Non-Standard Finite Difference Schemes for Michaelis-Menten type Reaction Diffusion Equations

Transcription

1 Non-Sandard Finie Difference Schees for Michaelis-Menen ype Reacion Diffsion Eqaions Michael Chapwanya, Jean M-S Lba and Ronald E Mickens 2 Deparen of Maheaics and Applied Maheaics, Universiy of Preoria Preoria 2, Soh Africa, (.chapwanya@p.ac.za, jean.lba@p.ac.za) 2 Deparen of Physics, Clark Alana Universiy Alana GA 334, USA, (rohrs@ah.gaech.ed) Absrac We copare and invesigae he perforance of he exac schee of he Michaelis-Menen (M-M) ordinary differenial eqaion wih several new non-sandard finie difference (NSFD) schees ha we consrc by sing Mickens rles. Frherore, he exac schee of he M-M eqaion is sed o design several dynaically consisen NSFD schees for relaed reaciondiffsion eqaions, advecion-reacion eqaions and advecion-reacion-diffsion eqaions. Nerical silaions ha sppor he heory and deonsrae copaionally he power of NSFD schees are presened. AMS Sbjec Classificaion (2): 65L2; 65L99; 65M6; 65M99 Keywords: nonsandard finie difference ehod, exac schee, Laber W fncion, Michaelis- Menen eqaion, advecion-reacion-diffsion eqaions Inrodcion The Michaelis-Menen (M-M) ordinary differenial eqaion d d = +, is of grea ineres in applicaions. I odels he enzye kineics in pharacology and was solved in [] and [2] wih a singlar perrbaion echniqe where is one of he leading order ers of he expansion of he oer solion. The righ-hand side of he M-M eqaion also arises in predaor-prey and infecios disease odels as a fncional response in specific exended fors ranging fro he Michaelis-Menen-ype, he raio-dependen-ype, he Beddingon-De Angelisype, o he general Holling ype II considered in he Rosenzweig-Mac Arhr odel (see, for insance, [3] and [4]). The M-M eqaion is siple in aheaical srcre. However, is solions canno be fond explicily. Ths, here has been a srge in he consrcion of nerical ehods ha can provide sefl and reliable inforaion on his differenial odel. The pariclar ype of inforaion and properies on which any researchers have focsed are he linear sabiliy of he he hyperbolic criical poin ũ = and he posiiviy of he solions. Nonsandard Finie Difference (NSFD) schees, which are eleenary sable and preserve he posiiviy of he exac solions have been exensively invesigaed for general dynaical syses inclding he M-M eqaion and he cobsion odel (see he books [5], [6] and [7] and he references herein). The poin of deparre of he crren work is he exac schee of he M-M eqaion, which was designed only recenly by Mickens [8]. The exac schee is based on a general exisence heore in [5, p. 7] ha is applicable in his case, hanks o he so-called Laber W fncion which peris s o wrie he solion of he M-M eqaion in a copac for [9]. Frherore, given he specific nare of he M-M fncional response, we consrc hree new NSFD schees, which are Corresponding ahor: jean.lba@p.ac.za

2 all opologically dynaically consisen in he sense of []. The analysis is aken one sep frher by designing varios NSFD schees for he ajor relaed reacion-parial differenial eqaions of which he M-M odel is he lii eqaion in he space independen case. This is done by sing Mickens rles [5]. One focs of he work is o provide he deails of he nerical solions for he M-M eqaion by coparing and invesigaing he perforance of he exac schee, he exac schee-relaed ehods and oher NSFD schees. The focs on deailed heoreical and copaional proofs disingishes or work fro previos work in he lierare, where he ephasis is js on plgging he M-M eqaion ino one of he sandard ODE solvers. In his regard, we can enion he relaively recen work [] (and he references herein), where nerically inegraed solions of he M-M eqaion are considered, wiho deails. The res of he paper is organized as follows. In he nex secion, we recall he definiion of he Laber W fncion, he ain ool we se in his work. The M-M eqaion is discssed a lengh in Sec. 3 where we sdy is exac schee and copare he perforance wih hree new NSFD schees. NSFD schees for he M-M conservaive oscillaor or Hailonian syse are invesigaed in Sec. 4. Secs. 5, 6 and 7 are devoed o he analysis of NSFD schees for he M-M reacion-diffsion eqaion, he M-M advecion-reacion eqaion and he M-M advecionreacion-diffsion eqaion, respecively. Nerical silaions ha illsrae he power of all or NSFD schees are presened in Sec. 8. Sec 9 provides conclding rearks peraining o how he new NSFD schees fi in he lierare and how he sdy can be exended. 2 The Laber W Fncion Special fncions play sch an iporan role in science and engineering ha here are any handbooks on he in he lierare and os of hese are ser friendly (e.g. [2, 3]). In his work, we will exensively se he Laber W fncion, also called he Oega fncion or he prodc log, given by he relaion z = W (z) exp(w (z)), () as he livaled inverse of he non-injecive coplex-valed fncion w w exp(w). The canonical reference for he Laber W fncion is [9] where i is coprehensibly sdied. In he nerical silaions, we will se he Malab bil-in fncion laberw(*). Since we will be dealing wih he dynaics of syses ha are defined on R +,we resric orselves o nonnegaive real argens z and hs are ineresed in he (principal) branch saisfying W (z), which is a single-valed fncion. Ofen special fncions are characerized by differenial eqaions. In he case nder consideraion, iplici differeniaion of () shows ha W saisfies he differenial eqaion 3 The Michaelis-Menen ODE dw dz = W (z) z( + W (z)). (2) The Michaelis-Menen (M-M) ordinary differenial eqaion (ODE) is wih iniial condiion d d = a, a >, b, (3) + b () =. (4) Eqaion (3) arises in pharacology odels of he basic enzye reacion. The heory of enzye kineics and of he M-M eqaion, which coes fro a singlarly perrbed syse of ODE s, can be fond in [4]. The decay eqaion d d = a, a >, (5) 2

3 is he linearized for of (3) abo is niqe.fixed-poin ũ =. In wha follows, i is ore insrcive o look a (5) as he coparison eqaion associaed wih (3) (see [5]). Considered wih he sae iniial condiion in (4), he eqaions (5) and (3) have exac solions () = exp( a), (6) and () = b W ( (b exp( a) exp(b ) ), (7) respecively. The closed for expression (7), based of he Laber W fncion, for he solion of he M-M eqaion is esablished in [9]. Fro hese exac solions, we dedce he following inrinsic properies of he M-M odel: Theore For a given iniial vale, we have and () (),, (8) li () = (),, (9) b () as, () where he laer noaion eans ha he fncion () decreases onoonically o he hyperbolic fixed-poin as +. Reark 2 In view of he sign of he righ hand side of Eq. (3), he hyperbolic fixed poin ũ = is locally asypoically sable. However, Theore says ore han his local propery. In pariclar he differenial eqaion (3) defines a dynaical syse on he posiive inerval [ ), wih he fixed-poin ũ = being globally asypoically sable. W (z) The propery (9) is obained by sing () where li =. Noice ha he relaion z z () () in (8) is an illsraion of a well-known onooniciy resl (see Theore 8.XI in [6]). Ths for < <, all rajecories are bonded and decrease onoonically o zero and is locally asypoically sable fixed-poin. Le s consider a seqence { k = k } k of eqally-spaced ie poins where he paraeer > is he sep size. We denoe by k an approxiaion o he solion a he poin = k. The design of an exac schee for he M-M odel has been considered only recenly, naely in Mickens [8]. Seing fro (7), k = ( k ), Mickens proposed for he M-M odel, along he lines of [5], he exac schee ha reads as follows: = b W ( (b exp( a ) k exp(b k ) ). () In order o wrie () as a non-sandard finie difference (NSFD) schee, we observe ha he exac schee = exp( a ) k, (2) for he decay eqaion is eqivalen o or where k ϕ ( ) k ϕ 2 ( ) = a, (3) = a k, (4) ϕ ( ) = exp(a ) and ϕ 2 ( ) = exp( a ). a a (5) In view of he propery (9), ha s be replicaed by he nerical ehods, he eqivalen for k = W ( (b exp( a ) k exp(b k ) ) b k, ϕ 2 ( ) bϕ 2 ( ) (6) 3

4 which is indeed a NSFD ehod, is obained for he exac schee () (see [8]). Wih he forlaion (6), i is readily seen fro he ean-vale heore and (2) ha W ( (b exp( a ) k exp(b k ) ) b k = W ( (b exp( a ) k exp(b k ) ) W ( b k exp(b k ) ) bϕ 2 ( ) bϕ 2 ( ) a( k) + b( k ). Exac schees ha are effecive in applicaions do no exis in any cases. Therefore, sing Mickens rles [5], we also consider he following NSFD schees for he M-M eqaion: k ϕ ( ) = ak+ + b k ; (7) k ϕ ( ) k ϕ 2 ( ) = ak+ ; (8) + bk+ = ak + b k. (9) By analogy wih he coninos odel, we denoe he seqence ( k ) in (7) (9) by ( k ) when b =. Ths, each k is a discree solion of he decay eqaion (5). Despie is apparen iplici srcre, he NSFD schee (8), where we are ineresed in posiive discree solions, adis he eqivalen explici for in (2) below. Or ai is o invesigae and copare he perforance of he NSFD schees (7), (8) and (9) wih he exac schee (6). In pariclar, we are ineresed in deerining if he properies in Theore hold. Theore 3 The NSFD schees (7) (9) are dynaically consisen wih all he properies saed in Theore. More precisely, for and any >, we have k k, k, and li b k = k, k k as k. Proof. We prove he heore for he NSFD schee (8), he siaion being ch easier for he oher wo schees (7) and (9). Eqaion (8), wih he explici expression of he denoinaor fncion in (5), is qadraic in in he sense ha b( ) 2 + (e a b k ) k =. (2) We asse ha k >. Then, he algebraic eqaion (2) has a negaive and a posiive roos. The reqired posiive roo or solion of (8) is b k e a + (b k e a ) 2 + 4b k = if b > 2b (2) k e a if b =. Fro (8) and he definiion of ϕ in (5), we have which, for k =, yields e a k = k + (e a )( + b ) k, (22) = e a. Proceeding by aheaical indcion on k, we obain he firs clai in he heore. The oher wo clais follow fro (2) and he decreasing propery of he seqence ( k ) in (22). 4

5 Reark 4 For os of he schees sdied below, we will priarily be ineresed in he preservaion of he posiiviy of solions, which is he ore relevan propery fro he pracical poin of view. However, siilarly o Reark 2 for he coninos case, Theore 3 saes ch ore han he local propery of eleenary sabiliy of all he NSFD schees (7) (9), which siply eans ha hese schees have ũ = only as a fixed-poin and his fixed-poin is locally asypoically sable as for he coninos odel (see, e.g. [7] and [5]). In pariclar, ũ = is globally asypoically sable. Frherore, since each one of he NSFD schees (7) (9) can be wrien in he for = G( ; k ) where in each case he fncion G saisfies he condiions G(; ) = and dg( ; ) d we have in fac for any vale of he replicaion of all opological dynaical properies, a concep inrodced in [], where i is referred o as opological dynaic consisency. On he conrary, he sandard finie difference schee k >, = ak + b k, (23) is known o be eleenary nsable (see, e.g., [5]). I is clear ha his sandard schee fails o have posiive solions and o saisfy all he above properies for arbirary vales of. On he oher hand, he NSFD schees (7) (9) are no consrced in a rando way. For insance, if insead of (7), we consider he NSFD schee k ϕ ( ) = ak + b, hen, on wriing i in he eqivalen for siilar o (2), i easy o check ha here exiss no posiive solion when k >. Reark 5 While here is acally no need for all hree NSFD schees, each is, however, a valid schee in is own and of heselves. Neverheless, we oped o display he hree schees for he prposes of copleeness, specifically ha concerns on syseaic ehodologies of consrcing NSFD schees are ofen raised in he lierare. Aong oher hings, he schees show he naral way of consrcing he denoinaor fncions ϕ i ( ). 4 The M-M Hailonian syse The seady-sae of he reacion diffsion eqaions o be considered in his work is governed by he second-order differenial eqaion d 2 dx 2 a =, a >, b, (24) + b which involves he Michaelis-Menen reacion er. The rivial seady-sae ũ = is fond o be nsable, since he hyperbolic eqilibri (, ) of he eqivalen aonoos syse d dx d 2 dx = 2 = a + b, (25) where = and 2 =, is a saddle-poin. Eqaion (24) is also eqivalen o he conservaion law H H[ (x), (x)] := 2 ( ) 2 d + K() = consan. (26) dx 5

6 Here, H is he Hailonian of he syse and ( ) b ln( + b) a K() = b 2 if b > a 2 if b = 2. (27) Wih x = x ( Z, x > ) being he space node poins and an approxiaion of a he poin x = x, he linear eqaion has he exac schee (see [5]) where d 2 a =, (28) dx ψ( x) 2 a =, (29) ψ( x) = 2 ρ sinh(ρ x 2 ), ρ = a. (3) We follow he ehodology in [8, 7] o consider, for (24), he NSFD schee ( ) b(+ ) ln( + b + ) + ln( + b ) ψ( x) 2 a b 2 =, (3) ( + ) where, by he ean-vale heore, we ake b( + ) [ln( + b + ) ln( + b )] b 2 ( + ) Noe ha by again he ean-vale heore, since b( + ) ln( + b + ) + ln( + b ) b 2 ( + ) d[b ln( + b)] d = + b if + = or b =. (32) = b2 for. + b The nderlying poin of he NSFD schee (3) is is eqivalence o a discree analoge of he conservaion law (26). More precisely, we have he relaion 2 ( ) K x( ) = ψ( x) 2, ( ) 2 + K x( ), (33) ψ( x) where K x ( ), wih an iniial gess K x ( ), is derived fro he ideniy K x ( ) = K x ( ) + K( +) K( ). 2 For insance if K x ( ) =, we have K x ( ) = i= i= K( i ) K( i ) 2 K( +i ) K( ++i ) 2 = K( +) K( ) 2 if <. for (34) We will no consider he nerical ipleenaion of he NSFD schee (3) and of is eqivalen forlaion (33), since his is done, for insance, in [7, 9] for siilar discree conservaive syses, 6

7 where Mickens rle on he nonlocal approxiaion of nonlinear ers is essenial. Frherore, he NSFD schee (3) is eqivalen o he syse of difference eqaions,+, = 2,+ ψ( x) ( 2,+ 2, b(,+, ) [ln( + b,+ ) ln( + b, )] = a ψ( x) b 2 (,+, ) which, as a discree analoge of he syse (25), is an eleenary sable NSFD schee becase he schee redces o he exac schee (29) when b =. Since or priary focs is no on he replicaion of he conservaion law (26), we shall also consider he NSFD schee ψ( x) 2 a =, + b which, hogh no preserving his propery becase of he violaion of Mickens rle on he nonlocal approxiaion of nonlinear ers, corresponds o an eleenary sable NSFD schee of he for siilar o (35). 5 The M-M PDE The reacion diffsion eqaion wih he M-M er is ), (35) = a + b + 2 x 2. (36) Is coparison eqaion or is linearized eqaion abo he eqilibri solion ũ = is he classical linear parabolic eqaion = a + 2 x 2. (37) We consider he sae iniial fncion (x) for he probles (36) and (37): (, x) = (x) = (, x). (38) a + b saisfies a + b a on R + and has bonded derivaive The real-valed fncion a ( + b) 2 on R +. On he oher hand, he Sr-Lioville proble defined by he operaor in he righ-hand side of (37), wih bondary condiions ϕ() = ϕ() =, has eigenvales and associaed (L 2 -orhonoral) eigenfncions λ n = a n 2 π 2 and ϕ n (x) = 2 sin(nπx), n =, 2,, respecively. Ths, he sandard exisence, niqeness, coparison and sabiliy resls (see for exaple [2, 2]) apply o he proble (36), as specified in he nex heore. Theore 6 For a given bonded and coninos fncion (x), he iniial vale probles (36), (37) and (38) adi niqe solions (, x) and (, x) sch ha (x) = (, x) (, x) M := sp (x). (39) x The eqilibri solion ũ = of (36) and (38) is L asypoically sable. Frherore, if he iniial-vale proble is copled wih he Dirichle bondary condiion (, ) = (, ) =, hen he eqilibri solion ũ = is L 2 asypoically sable in he precise anner ha he kineic energy decreases onoonically o as. E(()) := 2 (, x) 2 dx, (4) 7

8 Or ai is o consider for (36) varios NSFD schees and o deonsrae heoreically and copaionally ha hey are dynaically consisen wih he propery (39). The sabiliy properies will be deonsraed copaionally. The general sraegy, which sricly speaking akes sense when he so-called lped paraeer asspion [2] is valid, consiss in siably cobining he space-independen NSFD schees in Sec. 3 wih he ie-independen schees in Sec. 4. Firsly, we deal wih he schees (7), (9), (4) and (3) o obain he following explici NSFD schees, respecively: and and k ϕ ( ) k ϕ 2 ( ) k ϕ ( ) = a k+ k = ak+ + b k + k + 2 k + k ψ( x) 2, (4) ( b( k + k ) ln( + b k +) + ln( + b k ) ) b 2 ( k + k ) + k + 2 k + k ψ( x) 2, (42) k ϕ 2 ( ) = ak + b k + k + 2 k + k ψ( x) 2, (43) ( b( k = a + k ) ln( + b k +) + ln( + b k ) ) b 2 ( k + k ) + k + 2 k + k ψ( x) 2. (44) Secondly, we are ineresed in cobining he exac schee (6) wih (4) o obain he explici NSFD schee k ϕ 2 ( ) = W ( (b exp( a ) k exp(b k ) ) b k bϕ 2 ( ) + k + 2 k + k ψ( x) 2. (45) In ers of he Laber W -fncion, he following explici NSFD schee, which is ailored fro he schee (42) and (45), preserves he conservaion law in he saionary case: { k = a k+ ϕ 2 ( ) k b + ln [ ( + W (b exp( a ) k + exp(b k +) )] } + ln( + b k ) b [ W ( (b exp( a ) k + exp(bk + )) ] b k + k + 2 k + k ψ( x) 2. (46) Noice ha boh NSFD schees (45) and (46) can, in view of he ean-vale heore, redce o: k ϕ 2 ( ) = a k+ + b k + k + 2 k + k ψ( x) 2. (47) As observed firs by Mickens [22], he nex resl confirs he relevance of a fncional relaion beween sep sizes in order for an explici NSFD schee o preserve he posiiviy and bondedness propery. Theore 7 Asse ha ψ( x) is chosen sch ha he fncional relaion ϕ i ( ) ψ( x) 2 = 2 occrs beween he sep sizes. Then, we have he following posiiviy and bondedness propery for he explici NSFD schees (4) and (42), (43) and (44), (45) and (46): M = k k M, k, Z, and S k as k Here, k denoes he discree solion for he eqaion (37), resling on seing b = in each one of he above-enioned NSFD schees and S k := sp k. (48) 8

9 Proof. Observe firs ha (48) is siilar o a ypical condiion of Lax-Richyer sabiliy of he involved explici finie difference schees for he linear eqaion (37) (see e.g. [23]). This condiion allows s o re-wrie each one of he above NSFD schees in he copac for for soe fncion g saisfying = k + + k 2[ + g(b)] = k + + k 2[ + g()] g() g(b). This iplies ha (S k ) and (S k ) are nonnegaive onoonic decreasing seqences sch ha and by aheaical indcion on k, we have S k as k k k M. Reark 8 We will no invesigae he cobinaion of (8) wih (4) and (3), which leads o he following iplici NSFD schees ha are no easy o solve in : and k ϕ ( ) k ϕ ( ) = a = ak+ + b + k + 2 k + k ψ( x) 2. (49) ( b( k+ + k+ ) ln( + bk+ + ) + ln( + bk+ b 2 ( + k+ ) ) + k + 2 k + k ψ( x) 2. (5) ) As far as he posiiviy of discree solions is concerned, i is possible o achieve i by considering iplici NSFD schees in which ( x) 2 is sed in place of ψ( x) 2 and he condiion (48) is no needed. The precise resl reads as follows: Theore 9 Asse ha Dirichle bondary condiions are prescribed for he proble (36), naely (, p) = α and (, p) = β, (wih p >, α and β ) so ha a finie nber of nodes x = x, =, ±,, ±N, are considered where N 2 is an ineger and x = p/n. Then he posiiviy of he solions of (36) and of he coparison eqaion (37) is preserved by all he iplici NSFD schees obained fro (4), (42), (43), (44), (45) and (46) by replacing k + 2 k + k ψ( x) 2 wih k+ + 2k+ + ( x) 2. (5) Proof. To be specific, we focs only on he following iplici NSFD schee obained fro (4), alhogh he analysis below applies o all of he: k ϕ ( ) = ak+ + b k + k+ + 2k+ + ( x) 2. (52) Consider he coln vecor U k+ = [ N+ k+ T N ]. 9

10 Then he ehod (52) is eqivalen o solving a each sep he algebraic linear syse A b U k+ = F k (53) where A b = + 2c + aϕ ( ) +b k N c c + 2c + aϕ ( ) +b k 2 N c + 2c + aϕ ( ) +b k 3 N.. c. c + 2c + aϕ ( ) +b k N. c and c = ϕ ( ) ( x) 2 F k = [ k N+ + αc, k N+2, k N 2, k N + βc ] T. The ridiagonal arix A b is an M-arix, being a sricly diagonally doinan arix wih posiive diagonal and non-posiive off diagonal [25]. Ths Frherore, since A b A, we have [26] which by aheaical indcion yields U k = U k+ = A b F k. U = A F A b F = U, U k U k. 6 The M-M advecion reacion eqaion The advecion eqaion wih a M-M reacion er is + d x = a, d >. (54) + b Considering separaely he space and he ie independen fors of (54), which are boh of he ype (3), i akes sense o cobine siably he NSFD schees presened in Secion 3. We obain he nerical schees k ϕ ( ) k ϕ 2 ( ) k ϕ 2 ( ) + d k k dϕ ( x/d) + d k k dϕ 2 ( x/d) + d k k dϕ 2 ( x/d) which, on reqiring he fncional relaion redce o he following NSFD schees: = ak+ + b k, = ak + b k, = W ( (b exp( a ) k exp(b k ) ) b k, (55) bϕ 2 ( ) ϕ i ( ) = ϕ i (d x) i.e. x = d, i =, 2, (56) k ϕ ( ) = ak+ + b k, (57)

11 and k ϕ 2 ( ) k = ak ϕ 2 ( ) + b k, (58) = W ( (b exp( a ) k exp(b k ) ) b k. (59) bϕ 2 ( ) Observe ha he schees (57), (58) and (59) depend only on a general resl obained by Mickens [24] for he PDE + d x = f(). and k, in accordance wih The reason for his is ha he solions shold only ove along he characerisics and, in fac along he characerisics, he advecion-reacion eqaion (54) becoes, for a fixed x R, he M-M ODE: du d = au + bu, (6) where, x x + d, U() := (, x + d). (6) Noe ha Eq. (54) is js Eq. (3) wih a linear advecion er. I ay be copared o + d x = a o exaine he exisence, bondedness and oher solion properies. Also, in view of he connecion wih he M-M ODE (6), he properies of he solions of (54) along he characerisics are siilar o hose saed in Theore. In he nex heore, we sae how soe of hese properies are preserved by he NSFD schees (57), (58) and (59) along he lines of Theores 3 and 7. Theore Under he relaion (56), we have M = k M, k, Z, and S k as k. Proof. For a fixed space grid poin x k, we have x k = x d k by (56). The asspion on he iniial vales in he heore iplies ha k M which, for he fncion U associaed wih x = x k and given by (6)-(6), eans ha U M. I follows fro Theore 3 and (6) ha U k U k U M or k k k M. Ths S k S k S M and S k. 7 The M-M advecion reacion-diffsion eqaion The advecion-diffsion eqaion wih a M-M reacion er is + d x = a + b + 2 x 2 (62) and he physical solions shold have he posiiviy propery (39). In order o consrc a schee ha preserves he propery (39), i sees naral o cobine he approach in Secions 5 and 6. However, his wold reqire he iposiion of condiions (48) and (56), which aken silaneosly are incopaible. Firs, le s avoid he condiion (48) and keep condiion (56). Using (5), one possible NSFD schee is k ϕ 2 ( ) = W ( (b exp( a ) k exp(b k ) ) b k bϕ 2 ( ) + k+ + 2k+ + ( x) 2. (63) Analogosly o Theore 9, if bondary condiions are specified, he NSFD schee (63) redces o a workable forlaion siilar o (53), wih an M arix sch ha he posiiviy propery is preserved.

12 Secondly, we wan o avoid boh condiions (48) and (56). To his end, we follow an idea in [27, 28]. Given x = x, we inrodce x := x d he backrack poin of x, which is no in general on he grid. For any x, here exiss an ineger sch ha x < x x. Using he linear Lagrange inerpolaion a he poins (x, k ) and (x, k ), le s consider he following poin on his line: ū k := x x x k + x x k x. Noice ha ū k if k and k are nonnegaive. One possible NSFD schee for (62) is ū k ϕ 2 ( ) = W ( (b exp( a )ū k exp(bū k ) ) bū k bϕ 2 ( ) + k+ + 2k+ + ( x) 2, (64) which enjoys he sae dynaic consisency properies as he schee (63). Finally, he explici NSFD schee ū k ϕ 2 ( ) = W ( (b exp( a )ū k exp(bū k ) ) bū k bϕ 2 ( ) + k + 2ū k + k ψ( x) 2, (65) can be sed. The lef-hand sides of (64) and (65) approxiae he lef-hand side of (62), following a rewriing siilar o (55). In sary, we have he following resl: Theore Wih siable bondary condiions, he NSFD schees (63) and (64) preserve he propery (39). The sae propery is preserved by he NSFD schee (65) nder he relaion (48). 8 Nerical silaions In his secion, we deonsrae copaionally he heoreical resls obained in he previos secions. The resls confir in pariclar ha he NSFD schees consrced by sing Mickens classical rles are reliable and hs hey can be sed when exac schees are no available. We have prposely decided no coen oo ch on sandard finie difference schees, since heir poor perforance has been exensively repored in he lierare (see, for insance, [7], [27], [5], [6] and [7]). In all he exaples presened below, we ake a =.; he ie sep is = 2, which is considered o be large for sandard nerical schees. In he silaion involving Laber W fncion, we se he Malab bil-in fncion laberw(*). Figre sppors he heory in Theore 3 and Reark 2. I is seen ha he NSFD schees (7), (8) and (9) perfor as efficienly as he exac schee () or (6) regarding he preservaion of he dynaics of he syse saed in Theore when b = and b =, respecively. Aong oher hings, he sae figre displays he global asypoic sabiliy of he eqilibri solion ũ = given in Reark 4. The conras wih he sandard schee (23) and he Malab ODE45 solver is visalized in Figre 2. The apparen good resls by he ODE45 solver are a he high cos of very sall ie seps, which is a serios concern when one is ineresed in he long er behavior of he solion, or for siff/chaoic probles, [9]. 2

13 .8 Eqn. 7, b = Eqn. 7, b = Eqn., b = Eqn. 8, b = Eqn. 8, b = Eqn., b = Eqn. 9, b = Eqn. 9, b = Eqn., b = Figre : NSFD schees (7), (8) and (9) verss he exac schee () 3

14 .8.6 Eqn. 23, b = Eqn. 23, b = Eqn., b =.8 ode45, b = ode45, b = Exac, b = Exac, b = (a) (b). Figre 2: Sandard schee (23) in (a) and ODE45 Solver in (b). 4

15 Figres 3, 6, 7 and 8 illsrae Theore 7 for he explici NSFD schees (4), (42), (43) and (44), respecively. The iniial fncion is (x) = e x2, x R. The vale of he space size x is obained according o (48), i.e. ψ( x) = 2ϕ i ( ) = 2ϕ i (2), where i =, 2, while we ake b = and b =. I is seen on he picres ha he NSFD schees copare nicely wih he exac schee-relaed ehod (45), as far as he replicaion of he properies in Theore 6 is concerned. This incldes he saed L -sabiliy of he eqilibri solion ũ =, while he decay o of he kineic energy is displayed on Figre 5 (a). The poor perforance of he sandard finie difference ehod is shown on Figre 4, which exhibis negaive solions, and Figre 5 (b). Here and afer, he associaed Figre (a) shows, in each case, a side-view profile corresponding o Figre (b) aken a x =. The excellence perforance of he NSFD schees is frher confired by he vales of he solions ablaed in Tables, 2, 3 and Eqn. 4, b = Eqn. 4, b = Eqn. 45, b = (a)..5 x (b) Figre 3: NSFD schee (4) verss he exac schee-relaed ehod (45). x Exac Table : Coparaive resls corresponding o Figre Sd. schee, b= Sd. schee, b= Eqn. 45, b= (a)..5 2 x 2 (b) Figre 4: Sandard schee corresponding o (43) for ϕ i ( ) = = 2, ψ( x) = x = 2. 5

16 Kineic energy (a). Kineic energy (b). Figre 5: Kineic energy profiles for: (a) NSFD schee (43) and, (b) he sandard finie difference schee relaed o (43) for ϕ i ( ) =, ψ( x) = x. 6

17 Eqn. 42, b = Eqn. 42, b = Eqn. 45, b = (a)..5 x (b) Figre 6: NSFD schee (42) verss he exac schee-relaed ehod (45). x Exac Table 2: Coparaive resls corresponding o Figre 6 7

18 Eqn. 43, b = Eqn. 43, b = Eqn. 45, b = (a)..5 x (b) Figre 7: NSFD schee (43) verss he exac schee-relaed ehod (45). x Exac Table 3: Coparaive resls corresponding o Figre 7 8

19 Eqn. 44, b = Eqn. 44, b = Eqn. 45, b = (a)..5 x (b) Figre 8: NSFD schee (44) verss he exac schee-relaed ehod (45). x Exac Table 4: Coparaive resls corresponding o Figre 8 9

20 Mike: we shold add wo figres: one on he decay of he kineic energy and anoher one on sandard finie difference ehod. Figre 9 illsraes Theore 9 for he preservaion of he posiiviy of solions when he iplici NSFD schee (52) is sed. Fro now on, he sandard schee is no longer ploed de o is poor perforance enioned and observed above. We ake p = and consider hoogenos Dirichle bondary condiions i.e. α = β =. The iniial vale is (x) = sin πx for x [, ] and (x) = for x >. The space sep size is x =., which varies independenly fro = 2 in he sense ha he reqireen (48) is no e. Apar fro he posiiviy of he discree solions, Table 5 confirs he expeced coparison beween he solions k for b = and k for b = Eqn. 52, b = Eqn. 52, b = (a)..5 x (b) Figre 9: Posiiviy of he iplici NSFD schee(52). x Table 5: Coparaive resls corresponding o Figre 9 2

21 Figres and illsrae he posiiviy and bondedness propery in Theore for he explici NSFD schees (57) and (58) wih iniial vale (x) = e x2, x R. Wih d =. and b =, we ake he space sep size in accordance wih (56), i.e. x =.2. The resls copare nicely wih he exac schee-relaed ehod (59), as also shown on Tables 6 and Eqn. 57, b = Eqn. 59, b = (a)..5 x (b) Figre : Posiiviy and bondedness of he NSFD schee (57) verss he exac schee-relaed ehod (59). x Exac Table 6: Coparaive resls corresponding o Figre 2

22 Eqn. 58, b = Eqn. 59, b = (a)..5 x (b) Figre : Posiiviy and bondedness of he NSFD schee (58) verss he exac schee-relaed ehod (59). x Exac Table 7: Coparaive resls corresponding o Figre 22

23 Finally, Figre 2 illsraes he posiiviy of solions in Theore for he schee (65) for he iniial condiion (x) = e x2, b =, d =. and x obained according o (48) i.e. ψ( x) = 2ϕ 2 (2). Eqn (a)..5 x (b) Figre 2: Posiiviy of he NSFD schee (65). 23

24 9 Conclsion This paper is oivaed by he exac schee of he Michaelis-Menen ordinary differenial eqaion designed recenly in [8]. Using Mickens rles [5] and he specific for of he righ-hand side of he M-M eqaion, we inrodced several new NSFD schees for he M-M eqaion as well as for associaed reacion-diffsion-ype eqaions. We deonsraed heoreically and copaionally ha he new NSFD schees are dynaically consisen wih he coninos odel. They all perfor as efficienly as he exac schee and exac-relaed ehods, confiring hs he power of he nonsandard approach. Or fre ineres is on he applicaion of he resls of his sdy o he design of NSFD schees for epideiological odels where he conac beween he sscepible individals S(), and he infeced individals I(), in he oal poplaion N(), is expressed by a nonlinear er of he for S()I(), which is siilar o ha of he M-M eqaion (see [29]). Preliinary work in N() his direcion is done in [3]. Frherore, we are invesigaing how he sdy cold be exended o he cobsion odel d d = 2 ( ) whose exac schee can be dedced fro [9] as Acknowledgens = { + W [ ( k ) exp ( ( k ) )]}. The work was iniiaed when JM-SL was on sabbaical leave a Clark Alana Universiy in Febrary 29. JM-SL is graefl o he Soh African Naional Research Fondaion for fnding he visi as well as o R.E. Mickens and researchers in he Physics Deparen for aking his visi he os pleasan and prodcive. We are also sincerely graefl o he anonyos reviewers whose coens have significanly iproved his work. References [] R.E. O Malley, Jr., Singlar perrbaion ehods for ordinary differenial eqaions, Springer- Verlag Applied Maheaical Sciences Series, New york,99. [2] L.A. Segel and M. Slerod, The qasi-seady-sae asspion: a case sdy in perrbaion, SIAM Review, 3 (989), [3] D.T. Diirov and H.V. Kojoharov, Posiive and eleenary sable nonsandard nerical ehods wih applicaions o predaor-prey odels, Jornal of Copaional and Applied Maheaics, 89 (26), [4] D.T. Diirov and H.V. Kojoharov, Nonsandard nerical ehods for a class of predaorprey odels wih predaor inerference, Proceedings of he 6h Mississippi Sae-UBA Conference on Differenial Eqaions and Copaional Silaions, 67-75, Elecronic Jornal of Differenial Eqaions Conference, 5, Sohwes Texas Sae Universiy, San Marco, 27. [5] R.E. Mickens, Nonsandard finie difference odels of differenial eqaions, World Scienific, Singapore, 994. [6] R.E. Mickens (Ed.), Applicaions of nonsandard finie difference schees, World Scienific, Singapore, 2. [7] R.E. Mickens (Ed.), Advances in he applicaions of nonsandard finie difference schees, World Scienific, Singapore, 25 [8] R.E. Mickens, An exac discreizaion of he Michaelis-Menen ype eqaions wih applicaions, Jornal of Biological Dynaics, In press. [9] R.M. Corless, G.H. Gonne, D.E.G. Hare, D.J. Jeffrey and D.E. Knh, On he Laber W fncion, Advances in Copaional Maheaics, 5 (996),

25 [] R. Angelov, J. M.-S. Lba, M. Shillor, Topological dynaic consisency of nonsandard finie difference schees for dynaical syses, Jornal of Difference Eqaions and Applicaions, 7 (2), [] S. Schnell and P.K. Maini, Enzye kineics a high enzye concenraion, Bllein of Maheaical Biology, 62 (2), [2] M. Abraowiz and I.A. Segn (ediors), Handbook of aheaical fncions wih forlas, graphs and aheaical ables, Dover Pblicaions, New York, 9h ediion, 972. [3] W.H. Beyer (ed), CRC sandard aheaical ables, 28h ediion, 987. [4] J.D. Mrray, Maheaical biology I: an inrodcion, Springer, New York, 22. [5] R.K. Miller, Nonlinear Volerra inegral eqaions, Maheaics Lecre Noe Series, W.A. Benjain, Inc., Philippines, 97. [6] W. Waler, Differenial and inegral ineqaliies, Springer-Verlag, Berlin, 97. [7] R. Angelov and J.M-S. Lba, Conribions o he aheaics of he nonsandard finie difference ehod and applicaions, Nerical Mehods for Parial Differenial Eqaions, 7 (2), [8] R. Angelov, P. Kaa and J.M.-S. Lba, On non-sandard finie difference odels of reacion-diffsion eqaions, Jornal of Copaional and Applied Maheaics, 75 (25) -29. [9] Y. Don and J.M.-S. Lba, Non-sandard finie difference ehods for vibro-ipac probles, Proceedings of he Royal Sociey of London Series A: Maheaical, Physical and Engineering Sciences, 46A (25), [2] J.D. Logan, Nonlinear differenial eqaions, Wiley-Inerscience, New York, 994. [2] J. Soller, Shock waves and reacion diffsion eqaions, Springer-Verlag, 983. [22] R.E. Mickens, Relaion beween he ie and space sep-sizes in nonsandard finie-difference schees for he Fisher eqaion, Nerical Mehods for Parial Differenial Eqaions 3 (997), [23] K.W. Moron and D.F. Mayers, Nerical solions of parial differenial eqaions, Cabridge Universiy Press, London, 994. [24] R. E. Mickens, Nonsandard finie difference schee for scalar advecion-reacion parial differenial eqaions, Neral, Parallel, and Scienific Copaions, 2 (24), [25] R.S. Varga, Marix ieraive analysis, Prenice-Hall, Englewood Cliffs, NJ, 962. [26] A. Beran and R. J. Pleons, Nonnegaive arices in he aheaical sciences, Acadeic Press, New York, 979. [27] H.V. Kojoharov and B.M. Chen, Nonsandard ehods for advecion-diffsion-reacion eqaions, In: R.E. Mickens (Edior), Applicaions of nonsandard finie difference schees, World Scienific, Singapore, 2, pp [28] J.M-S Lba and K.C. Paidar, Solving singlarly perrbed advecion reacion eqaion via non-sandard finie difference ehods, Maheaical Mehods in he Applied Sciences, 3 (27), [29] S.M. Garba, A.B Gel and J.M.-S Lba, Dynaically-consisen nonsandard finie difference ehod for an epideic odel, Maheaical and Coper Modeling, 53 (2), 3-5 (doi:.6/j.c ). [3] M. Chapwanya, J.M.-S Lba and R.E. Mickens, Fro Enzye Kineics o Epideiological Models wih Michaelis-Menen Conac Rae: Design of Nonsandard Finie Difference Schees, Copers and Maheaics wih Applicaions, 22, in press, (doi:.6/j.cawa ). 25