 5.95E- Faska et al [] /.075 E E E -.78 E E -5 /.550 E E E E E - / E E -08. E E E - /8.09 E E -07. E E E - /5.5 E E -07. E E E - Arsad and Pooa [0] /.09E-08.9E-0.5E-.59E-.8E- /.085E-07.75E-09.80E- 5.7E-.97E- / 5.E-0.7E-08.9E-0.97E-.0E- /8.0E-05.9E-07.7E-09.79E-.87E- /5.9E-0.077E-0.E-08.0E-0.55E- 5. Dsusson and Conluson Te tent order ompat fnte dfferene metod as been presented for solvng sngularl perturbed reaton-dffuson equatons wt drlet boundar ondtons. Dervatves appearng n te gven dfferental equaton are replaed b fnte dfferene approxmatons obtaned b Talor seres expansons at te grd ponts. Ts gves a large algebra tr-dagonal sstem of equatons to be solved b Tomas algortm, and to obtan te solutons at te mes ponts usng MATLAB software. Tree model examples are gven to demonstrate te effen of te proposed metod. Te maxmum absolute errors tabulated n (Tables ) for dfferent values of te perturbaton parameter and mes sze are ompared wt some prevous fndngs of oter metods reported n te lterature. As t an be observed from te tables, te proposed metod mproved te fndngs reported b autors gven n [] and [0]. Referenes [] Farrell, A. F., Mller, J. J. H., O Rordan, E and Sskn, G. I., Robust Computatonal Tenques for Boundar Laers. Capman-Hall/CRC, New York, 000. [] Roos, H.G., Stnes, M. and Tobska, L.,Robust Numeral Metods for Sngularl Perturbed Dfferental Equatons, Sprnger-Verlag, Berln Hedelberg, 008. [] Boglaev, I. P., A varaton dfferene seme for boundar value problems wt a small parameter n te gest dervatve. Journal of Computatonal Matemats and Matematal Pss, 7 8, 98. [] Faska W., Gemes F., and Tesfae A., Sxt-order ompat fnte dfferene metod for sngularl perturbed D reaton dffuson problems, Journal of Taba Unverst for Sene (Artle n Press), 0.

10 [5] Kadalbaoo, M. K and Kumar, D., Parameter-unform ftted operator B-splne olloaton metod for self-adont sngularl perturbed two-pont boundar value problems. Eletron Transatons on Numeral Analss 0, -58, 008. [] Msra, H. K., Kumar, M and Sng, P., Intal value tenque for self-adont perturbaton boundar value problems. Computatonal Matemats and Modelng 0,07-7, 009. [7] Roos, H. G., Stnes, M and Tobska, L., Numeral Metods for Sngularl Perturbed Dfferental Equatons: Conveton-Dffuson and Flow Problems. Sprnger Verlag, 99. [8] Gupta,Y and Panka, S. A.,Computatonal metod for solvng two pont boundar problems of order four. Internatonal Journal of Computer Tenolog and Applatons,, -, 0. [9] Moant, R. K., Ja, N, A lass of varable mes splne n ompresson metods for sngularl perturbed two-pont sngular boundar-value problems. Appl. Mat. Comput.8, 70-7, 005. [0] Arsad. K. and Pooa. K., Non-Polnomal sext splne soluton of sngularl perturbed value problems, Internatonal Journal of omputer matemats, 9:5, 0. [] Kadalbaoo, M. K. and Kumar. D., Varable mes fnte dfferene metod for self- adont sngularl perturbed two-pont boundar value problems. Journal of Computatonal Matemats 8, 7-7, 00. [] Terefe A., Gemes F., and Tesfae A., Fourt-order stable entral dfferene metod for self-adont sngular perturbaton problems, Etop. J. S. & Tenol. 9, 5-8, 0. [] Rasdna J., and Moammad Z., Non-Polnomal Splne approxmatons for te soluton of a sngularl-perturbed boundar-value problems, TWMS J. Pure Appl. Mat, -5, 00.