Pathan Mahabub Basha *, Vembu Shanthi
|
|
- Suzanna Casey
- 5 years ago
- Views:
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 (
Solution for singularly perturbed problems via cubic spline in tension
ISSN 76-769 England UK Journal of Informaton and Computng Scence Vol. No. 06 pp.6-69 Soluton for sngularly perturbed problems va cubc splne n tenson K. Aruna A. S. V. Rav Kant Flud Dynamcs Dvson Scool
More informationApplication of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems
Mathematca Aeterna, Vol. 1, 011, no. 06, 405 415 Applcaton of B-Splne to Numercal Soluton of a System of Sngularly Perturbed Problems Yogesh Gupta Department of Mathematcs Unted College of Engneerng &
More informationA Spline based computational simulations for solving selfadjoint singularly perturbed two-point boundary value problems
ISSN 746-769 England UK Journal of Informaton and Computng Scence Vol. 7 No. 4 pp. 33-34 A Splne based computatonal smulatons for solvng selfadjont sngularly perturbed two-pont boundary value problems
More informationNumerical Solution of Singular Perturbation Problems Via Deviating Argument and Exponential Fitting
Amercan Journal of Computatonal and Appled Matematcs 0, (): 49-54 DOI: 0.593/j.ajcam.000.09 umercal Soluton of Sngular Perturbaton Problems Va Devatng Argument and Eponental Fttng GBSL. Soujanya, Y.. Reddy,
More informationSolving Singularly Perturbed Differential Difference Equations via Fitted Method
Avalable at ttp://pvamu.edu/aam Appl. Appl. Mat. ISSN: 193-9466 Vol. 8, Issue 1 (June 013), pp. 318-33 Applcatons and Appled Matematcs: An Internatonal Journal (AAM) Solvng Sngularly Perturbed Dfferental
More informationNumerical Simulation of One-Dimensional Wave Equation by Non-Polynomial Quintic Spline
IOSR Journal of Matematcs (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 14, Issue 6 Ver. I (Nov - Dec 018), PP 6-30 www.osrournals.org Numercal Smulaton of One-Dmensonal Wave Equaton by Non-Polynomal
More informationThe Finite Element Method: A Short Introduction
Te Fnte Element Metod: A Sort ntroducton Wat s FEM? Te Fnte Element Metod (FEM) ntroduced by engneers n late 50 s and 60 s s a numercal tecnque for solvng problems wc are descrbed by Ordnary Dfferental
More informationMaejo International Journal of Science and Technology
Maejo Int. J. Sc. Technol. () - Full Paper Maejo Internatonal Journal of Scence and Technology ISSN - Avalable onlne at www.mjst.mju.ac.th Fourth-order method for sngularly perturbed sngular boundary value
More informationTR/95 February Splines G. H. BEHFOROOZ* & N. PAPAMICHAEL
TR/9 February 980 End Condtons for Interpolatory Quntc Splnes by G. H. BEHFOROOZ* & N. PAPAMICHAEL *Present address: Dept of Matematcs Unversty of Tabrz Tabrz Iran. W9609 A B S T R A C T Accurate end condtons
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationThe finite element method explicit scheme for a solution of one problem of surface and ground water combined movement
IOP Conference Seres: Materals Scence and Engneerng PAPER OPEN ACCESS e fnte element metod explct sceme for a soluton of one problem of surface and ground water combned movement o cte ts artcle: L L Glazyrna
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationTR/28. OCTOBER CUBIC SPLINE INTERPOLATION OF HARMONIC FUNCTIONS BY N. PAPAMICHAEL and J.R. WHITEMAN.
TR/8. OCTOBER 97. CUBIC SPLINE INTERPOLATION OF HARMONIC FUNCTIONS BY N. PAPAMICHAEL and J.R. WHITEMAN. W960748 ABSTRACT It s sown tat for te two dmensonal Laplace equaton a unvarate cubc splne approxmaton
More information1 Introduction We consider a class of singularly perturbed two point singular boundary value problems of the form: k x with boundary conditions
Lakshm Sreesha Ch. Non Standard Fnte Dfference Method for Sngularly Perturbed Sngular wo Pont Boundary Value Problem usng Non Polynomal Splne LAKSHMI SIREESHA CH Department of Mathematcs Unversty College
More informationStanford University CS254: Computational Complexity Notes 7 Luca Trevisan January 29, Notes for Lecture 7
Stanford Unversty CS54: Computatonal Complexty Notes 7 Luca Trevsan January 9, 014 Notes for Lecture 7 1 Approxmate Countng wt an N oracle We complete te proof of te followng result: Teorem 1 For every
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationOn a nonlinear compactness lemma in L p (0, T ; B).
On a nonlnear compactness lemma n L p (, T ; B). Emmanuel Matre Laboratore de Matématques et Applcatons Unversté de Haute-Alsace 4, rue des Frères Lumère 6893 Mulouse E.Matre@ua.fr 3t February 22 Abstract
More informationALGORITHM FOR THE CALCULATION OF THE TWO VARIABLES CUBIC SPLINE FUNCTION
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LIX, 013, f.1 DOI: 10.478/v10157-01-00-y ALGORITHM FOR THE CALCULATION OF THE TWO VARIABLES CUBIC SPLINE FUNCTION BY ION
More informationMultivariate Ratio Estimator of the Population Total under Stratified Random Sampling
Open Journal of Statstcs, 0,, 300-304 ttp://dx.do.org/0.436/ojs.0.3036 Publsed Onlne July 0 (ttp://www.scrp.org/journal/ojs) Multvarate Rato Estmator of te Populaton Total under Stratfed Random Samplng
More informationPolynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
More informationResearch Article Numerov s Method for a Class of Nonlinear Multipoint Boundary Value Problems
Hndaw Publsng Corporaton Matematcal Problems n Engneerng Volume 2012, Artcle ID 316852, 29 pages do:10.1155/2012/316852 Researc Artcle Numerov s Metod for a Class of Nonlnear Multpont Boundary Value Problems
More informationA Discrete Approach to Continuous Second-Order Boundary Value Problems via Monotone Iterative Techniques
Internatonal Journal of Dfference Equatons ISSN 0973-6069, Volume 12, Number 1, pp. 145 160 2017) ttp://campus.mst.edu/jde A Dscrete Approac to Contnuous Second-Order Boundary Value Problems va Monotone
More informationOn Pfaff s solution of the Pfaff problem
Zur Pfaff scen Lösung des Pfaff scen Probles Mat. Ann. 7 (880) 53-530. On Pfaff s soluton of te Pfaff proble By A. MAYER n Lepzg Translated by D. H. Delpenc Te way tat Pfaff adopted for te ntegraton of
More informationAsymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton
More informationLecture 13 APPROXIMATION OF SECOMD ORDER DERIVATIVES
COMPUTATIONAL FLUID DYNAMICS: FDM: Appromaton of Second Order Dervatves Lecture APPROXIMATION OF SECOMD ORDER DERIVATIVES. APPROXIMATION OF SECOND ORDER DERIVATIVES Second order dervatves appear n dffusve
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationThe Study of Teaching-learning-based Optimization Algorithm
Advanced Scence and Technology Letters Vol. (AST 06), pp.05- http://dx.do.org/0.57/astl.06. The Study of Teachng-learnng-based Optmzaton Algorthm u Sun, Yan fu, Lele Kong, Haolang Q,, Helongang Insttute
More informationUniqueness of Weak Solutions to the 3D Ginzburg- Landau Model for Superconductivity
Int. Journal of Math. Analyss, Vol. 6, 212, no. 22, 195-114 Unqueness of Weak Solutons to the 3D Gnzburg- Landau Model for Superconductvty Jshan Fan Department of Appled Mathematcs Nanjng Forestry Unversty
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informationPositivity Preserving Interpolation by Using
Appled Matematcal Scences, Vol. 8, 04, no. 4, 053-065 HIKARI Ltd, www.m-kar.com ttp://dx.do.org/0.988/ams.04.38449 Postvty Preservng Interpolaton by Usng GC Ratonal Cubc Splne Samsul Arffn Abdul Karm Department
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationCubic Trigonometric B-Spline Applied to Linear Two-Point Boundary Value Problems of Order Two
World Academy of Scence Engneerng and echnology Internatonal Journal of Mathematcal and omputatonal Scences Vol: No:0 00 ubc rgonometrc B-Splne Appled to Lnear wo-pont Boundary Value Problems of Order
More informationThe Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method
Journal of Electromagnetc Analyss and Applcatons, 04, 6, 0-08 Publshed Onlne September 04 n ScRes. http://www.scrp.org/journal/jemaa http://dx.do.org/0.46/jemaa.04.6000 The Exact Formulaton of the Inverse
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationInitial-value Technique For Singularly Perturbed Two Point Boundary Value Problems Via Cubic Spline
Unversty of Central Florda Electronc Teses and Dssertatons Masters Tess (Open Access) Intal-value Tecnque For Sngularly Perturbed Two Pont Boundary Value Problems Va Cubc Splne 010 Lus G. Negron Unversty
More informationA Linearized Finite Difference Scheme with Non-uniform Meshes for Semi-linear Parabolic Equation
Jun Zou A Lnearzed Fnte Dfference Sceme wt Non-unform Meses for Sem-lnear Parabolc Equaton Jun Zou Yangtze Normal Unversty Scool of Matematcs and Computer Scence Congqng 40800 Cna flzjzklm@6.com Abstract:
More informationThe Quadratic Trigonometric Bézier Curve with Single Shape Parameter
J. Basc. Appl. Sc. Res., (3541-546, 01 01, TextRoad Publcaton ISSN 090-4304 Journal of Basc and Appled Scentfc Research www.textroad.com The Quadratc Trgonometrc Bézer Curve wth Sngle Shape Parameter Uzma
More informationSolution of Singularly Perturbed Differential Difference Equations Using Higher Order Finite Differences
Amercan Journal of Numercal Analyss, 05, Vol. 3, No., 8-7 Avalable onlne at http://pubs.scepub.com/ajna/3// Scence and Educaton Publshng DOI:0.69/ajna-3-- Soluton of Sngularly Perturbed Dfferental Dfference
More informationInexact Newton Methods for Inverse Eigenvalue Problems
Inexact Newton Methods for Inverse Egenvalue Problems Zheng-jan Ba Abstract In ths paper, we survey some of the latest development n usng nexact Newton-lke methods for solvng nverse egenvalue problems.
More informationDUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More information5 The Laplace Equation in a convex polygon
5 Te Laplace Equaton n a convex polygon Te most mportant ellptc PDEs are te Laplace, te modfed Helmoltz and te Helmoltz equatons. Te Laplace equaton s u xx + u yy =. (5.) Te real and magnary parts of an
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationA New Recursive Method for Solving State Equations Using Taylor Series
I J E E E C Internatonal Journal of Electrcal, Electroncs ISSN No. (Onlne) : 77-66 and Computer Engneerng 1(): -7(01) Specal Edton for Best Papers of Mcael Faraday IET Inda Summt-01, MFIIS-1 A New Recursve
More informationTHE STURM-LIOUVILLE EIGENVALUE PROBLEM - A NUMERICAL SOLUTION USING THE CONTROL VOLUME METHOD
Journal of Appled Mathematcs and Computatonal Mechancs 06, 5(), 7-36 www.amcm.pcz.pl p-iss 99-9965 DOI: 0.75/jamcm.06..4 e-iss 353-0588 THE STURM-LIOUVILLE EIGEVALUE PROBLEM - A UMERICAL SOLUTIO USIG THE
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationGeneral viscosity iterative method for a sequence of quasi-nonexpansive mappings
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,
More informationOn Finite Rank Perturbation of Diagonalizable Operators
Functonal Analyss, Approxmaton and Computaton 6 (1) (2014), 49 53 Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: http://wwwpmfnacrs/faac On Fnte Rank Perturbaton of Dagonalzable
More information2.29 Numerical Fluid Mechanics
REVIEW Lecture 10: Sprng 2015 Lecture 11 Classfcaton of Partal Dfferental Equatons PDEs) and eamples wth fnte dfference dscretzatons Parabolc PDEs Ellptc PDEs Hyperbolc PDEs Error Types and Dscretzaton
More informationTR/01/89 February An O(h 6 ) cubic spline interpolating procedure for harmonic functions. N. Papamichael and Maria Joana Soares*
TR/0/89 February 989 An O( 6 cubc splne nterpolatng procedure for armonc functons N. Papamcael Mara Joana Soares* *Área de Matematca, Unversdade do Mno, 4700 Braga, Portugal. z 6393 ABSTRACT An O( 6 metod
More informationLecture 2: Numerical Methods for Differentiations and Integrations
Numercal Smulaton of Space Plasmas (I [AP-4036] Lecture 2 by Lng-Hsao Lyu March, 2018 Lecture 2: Numercal Methods for Dfferentatons and Integratons As we have dscussed n Lecture 1 that numercal smulaton
More informationLecture 16 Statistical Analysis in Biomaterials Research (Part II)
3.051J/0.340J 1 Lecture 16 Statstcal Analyss n Bomaterals Research (Part II) C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ x = σ x amed for Brtsh statstcan
More informationLOW BIAS INTEGRATED PATH ESTIMATORS. James M. Calvin
Proceedngs of the 007 Wnter Smulaton Conference S G Henderson, B Bller, M-H Hseh, J Shortle, J D Tew, and R R Barton, eds LOW BIAS INTEGRATED PATH ESTIMATORS James M Calvn Department of Computer Scence
More informationMultigrid Methods and Applications in CFD
Multgrd Metods and Applcatons n CFD Mcael Wurst 0 May 009 Contents Introducton Typcal desgn of CFD solvers 3 Basc metods and ter propertes for solvng lnear systems of equatons 4 Geometrc Multgrd 3 5 Algebrac
More informationThe lower and upper bounds on Perron root of nonnegative irreducible matrices
Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College
More informationIrregular vibrations in multi-mass discrete-continuous systems torsionally deformed
(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected
More information( ) [ ( k) ( k) ( x) ( ) ( ) ( ) [ ] ξ [ ] [ ] [ ] ( )( ) i ( ) ( )( ) 2! ( ) = ( ) 3 Interpolation. Polynomial Approximation.
3 Interpolaton {( y } Gven:,,,,,, [ ] Fnd: y for some Mn, Ma Polynomal Appromaton Theorem (Weerstrass Appromaton Theorem --- estence ε [ ab] f( P( , then there ests a polynomal
More information(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate
Internatonal Journal of Mathematcs and Systems Scence (018) Volume 1 do:10.494/jmss.v1.815 (Onlne Frst)A Lattce Boltzmann Scheme for Dffuson Equaton n Sphercal Coordnate Debabrata Datta 1 *, T K Pal 1
More informationHongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k)
ISSN 1749-3889 (prnt), 1749-3897 (onlne) Internatonal Journal of Nonlnear Scence Vol.17(2014) No.2,pp.188-192 Modfed Block Jacob-Davdson Method for Solvng Large Sparse Egenproblems Hongy Mao, College of
More informationMATH 5630: Discrete Time-Space Model Hung Phan, UMass Lowell March 1, 2018
MATH 5630: Dscrete Tme-Space Model Hung Phan, UMass Lowell March, 08 Newton s Law of Coolng Consder the coolng of a well strred coffee so that the temperature does not depend on space Newton s law of collng
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,
More information[7] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Clis, New Jersey, (1962).
[7] R.S. Varga, Matrx Iteratve Analyss, Prentce-Hall, Englewood ls, New Jersey, (962). [8] J. Zhang, Multgrd soluton of the convecton-duson equaton wth large Reynolds number, n Prelmnary Proceedngs of
More informationNew Method for Solving Poisson Equation. on Irregular Domains
Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad
More informationHomogenization of reaction-diffusion processes in a two-component porous medium with a non-linear flux-condition on the interface
Homogenzaton of reacton-dffuson processes n a two-component porous medum wth a non-lnear flux-condton on the nterface Internatonal Conference on Numercal and Mathematcal Modelng of Flow and Transport n
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More information6.3.4 Modified Euler s method of integration
6.3.4 Modfed Euler s method of ntegraton Before dscussng the applcaton of Euler s method for solvng the swng equatons, let us frst revew the basc Euler s method of numercal ntegraton. Let the general from
More informationOptimal Pursuit Time in Differential Game for an Infinite System of Differential Equations
Malaysan Journal of Mathematcal Scences 1(S) August: 267 277 (216) Specal Issue: The 7 th Internatonal Conference on Research and Educaton n Mathematcs (ICREM7) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationNumerical Solutions of a Generalized Nth Order Boundary Value Problems Using Power Series Approximation Method
Appled Mathematcs, 6, 7, 5-4 Publshed Onlne Jul 6 n ScRes. http://www.scrp.org/journal/am http://.do.org/.436/am.6.77 umercal Solutons of a Generalzed th Order Boundar Value Problems Usng Power Seres Approxmaton
More informationA new construction of 3-separable matrices via an improved decoding of Macula s construction
Dscrete Optmzaton 5 008 700 704 Contents lsts avalable at ScenceDrect Dscrete Optmzaton journal homepage: wwwelsevercom/locate/dsopt A new constructon of 3-separable matrces va an mproved decodng of Macula
More information2.29 Numerical Fluid Mechanics Fall 2011 Lecture 12
REVIEW Lecture 11: 2.29 Numercal Flud Mechancs Fall 2011 Lecture 12 End of (Lnear) Algebrac Systems Gradent Methods Krylov Subspace Methods Precondtonng of Ax=b FINITE DIFFERENCES Classfcaton of Partal
More informationis the calculated value of the dependent variable at point i. The best parameters have values that minimize the squares of the errors
Multple Lnear and Polynomal Regresson wth Statstcal Analyss Gven a set of data of measured (or observed) values of a dependent varable: y versus n ndependent varables x 1, x, x n, multple lnear regresson
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationNumerical Solution of two dimensional coupled viscous Burgers Equation using the Modified Cubic B-Spline Differential Quadrature Method
umercal Soluton of two dmensonal coupled vscous Burgers Equaton usng the odfed Cubc B-Splne Dfferental Quadrature ethod H. S. Shukla 1, ohammad Tamsr 1*, Vneet K. Srvastava, Ja Kumar 3 1 Department of
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationHaar wavelet collocation method to solve problems arising in induction motor
ISSN 746-7659, England, UK Journal of Informaton and Computng Scence Vol., No., 07, pp.096-06 Haar wavelet collocaton method to solve problems arsng n nducton motor A. Padmanabha Reddy *, C. Sateesha,
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationProcedia Computer Science
Avalable onlne at www.scencedrect.com Proceda Proceda Computer Computer Scence Scence 1 (01) 00 (009) 589 597 000 000 Proceda Computer Scence www.elsever.com/locate/proceda Internatonal Conference on Computatonal
More informationA boundary element method with analytical integration for deformation of inhomogeneous elastic materials
Journal of Physcs: Conference Seres PAPER OPEN ACCESS A boundary element method wth analytcal ntegraton for deformaton of nhomogeneous elastc materals To cte ths artcle: Moh. Ivan Azs et al 2018 J. Phys.:
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationTwo point fuzzy boundary value problem with eigenvalue parameter contained in the boundary condition
Malaya Journal of Matemat Vol 6 No 4 766-773 8 ttps://doorg/6637/mjm64/ wo pont fuzzy oundary value prolem wt egenvalue parameter contaned n te oundary condton ar Ceylan * and Nat Altınıs ı Astract In
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationA PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.
Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR
More informationThe internal structure of natural numbers and one method for the definition of large prime numbers
The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract
More informationResearch Article Green s Theorem for Sign Data
Internatonal Scholarly Research Network ISRN Appled Mathematcs Volume 2012, Artcle ID 539359, 10 pages do:10.5402/2012/539359 Research Artcle Green s Theorem for Sgn Data Lous M. Houston The Unversty of
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationProjective change between two Special (α, β)- Finsler Metrics
Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 2394-9333 www.jtrd.com Projectve change between two Specal (, β)- Fnsler Metrcs Gayathr.K 1 and Narasmhamurthy.S.K 2 1 Assstant
More information, rst we solve te PDE's L ad L ad n g g (x) = ; = ; ; ; n () (x) = () Ten, we nd te uncton (x), te lnearzng eedbac and coordnates transormaton are gve
Freedom n Coordnates Transormaton or Exact Lnearzaton and ts Applcaton to Transent Beavor Improvement Kenj Fujmoto and Tosaru Suge Dvson o Appled Systems Scence, Kyoto Unversty, Uj, Kyoto, Japan suge@robotuassyoto-uacjp
More informationFuzzy Boundaries of Sample Selection Model
Proceedngs of the 9th WSES Internatonal Conference on ppled Mathematcs, Istanbul, Turkey, May 7-9, 006 (pp309-34) Fuzzy Boundares of Sample Selecton Model L. MUHMD SFIIH, NTON BDULBSH KMIL, M. T. BU OSMN
More informationIN the theory of control system, an optimal control problem
IAENG Internatonal Journal of Appled Mathematcs, 46:3, IJAM_46_3_3 An Iteratve Non-overlappng Doman Decomposton Method for Optmal Boundary Control Problems Governed by Parabolc Equatons Wenyue Lu, Keyng
More informationOptimal Dynamic Proportional and Excess of Loss Reinsurance under Dependent Risks
Modern Economy, 6, 7, 75-74 Pulsed Onlne June 6 n ScRes. ttp://www.scrp.org/journal/me ttp://dx.do.org/.436/me.6.7675 Optmal Dynamc Proportonal Excess of Loss Rensurance under Dependent Rsks Crstna Goso,
More information