Pathan Mahabub Basha *, Vembu Shanthi

Transcription

1 Amercan Journal of umercal Analyss 5 Vol 3 o Avalable onlne at ttp://pubsscepubcom/ajna/3// Scence Educaton Publsng DOI:69/ajna-3-- A Unformly Convergent Sceme for A System of wo Coupled Sngularly Perturbed Reacton-Dffuson Robn ype Boundary Value Problems wt Dscontnuous Source erm Patan Maabub Basa Vembu Sant Department of Matematcs atonal Insttute of ecnology rucrappall Inda Correspondng autor: pmbasa9@gmalcom Receved January 5 Revsed February 5 5; Accepted September 8 5 Abstract In ts paper a unformly convergent sceme for a system of two coupled sngularly perturbed reactondffuson Robn type mxed boundary value problems (MBVPs wt dscontnuous source term s presented A ftted mes metod as been used to obtan te dfference sceme for te system of MBVPs on a pecewse unform Sskn mes A cubc splne sceme s used for Robn boundary condtons te classcal central dfference sceme s used for te dfferental equatons at te nteror ponts An error analyss s carred out numercal results are provded to sow tat te metod s unformly convergent wt respect to te sngular perturbaton parameter wc supports te teoretcal results Keywords: sngular perturbaton problem weakly coupled system dscontnuous source term Robn boundary condtons Sskn mes ftted mes metod unform convergence Cte s Artcle: Patan Maabub Basa Vembu Sant A Unformly Convergent Sceme for A System of wo Coupled Sngularly Perturbed Reacton-Dffuson Robn ype Boundary Value Problems wt Dscontnuous Source erm Amercan Journal of umercal Analyss vol 3 no (5: do: 69/ajna-3-- Introducton Sngular perturbaton problems (SPPs arse n varous felds of scence engneerng wc nclude flud mecancs flud dynamcs quantum mecancs control teory semconductor devce modelng cemcal reactor teory elastcty ydrodynamcs gas porous electrodes teory etc SPPs are caracterzed by te presence of a small parameter ( < tat multples te gest dervatve term s leads to boundary /or nteror layers n te soluton of suc problems A muc attenton as been drawn on tese problems to obtan good approxmate solutons for te past few decades Snce classcal numercal metods fal to produce good approxmatons for tese equatons t s nevtable to go for non-classcal metods ere are several artcles avalable at te lterature but tey are manly based on sngularly perturbed problems contanng one equaton Some autors ave developed robust numercal metods for a system of sngularly perturbed convecton-reactondffuson problems on smoot data Very few researcers can be seen for problems wt non-smoot data wc frequently arses n electro analytc cemstry predatorprey populaton dynamcs etc as a perfect applcaton Oseen equatons form a convecton-dffuson system were as lnearzed aver-stokes equatons yeld a reacton-dffuson system at large Reynolds number For a parameter-unform metods pertanng to sngular perturbaton problems one can refer te books [3] A stard fnte dfference metod s proved unformly convergent on a ftted pece wse unform Sskn mes for a sngle equaton reacton-dffuson problem [] e same approac for coupled system of two sngularly perturbed reacton-dffuson problems wt dffuson coeffcents was orgnally proposed by Sskn [4] dentfed tree dfferent cases < = ; < = ; < For case- ( Mattews et al [5] proved almost frst order convergence usng classcal fnte dfference sceme on Sskn mes for a system of sngularly perturbed reacton-dffuson equatons subject to Drclet boundary condtons amlselvan et al [6] developed a numercal metod usng ftted pecewse unform Sskn mes for te coupled system of sngularly perturbed reacton- wt dscontnuous source dffuson equatons for case- term subject to Drclet boundary condtons obtaned almost frst order unform convergence Sngularly perturbed lnear second order ordnary dfferental equatons of reacton-dffuson type wt dscontnuous source term subject to Drclet boundary condtons avng dffuson parameters wt dfferent

2 Amercan Journal of umercal Analyss 4 magntudes was studed by Paramasvam et al [7] In tat paper te autors constructed a numercal metod usng classcal fnte dfference sceme on Sskn mes wt frst order parameter-unform accuracy Usng mes equdstrbuton tecnque Das atesan [8] studed te sngularly perturbed system of reacton-dffuson problems subject to Drclet boundary condtons on smoot data avng dffuson parameters wt dfferent magntudes In tat artcle te central dfference sceme s used to dscretze te problem on adaptvely generated mes obtaned an optmal second order parameter unform convergence In recent years system of sngularly perturbed Robn type reacton-dffuson problems as attracted a lot of attenton for many researcers Das atesan [9] aceved perfect second order accuracy for a sngle second order Robn type reacton-dffuson problems usng adaptvely generated grd for smoot case In tat artcle te autors proposed te cubc splne dfference sceme for mxed boundary condtons te classcal central dfference sceme for te dfferental equaton at te nteror ponts to get second order parameter unform convergence Das atesan [] also proposed an effcent ybrd numercal sceme wc uses cubc splne dfference sceme n te nner regon central dfference sceme n te outer regon for sngularly perturbed system of Robn type reacton-dffuson problems on Sskn me for smoot case It as been sown tat te sceme s -unform convergent wt almost second order accuracy wo ybrd dfference scemes on te Sskn mes were constructed by Mytl Pryadarsn Ramanujam [] for solvng te sngularly perturbed coupled system of convectondffuson equatons wt mxed type boundary condtons on smoot data wc generate -unform convergent numercal approxmatons to te soluton Recently Maabub Basa Sant [] ave consdered a numercal metod for sngularly perturbed coupled system of convecton-dffuson Robn type boundary value problems wt dscontnuous source term Motvated by te above works n ts artcle we ave developed a unformly convergent numercal metod on te Sskn mes for a system of two coupled sngularly perturbed reacton-dffuson Robn type boundary value problems wt dscontnuous source term s paper s organzed as follows: In Secton- some analytcal results of te soluton of sngularly perturbed MBVP wt dscontnuous source term are presented e numercal metod s descrbed n Secton-3 Error analyss s carred out n Secton-4 umercal examples are provded n Secton-5 conclusons are gven n Secton-6 rougout ts paper denotes a generc postve constant ndependent of te sngular perturbaton parameter te nodal ponts x te number of mes ntervals wc may not be same at eac occurrence Let y: D= [ ab ] e norm wc s sutable for studyng te convergence of numercal soluton to te exact soluton of te sngular perturbaton problem s te y = sup y x Furter maxmum norm D x D yx = ( y( x y( x y = max { y y } x D Contnuous Problem Statement of te Problem Fnd y y Y C ( Ω C ( Ω Ω Py( x suc tat Ω Ω Ω Ω y x a x y x a x y x = f x x Py x y x a x y x a x y x = f x x wt te boundary condtons α β ( ( B y y y = p (3 γ δ B y y y = q α β B y y y = r γ δ B y y y = s (4 (5 (6 were s a small parameter ( < α α β β γ γ δ δ > a x a x a x a ( x are suc tat a x a ( x (7 a x > a x a x > a x x Ω also [ ] [ ] (8 f ( d C f d C (9 Here ( Ω = Ω = d Ω = d d Ω y = y y It s also assumed tat te source terms f f are suffcently smoot on Ω \{ d}; At te pont d Ω te functons f fave jump dscontnuty In general ts dscontnuty gves rse to nteror layers n te soluton of te problem Snce f f are dscontnuous at d te soluton y of (-(6 does not necessary to ave a contnuous second order dervatve at te pont d e y y C Ω But te frst dervatve of te soluton exsts s contnuous e above system (-(6 can be wrtten n matrx form as d Py Py y A x y = f x dx Py d y x A x y x = f x x Ω Ω dx

3 4 Amercan Journal of umercal Analyss wt te boundary condtons B y p By q = = B y r B y s a( x a( x f( x were A( x = f ( x = a( x a( x f( x e jump at d s denoted n any functon ω wt [ ω]( d ω( d ω( d = Remark-: e presence of multplyng te dervatve terms n te mxed boundary condtons amplfes te sgnfcance of te boundary layers at bot ends In te absence of te layers are suffcently weak [93] Some Analytcal Results In ts secton te exstence of a soluton te maxmum prncple stablty result are establsed for te MBVP (-(6 eorem-: e MBVP (-(6 as a soluton y = ( y y wt y y Y Proof: e proof s by constructon Let y y be te partcular solutons of te followng system of equatons ( y ( x A( x y ( x f ( x = x Ω ( y y ( x f ( x x A x = x Ω respectvely Also let φ ψ be te solutons of te followng MBVPs: φ ( x A( x φ ( x = x Ω αφ βφ = γφ δφ = ψ ( x A( x ψ ( x = x Ω αψ βψ = γψ δψ = respectvely φ = ( φ φ ψ = ( ψ ψ α = ( α α Here β = β β γ = γ γ δ = δ δ = ( ( ( = ( y en y can be wrtten as ( x were B ( y ( x ( φ K( ψ x Ω B B ( y ( x ( ψ K ( φ x Ω B ( α ( β ( ( α ( β ( ( α β ( α β ( α ( β ( ( α ( β ( ( α ( β ( ( α ( β ( = B y D y y D y = B y D y y D y = B y D y y D y = B y D y y D y K K are matrces wt constant entres < φψ < φψ cannot ave nternal On maxmum or mnmum [4] K Hence φ < ψ > x Coose te matrces k = k so tat y y C k K = k Ω e we mpose te condtons y( d = y( d y( d For te matrces K ψ ψ = y( d K to exst t requres ( d φ ( d ψ( d φ( d ( d φ ( d ψ ( d φ ( d s mples ( ψφ ψφ ( φψ ψφ > eorem-: (Maxmum prncple Suppose y y Y Furter suppose tat y ( y y B y ( B y ( By ( y ( d By ( P y( x P y( x Also let a ( x a ( x on exsts a functon Bt Bt Bt B t Pt ( x P t ( x = satsfes Ω en f tere t = t t t t Y suc tat t ( d ten y( x x Ω Proof: η max x Ω max x Ω t t Defne y y = max Assume tat te teorem s not true en η > tere exsts a pont x suc tat y y ( x = η or ( x = η or bot t t Furter x Ω Ω or x = d Also y ηt x = x Ω ( Case-(: ( y ηt ( x = for x = It mples tat ( y ηt attans a mnmum at x erefore B ( y ηt ( x < = α y ηt x β ( y ηt ( x contradcton wc s a

4 Amercan Journal of umercal Analyss 4 y t Case-(: ( x ( y ηt ( x = erefore ( η < P y t x = = η x Ω Ω e ( y t ( x a ( x ( y t ( x η η a ( x ( y ηt ( x snce ( ( y ηt attans a mnmum at x wc s a contradcton y Case-(: ( x = η x = d e t ( y ηt ( x = Snce ( y ηt attans a mnmum at x ten ( y ηt ( x y ( d η t ( d = < wc s a contradcton y ηt x = for x = It mples tat Case-(v: ( y ηt attans a mnmum at x erefore B ( y ηt ( x < = γ y ηt x δ ( y ηt ( x contradcton Case-(v: ( η wc s a y t x = x Ω Ω Smlar to Case-( t leads to a contradcton y ηt x = x = d Smlar to Case- Case-(v: ( t leads to a contradcton y ηt x = x = Smlar to Case- Case-(v: ( t leads to a contradcton y ηt x = x = Smlar to Case- Case-(v: (v t leads to a contradcton Hence y( x x Ω Corollary-: Consder te dfferental equatons (-( subject to te condtons (7-(8 Let t = ( t t were x d x Ω { d } t x d x Ω { } t x d x Ω { d } 8 8 x d x Ω { } 4 4 en te above maxmum prncple s true for te MBVP (-(6 Remark-: e MBVP (-(6 as a soluton t s unque eorem-3: (Stablty result Consder te dfferental equatons (-( subject to te quas-monotoncty dagonally domnant condtons (7-(8 If y y Y ten Ω = y x [max{ B y B y B y By Py P y }] x Ω Ω Ω Ω Proof: R = [max{ B y B y B y By Py P y }] Ω Ω Ω Ω Defne te functons ω ( Let ± ω x = ω x ω ( x were ( x Rt ( x y ( x ω ( x Rt ( x y ( x ± ± = ± = ± It s easy to prove tat ω ( d ± ± ± ± αω βω γω δω ± ± ω P x P x ω by a proper coce of erefore by te maxmum prncple te requred result follows Remark-3: e MBVP (-(6 s well-posed e te problem as a unque stable soluton 3 Dervatve Estmates In ts secton te dervatve estmates for te MBVP (-(6 are provded eorem-3: Let y be te soluton of te MBVP (- k (6 en for k= ( k x Ω\ {d} y ( 3 3 y Proof: s teorem can be proved by usng te results of [] [5] Remark-4: e sarper bounds on te dervatves of te soluton are obtaned by decomposng te soluton y nto smoot sngular components as y = v w were te smoot component v s gven by k = k Ω Ω = v( β v ( = A ( f ( ( = ( ( = ( v( δ v ( = A ( f ( Pv x f x x k α v d A d f d v d A d f d γ te sngular component w s gven by were k = Ω Ω = = = Pw x x k wd v d w d v d w( w ( = ( y( y ( ( αv( β v ( w( w ( = ( y( y ( ( γ v( δ v ( α β α β γ δ γ δ ( ( ( ( α = α α β = β β γ = γ γ δ = δ δ

5 43 Amercan Journal of umercal Analyss e soluton w can be constructed by te procedure gven n [6] erefore te sngular component s well defned eorem-4: e smoot sngular components v w of y satsfy te bounds were k l ( x ρ x Ω ( k v ( x k r ( x ρ x Ω k ( k l ( x ρ x Ω w ( x = k r ( x ρ x Ω ρ = mn { ρ ρ } ρ = mn a x a x x Ω { a ( x a ( x } ρ = mn { } x Ω x Ω Proof: s teorem can be proved by usng te results of [567] by followng te tecnque of [678] ote tat v v w w C v w v w C Ω 3 Dscrete Problem Ω but A ftted mes metod for problem (-(6 s now descrbed On Ω mes ntervals s constructed as follows: Ω a pecewse unform mes of e nterval Ω s subdvded nto tree subntervals τ [ τ d τ] [ d τ d] for some τ tat [ ] d satsfes τ 4 < On [ ] τ [ d τ d] a unform mes wt mes ntervals s placed wle [ τ d τ ] 8 as a unform mes wt mes ntervals e 4 dd τ [ d τ τ ] τ of subntervals [ ] [ ] Ω are treated analogously for some τ satsfyng ( d < τ 4 e nteror ponts of te mes are denoted by Ω x : x : Clearly x = d Ω = { } ote tat ts mes s x d d a unform mes wen τ = τ = e 4 4 transton parameters τ τ are functons of d are cosen as τ = mn / ρ ln 4 d τ = mn / ρ ln 4 e sx mes wdts are gven by 8τ 4( d τ 8τ = 3 = = 4 = 6 = 4 ( d τ 5 = On te pecewse unform mes Ω a cubc splne sceme s used for Robn boundary condtons te classcal central dfference sceme s used for te dfferental equatons at te nteror ponts en te ftted mes metod for MBVP (-(6 s: δ ( Ω P y y a x y a x y = f x x δ ( Ω P y y a x y a x y = f x x e δ at were P y A x y = f x = d te sceme s gven by [9] δ = P y d y d A d y d f d ( ( fj x fj x f ( d = j = ( wt te boundary condtons were δ α β B y y S y = p (3 γ δ B y y S y = q α β B y y S y = r γ δ B y y S y = s ( D y D y y y y = D y = y y D y x x x x = = = = ( S y S y sded lmts j j (4 (5 (6 can be obtaned from te one

6 Amercan Journal of umercal Analyss 44 yj x yj x S x = M j M j (7 3 6 y j x yj x S x = M j M j (8 6 3 for j= respectvely of te frst order dervatves of cubc splne functon gven n [9] Substtutng M M from M a ( x y a ( x y = f ( x M a x y a x y = f ( x to (7 (8 we get te approxmaton of te one sded frst order dervatves at bot boundary ponts Hence te dscretzaton of te Robn boundary condtons of (3- (6 reduce to 3 β α β β a ( x β 3 p β = β f x f x a x y 3 β a ( x β y a x y y ( x 3 δ a δ y 3 δ δa ( x y δ a ( x y 3 q δ = δ f ( x f ( x γ a x δ y 3 β α β β a ( x 3 r β = a x y 3 β a ( x β y βa ( x y y β f x f x ( x 3 δ a δ y 3 δ δa ( x y δ a ( x y 3 s δ = δ f ( x f ( x γ a x δ y (9 ( ( ( e followng dscrete maxmum prncple dscrete stablty result can be proved analogous to te contnuous results stated n eorem- eorem-3 eorem-5: (Dscrete maxmum prncple For any mes functon Ψ assume tat x Ψ ( Ψ ( Ψ ( Ψ ( Ψ( Ψ( Ω D D Ψ B x B x B x B x P x P x x Ψ Also let a x a ( x en f tere exsts a mes functon t suc tat D t D t > Bt > Bt > t > t > Ω Ψ x x Ω B t P P x ten Corollary-: Consder te dscrete problem (-(5 subject to te condtons (7-(8 Let t ( t t t t x d x d x d 8 8 x d 4 4 = were en te above dscrete maxmum prncple s true for (-(5 eorem-6: (Dscrete stablty resultif ( ( ( x ( Z x = Z Z x s any mes functon ten x Ω j ( y y j = Z x = [max{ B y B y B y By P P }] Ω Ω Ω Ω 4 Error Analyss Usng te results of eorem-4 te procedure adopted n [7] te basc deas of te proofs of some teorems presented n [6] for te dervaton of estmates for te truncaton error te followng nequaltes can be derved for te MBVP (-(6: At te pont ( Pk Pk y x x Ω k = x ( ln (3 = d usng te procedure adopted n [9] wt approprate barrer functons t s easy to see tat ( Pk Pk y d C k = ( ln (4

7 45 Amercan Journal of umercal Analyss e truncaton errors of te soluton y at boundary ponts x= for te dscrete problem (-(6 were Robn boundary condtons are dscretzed by usng splne approxmaton from (7-( lead to te followng estmates [9]: ( Pk Pk y ( x = k = (5 eorem-7: e error of te numercal sceme (- (6 at nner grd ponts Ω satsfes x yk x yk ( ln k = (6 for suffcently large Proof: Usng (3 (4 (5 te desred result follows Remark-5: (Adjont system Consder te MBVP (- (6 Suppose tat te quas-monotoncty condton (7 s not satsfed by te system en te followng system s adjoned to (-(: ( x 4 yˆ x a x yˆ x a x yˆ x a x yˆ x (7 = f ˆ 3 yˆ x a x yˆ x a x yˆ x a x yˆ x =f x x Ω Ω (8 yˆ x a x y x a 3 3 ˆ ˆ3 ˆ3 ˆ 4 y x a x y x a x y x a x y x = f x yˆ 4 ( x a ( x yˆ ( x a ( x yˆ 3 ( x a ( x y4 ( x = f ( x x Ω Ω ˆ (9 (3 α yˆ β yˆ = p γ yˆ δ yˆ = q (3 ( ˆ ( r ( ˆ ( αyˆ βy = γ yˆ δy = s (3 ( ( p yˆ α yˆ β yˆ = γ yˆ δ = q ( ( r yˆ (33 αyˆ 4 βyˆ 4 = γ yˆ 4 δ 4 = s (34 a ( x = a x f a x oterwse a a x a ( x a ( x a x f a x oterwse = a a x a x Snce y ( y y = s a soluton of (-(6 yˆ = ( y y y y s a soluton of te above adjont system (7-(34 e results derved for (-(6 stll old good even f te quas-monotoncty condton s not met 5 umercal Results In ts secton two examples are gven to llustrate te computatonal metods dscussed n ts paper Consder te followng sngularly perturbed Robn type boundary value problems wt dscontnuous source term: Example-: f y x y x y x = f x x Ω Ω y x y x y x = f x x Ω Ω 3y y = y y = 3y y = y y = were f ( x ( x x < x Example-: x < x 3 y x x y x x y x = f x π x x Ω Ω y x cos x y 4 x e y x = f x x Ω Ω y y = y y = y 3 y = y y = x e x 5 were f ( x 5 < x x x 5 f ( x 5 < x e maxmum errors te orders of convergence for te soluton of te above two examples are presented for varous values of n te able - able able 3- able 4 respectvely For a fnte set of values 5 = maxmum pont-wse errors { } E j are computed as j = were 89 j max x j j Ω E = y y for 89 y j s te pecewse lnear nterpolant of y onto [ ] te mes functon 89 j From tese values te unform maxmum error s calculated by E = max E j = Furter te order of j j E j convergence s computed by p j = log j = E j

8 Amercan Journal of umercal Analyss 46 able Maxmum pont-wse errors E unform error E unform order of convergence p for dfferent values of te mes ponts for te soluton y of Example- umber of mes ponts E-3 575E-3 43E-3 54E-4 59E-4 8E E-5-448E- 7483E E-3 853E-3 853E E-4 E E- 588E- 78E- 5659E-3 666E-3 54E E E- 4484E- 849E- 67E- 89E E-3 33E E- 4454E- 83E- 65E E-3 464E-3 476E E- 4454E- 83E- 65E E-3 464E-3 476E E- 4454E- 83E- 65E E-3 464E-3 476E E- 4454E- 83E- 65E E-3 464E-3 476E E- 4454E- 83E- 65E E-3 464E-3 476E E- 4454E- 83E- 65E E-3 464E-3 476E-3 E p 64E- 4484E- 849E- 67E E-3 464E-3 476E able Maxmum pont-wse errors E unform error E unform order of convergence p for dfferent values of te mes ponts for te soluton y of Example- umber of mes ponts E-3 534E-3 3E E-4 53E-4 8E-4 364E-5-446E- 7479E-3 377E-3 85E-3 853E E-4 E E- 588E- 78E- 5659E-3 666E-3 54E E E- 4484E- 849E- 67E- 89E E-3 33E E- 4454E- 83E- 65E E-3 464E-3 476E E- 4454E- 83E- 65E E-3 464E-3 476E E- 4454E- 83E- 65E E-3 464E-3 476E E- 4454E- 83E- 65E E-3 464E-3 476E E- 4454E- 83E- 65E E-3 464E-3 476E E- 4454E- 83E- 65E E-3 464E-3 476E-3 E p 64E- 4484E- 849E- 67E E-3 464E-3 476E able 3 Maxmum pont-wse errors E unform error E unform order of convergence p for dfferent values of te mes ponts for te soluton y of Example- umber of mes ponts E- 743E- 859E E E-3 83E E E- 53E- 6474E- 34E- 644E-3 634E E E- 454E E- 3988E- 936E- 858E E E- 974E- 468E- 43E E- 5598E- 8699E E- 948E- 43E- 36E- 6786E- 3738E- 889E E- 9458E- 37E- 35E E- 3737E- 889E E- 945E- 35E- 34E E- 3736E- 889E E- 9449E- 34E- 34E E- 3736E- 889E E- 9449E- 34E- 34E E- 3736E- 889E E- 9448E- 34E- 34E E- 3736E- 889E- E p 3866E- 974E- 468E- 36E- 6786E- 3738E- 889E

9 47 Amercan Journal of umercal Analyss able 4 Maxmum pont-wse errors E unform error E unform order of convergence p for dfferent values of te mes ponts for te soluton y of Example- umber of mes ponts E- 9453E- 4658E- 7346E E E E-4-73E- 99E E- 58E- 6E- 4566E-3 54E E- 556E- 3576E- 696E- 334E- 433E- 4877E E- 5E E- 988E- 9976E- 4448E- 5E E- 53E- 3495E- 365E- 777E- 5683E- 64E E- 566E- 3493E- 37E- 779E E- 644E E- 579E E- 374E- 78E- 5684E- 644E E- 584E E- 374E- 78E- 5684E- 645E E- 585E- 3494E- 374E- 78E- 5684E- 645E E- 586E- 3494E- 375E- 78E- 5684E- 645E- E p 6575E- 5E E- 375E- 78E- 5684E- 645E dscontnuous source term was examned A dfference sceme usng ftted mes metod on pecewse unform Sskn mes was constructed for solvng te problem wc gves unform convergence A cubc splne sceme s used for Robn boundary condtons te classcal central dfference sceme s used for te dfferental equatons at te nteror ponts From te obtaned numercal results t s noted tat te rate of convergence s approacng to almost te second order as ncreases are n agreement wt te teoretcal results Remark-6: e autors are n te process of extendng te same analyss for convecton-dffuson problems consdered n [] Fgure umercal solutons y y of Example- for 5 = = 56 Acknowledgement e autors are tankful to te anonymous referee for s constructve comments valuable suggestons n mprovng te qualty of te paper References Fgure umercal solutons y y of Example- for 4 = = 56 6 Conclusons A system of two coupled sngularly perturbed reactondffuson Robn type boundary value problem wt [] H-G Roos M Stynes L obska umercal Metods for Sngularly Perturbed Dfferental Equatons Sprnger Verlag ew York 996 [] JJH Mller E O Rordan GI Sskn Ftted numercal metods for sngular perturbaton problems(revsed Edton World Scentfc Publsng Co Sngapore ew Jersey London Hong Kong [3] PA Farrell AF Hegarty JJH Mller E O Rordan GI Sskn Robust computatonal tecnques for boundary layers Capman Hall/ CRC Boca Raton [4] GI Sskn Mes approxmaton of sngularly perturbed boundary value problems for systems of elptc parabolc equatons Comput Mats Mat Pys 35 ( [5] S Mattews JJH Mller E O Rordan GI Sskn A parameter robust numercal metod for a system of sngularly perturbed ordnary dfferental equatons n: JJH Mller GI Sskn L Vulkov edtors Analytcal umercal Metods for Convecton-Domnated Sngularly Perturbed Problems ew York ova Scence Publsers pp 9-4 [6] A amlselvan Ramanujam V Sant A numercal metod for sngularly perturbed weakly coupled system of two second order ordnary dfferental equatons wt dscontnuous source term Journal of Computatonal Appled Matematcs (7 3-6

10 Amercan Journal of umercal Analyss 48 [7] M Paramasvam JJH Mller S Valarmat Parameter-unform convergence for a fnte dfference metod for sngularly perturbed lnear reacton-dffuson system wt dscontnuous source terms Internaton Journal of umercal Analyss Modelng ( ( [8] P Das S atesan Optmal error estmate usng mes equdstrbuton tecnque for sngularly perturbed system of reacton-dffuson boundary value problems Appled Matematcs Computaton 49 ( [9] P Das S atesan Hger-order parameter unform convergent scemes for Robn type reacton-dffuson problems usng adaptvely generated grd Internatonal Journal of Computatonal Metods 9 (4 ( [] P Das S atesan A unformly convergent ybrd sceme for sngularly perturbed system of reacton-dffuson Robn type boundary value problems Journal of Appled Matematcs Computng 4 ( [] R Mytl Pryadarsn Ramanujam Unformly-convergent numercal metods for a system of coupled sngularly perturbed convecton-dffuson equatons wt mxed type boundary condtons Mat Model Anal 8 (5 ( [] P Maabub Basa V Sant A numercal metod for sngularly perturbed second order coupled system of convecton-dffuson Robn type boundary value problems wt dscontnuous source term Int J Appl Comput Mat (3 ( [3] AR Ansar AF Hegarty umercal soluton of a convectondffuson problem wt Robn boundary condtons Journal of Computatonal Appled Matematcs 56 (3-38 [4] MH Protter HF Wenberger Maxmum prncples n Dfferental Equatons Prentce-Hall Englewood Clffs ew Jersey 967 [5] Lnß Madden An mproved error estmate for a numercal metod for a system of coupled sngularly perturbed reactondffuson equatons Comput Metods Appl Mat 3 ( [6] PA Farrell JJH Mller E O Rordan GI Sskn Sngularly perturbed dfferental equatons wt dscontnuous source terms n: JJH Mller GI Sskn L Vulkov (Eds Proceedngs of Analytcal umercal Metods for Convecton-Domnated Sngularly Perturbed Problems Lozenetz Bulgara 998 ova Scence Publsers ew York USA pp 3-3 [7] S Mattews Parameter robust numercal metods for a system of two coupled sngularly perturbed reacton-dffuson equatons Master ess Scool of Matematcal Scences Dubln Cty Unversty [8] JJH Mller E O Rordan GI Sskn S Wang A parameterunform Scwarz metod for a sngularly perturbed reactondffuson problem wt an nteror layer Appl umer Mat 35 ( [9] S Cra Sekara Rao S Cawla Interor layers n coupled system of two sngularly perturbed reacton-dffuson equatons wt dscontnuous source term n: I Dmov I Farago L Vulkov (Eds Proceedngs of 5 t Internatonal Conference umercal Analyss ts Applcatons Lozenetz Bulgara June LCS pp [] C de Falco E O Rordan Interor layers n a reacton-dffuson equaton wt a dscontnuous dffuson coeffcent Int J umer Anal Model 7 ( [] Z Cen A ybrd dfference sceme for a sngularly perturbed convecton-dffuson problem wt dscontnuous convecton coeffcent Appled Matematcs Computaton 69 (