A Numerical Patching Technique for Singularly Perturbed Nonlinear Differential-Difference Equations with a Negative Shift

Transcription

1 Aled Matematcs, (: - DOI:.593/j.am..4 A Numercal Patcg Tecque for Sgularl Perturbed Nolear Dfferetal-Dfferece Equatos wt a Negatve Sft R. Nageswar Rao, P. Pramod Cakravart Deartmet of Matematcs, Vsvesvaraa Natoal Isttute of Tecolog, Nagur, 44, Ida Abstract I ts aer, we reset a umercal atcg tecque for solvg sgularl erturbed olear dfferetal-dfferece equato wt a small egatve sft. Te olear roblem s coverted to a sequece of lear roblems b quaslearzato rocess. After learzato, t s dvded to two roblems, amel er rego roblem ad outer rego roblem. Te boudar codto at te cuttg ot s obtaed from te teor of sgular erturbatos. Usg stretcg trasformato, a modfed er rego roblem s costructed ad s solved b usg te uwd fte dfferece sceme. Te outer rego roblem s solved b a Talor olomal aroac. We combe te solutos of bot roblems to obta a aromate soluto of te orgal roblem. Te roosed metod s teratve o te cuttg ot. Te rocess s reeated for varous coces of te cuttg ot, utl te soluto rofles stablze. Some umercal eamles ave bee solved to demostrate te alcablt of te metod. Te metod s aalzed for stablt ad covergece. Kewords Sgular Perturbatos, Numercal Patcg Tecque, Nolear Dfferetal-Dfferece Equato, Uwd Fte Dfferece Sceme, Talor Polomal Aroac. Itroducto A sgularl erturbed dfferetal-dfferece equato s a ordar dfferetal equato wc te gest dervatve s multled b a small arameter ad volvg at least oe dela term. Suc roblems are foud trougout te lterature o edemcs ad oulato damcs were tese small sfts la a mortat role te modelg of varous real lfe eomea[]. Boudar value roblems dfferetal dfferece equatos arse a ver atural wa studg varatoal roblems cotrol teor were te roblem s comlcated b te effect of tme delas sgal trasmsso[]. I te matematcal model for te determato of te eected frst-et tme te geerato of acto otetal erve cells b radom satc uts dedrtes, te sfts are due to te jums te otetal membrae wc are ver small[]. I[5] te autors C.G. Lage ad R.M. Mura gave a asmtotc aroac te stud of a class of boudar value roblems for lear secod order dfferetal-dfferece equatos wc te gest order dervatve s multled b a small arameter. I[4] te autors ave eteded te deas to te boudar value roblems for sgularl erturbed olear Corresodg autor: ramododla@aoo.co. (P. Pramod Cakravart Publsed ole at tt://joural.saub.org/f Corgt Scetfc & Academc Publsg. All Rgts Reserved dfferetal dfferece equatos ad dscussed te estece ad uqueess of ter solutos. I[3], te autors roosed a umercal metod to solve te boudar value roblems for sgularl erturbed lear dfferetal-dfferece equatos, wc worked well we te dela arameter s of O( or o(. To adle te dela argumet, te costructed a secal te of mes so tat te term cotag dela les o odal ots after dscretzato. Mustafa Gulsu ad Memet Sezer[6] ave roosed a Talor olomal aroac for solvg mt order lear dfferetal-dfferece equatos wt med codtos. Ts metod s based o frst takg te trucated Talor s easos of te fuctos te dfferetal-dfferece equatos ad te substtutg ter matr forms to te equato. Hece te result matr equato ca be solved ad te ukow Talor coeffcets ca be foud aromatel. I[], te autors M.K. Kadalbajoo ad K.K. Sarma reseted te umercal stud to solve te sgularl erturbed olear dfferetal dfferece equatos wt egatve sft. We te sft s o(, te term cotag te sft s eaded Talor seres ad we te sft s O(, a secal te of mes s costructed to adle te sft term. I[], te autors M.K.Kadalbajoo, Devedra Kumar roosed a B-Sle collocato metod for solvg a sgularl erturbed olear dfferetal dfferece equato wt egatve sft. A ecewse uform mes s used to gras te better aromato to te eact soluto te boudar laer rego. Talor seres s used to tackle te dela.

2 R. Nageswar Rao et al.: A Numercal Patcg Tecque for Sgularl Perturbed Nolear Dfferetal-Dfferece Equatos wt a Negatve Sft I ts aer, we reset a umercal atcg tecque for solvg sgularl erturbed olear dfferetal dfferece equatos wt egatve sft ad reset te umercal stud to suc roblems. I order to kow te beavour of te soluto of te sgularl erturbed dfferetal dfferece equatos te boudar laer rego, t s alwas suggestve to dvde te orgal roblem to two roblems amel, te er rego roblem ad te outer rego roblem ad solve tem searatel. Frst, we learze te olear boudar value roblem usg quaslearzato[8] ad obta a sequece of lear boudar value roblems. Te t s dvded to two roblems amel er rego roblem ad outer rego roblem. Te boudar codto at te cuttg ot s obtaed from te teor of sgular erturbatos. Usg stretcg trasformato, a modfed er rego roblem s costructed ad s solved b usg te uwd fte dfferece sceme. Te outer rego roblem s solved b a Talor olomal aroac gve b Mustafa Gulsu ad Memet Sezer[6]. We combe te solutos of bot te roblems to obta a aromate soluto to te orgal roblem. Te roosed metod s teratve o te cuttg ot. Te rocess s to be reeated for varous coces of te cuttg ot, utl te soluto rofles stablze. Te estece ad uqueess of te dscretzed roblem alog wt stablt estmates are dscussed. Some umercal eamles ave bee solved to demostrate te alcablt of te metod.. Numercal Patcg Tecque To descrbe te metod, we frst cosder a olear sgularl erturbed dfferetal dfferece equato of te form: "( = f (, (, '( δ, ( subject to te terval ad boudar codtos ( = φ(; δ ( = β ( were < << s a small erturbato arameter ad < δ <<. Te soluto ( of te boudar value roblem (-( s assumed to be cotuous o[,] ad cotuousl dfferetable o (,. It s assumed tat f (,, ' s smoot fucto satsfg te codtos: f f f f > M > ' ', were M s a ostve costat. v Te growt codto f (,, ' = O(' as ' for all [,] ad all real ad '. For δ=, uder te codtos lsted above te roblem (-( as a uque soluto[9]. To develo a umercal sceme for te boudar value roblem (-(, we frst learze te orgal olear roblem b usg quaslearzato rocess[8]. Te olear dfferetal equato s learzed aroud a omal soluto of te olear dfferetal equato wc satsfes te secfed boudar codtos. Let ( ( be te tal guess to te soluto of te roblem ( satsfg te boudar codtos ( ( = φ(, δ ; ( ( = β. Te a sequece of boudar value roblems ca be obtaed as follows: We ave (" (k + = f wt ( = φ( (k + (, (,( (k + ( δ (k +, δ ; ( = β Eadg te rgt ad sde of (3 Talor seres about, we obta (k + ( = f (, (,( ( δ (k f (( + ( δ ( ( δ (" ' (k + f ( ( ( +... B rearragg te terms of (4 we obta te recurrece relato of sgularl erturbed lear dfferetal-dfferece equatos (k + f (k + (" ( (' ( δ ' f (k + ( f (, (,(' ( δ = (5 f f ( (' ( δ ' (k + (k + wt ( = φ( o δ, ( = β For smlct, we deote f a ( ' =, f b ( =, F ( = f (, (,(' ( δ f f ( ( ( δ ' Te (5 ma be wrtte as (k + (" a (( (k + + ( δ + (6 (k + b ( ( = F ( wt (k + ( = φ( o δ, (k ( = β Te soluto of te reduced roblem of (-( s assumed to be te tal aromato ( (, ad te successve (3 (4 (7

3 Aled Matematcs, (: - 3 aromatos { (} k = are determed b (6-(7. Hece, stead of solvg te orgal olear roblem (-(, we solve te sequece of boudar value roblems for sgularl erturbed secod order lear dfferetal equatos wt egatve sft gve b (6-(7 for k=,,, For large values of k te solutos ( k ( coverge to te soluto ( of te orgal olear dfferetal equato (k + wle umercall, we requre tat ( ( < μ,, were µ s te rescrbed tolerace. Te terato ca be termated we te above codto s satsfed, ad ( k + te rofle ( s te umercal soluto of te olear boudar value roblem (-(. Due to te resece of te sgular erturbato arameter, te soluto of te roblem ebts boudar laer beavor. As, te order of te reduced roblem decreases b oe, terefore te soluto ebts boudar laer beavor at eter of te boudar ots.e., te boudar laer wll be o te left sde or te rgt sde of te doma of cosderato deedg o te sg of te coeffcet a ( k ( of te covecto term tat s accordg as a ( k ( M > or a ( k ( M < resectvel, were M s a ostve costat. We assume tat a ( k ( M > trougout te terval [, ]. Ts assumto mles tat te boudar laer wll be te egborood of =. We set δ=τ wt τ=o(. If τ s ot too large, te laer structure s modfed but mataed at te same ed [5]. We cosder = O( te cuttg ot or te tckess of te boudar laer. Now te learzed roblem s dvded to two roblems amel te er rego roblem ad te outer rego roblem. Te er rego roblem s defed te terval ad te outer rego roblem s defed te terval... Boudar Codto at te Cuttg Pot We assume tat { (} k = are suffcetl dfferet- able. B eadg te retarded term (' ( δ usg Talor seres, we obta ( ( δ (' ( δ(" ( Sce δ=τ, we ave ( ( δ (' ( τ(" ( (8 B substtutg (8 (6, we get [ a (τ]( ( + a (( ( + (9 b ( ( = F (. We sall seek a outer soluto as a asmtotc easo te form ( k ( ( = = ( were ( k ( are ukow fuctos to be determed. Substtutg ( (9 ad smlfg, we get a a (( wt ( + b ( (( [ a (τ] ( k+ Te soluto of ( s ( = F ( wt ( = β ad ( ( + b ( b (ξ F ( = e dξ a (ξ a ( ( =, =,,3... (s s b e (s a ( k+ ( = ( (ξ dξds + (ξ β Te fuctos (, ( ca be obtaed b solvg equato ( for =,,3. Tus te easo for ( k ( gve equato ( s obtaed. Hece te boudar codto at te cuttg ot ca be obtaed from ( ad deote ( = ( = γ (sa (3 = Sce te termal ot s commo to bot te er ad outer regos, t defes te er rego roblem as a boudar value roblem ( ( + a (( ( δ +, (4 b ( ( = F ( wt te terval ad boudar codtos ( = φ(; δ (5 ( = γ. ad te outer rego roblem as a boudar value roblem ( ( + a (( ( δ +, (6 b ( ( = F ( wt te boudar codtos ( = γ ad ( = β (7.. Ier Rego Problem From (4-(5 te er rego roblem s gve b ( ( + a (( ( δ +, (8 b ( ( = F ( wt te terval ad boudar codtos ( k ( = φ(; δ ( = γ. (9 We coose te trasformato =t to create a ew er rego roblem. B rescalg te equato (8 wt

4 4 R. Nageswar Rao et al.: A Numercal Patcg Tecque for Sgularl Perturbed Nolear Dfferetal-Dfferece Equatos wt a Negatve Sft ( = (t = Y (t (' ( = ( (t = (Y (t (" ( = ( (t = (Y (t a ( = a (t = A (t b ( = b (t = B (t F ( = F (t = Fˆ(t ( we obta te ew dfferetal-dfferece equato for te er rego soluto as (Y" (t + A B (t(y' (ty (t τ + (t = Fˆ(t ( δ were τ =. Te boudar codtos for te equato ( are determed b (9 ad ( as Y Y (t = Φ(t = φ(t; τ t (t = γ. ( We solve te ew er rego roblem (-( to obta te soluto over te terval t t. We costruct a umercal sceme for solvg (-( based o a uwd fte dfferece sceme. We dvde te terval [, t ] to equal arts. To tackle te dela term, we coose te mes τ arameter as =, were m=q, s a ostve teger ad m q s te matssa of τ. Te dfferece sceme for (-( s gve b D+ D Y + A D+ Y m + B Y = Fˆ (3, =,,3...,. ( k Y =Φ, = m, m+,., (4 ad Y ( k + = γ. were Y Y + Y + Y Y D + D Y =, + D Y + = ad A = A (t, B = B (t, Fˆ = Fˆ (t. Now (3-(4 become Y + ( B Y + Y + = Fˆ, for =,, m-. A ( Φ m+ Φ m Y + ( B Y + Y + + A Y =, for = m. Fˆ + A Φ Y + ( B Y + Y + + (5 A ( Y m+ Y m = Fˆ for = m+, m+, - wt Y = Φ = Φ( ad Y ( k + = γ (6 Te dscrete roblem (5-(6 reduces to a sstem of (+ lear dfferece equatos gve b G Y = H, (7 T were Y = [Y Y... Y, ],...,H [ H, H ] T H = ad G = [gj], te ozero etres of te sstem matr beg gve b g(, = g(, = + B for =,,...,m g(, + = H = Fˆ A [ Φ m Φ m] + ad g(, = g(, = + B g(, + = for = m,..., g(, m + = A g(, m = A H = Fˆ H = Φ, H = γ, g(, = = g(, Te sstem (7 s solved b Gauss elmato metod wt artal votg. I fact, a umercal metod or aaltcal metod ca be used..3. Outer Rego Problem Sce te termal ot s commo to bot te er ad outer regos, t defes te outer rego roblem as a boudar value roblem ( ( + a (( ( δ +, (8 b ( ( = F ( wt te boudar codtos ( = γ ad ( = β (9 We solve te outer rego roblem (8-(9 b emlog te Talor olomal aroac gve b Mustafa Gulsu ad Memet Sezer[6] to obta te soluto over te terval..4. Soluto of te Orgal Problem After gettg te soluto of te er rego roblem ad outer rego roblem, we combe bot to obta te aromate soluto of te orgal roblem (-( over te terval. We reeat te rocess for varous coces of, utl te soluto rofles do ot dffer materall from terato to terato. For comutatoal uroses we use a absolute error crtero, amel + ( Y (t ( Y (t σ; t t were ( Y (t s te t terate of te er rego soluto ad σ s te rescrbed tolerace boud.

5 Aled Matematcs, (: Error Estmate Now we sall fd te error estmate for te modfed er rego roblem (3 uder te boudar codtos (4. Trougout te aalss smlct we relace (k Y +, A, B, Fˆ wt Y, A, B, F resectvel. Case : We B (t θ <, were θ s a ostve costat. Lemma : (Dscrete mmum rcle Let π be a mes fucto tat satsfes π ad π ad L (π te π for all =,,,,. Proof: Let k {,,,..., } be suc tat πk = m π ad assume tat π k <. Clearl k {,}. Te for k m we ave We ave L (πk = D D πk B + + (t k πk πk πk + π = k + + B (t k πk = (πk + πk (πk πk + B (t k πk >, sce π B k + πk ad πk πk, (t k < Tus, we ave L ( πk > for k m wc cotradcts te otess tat L ( πk for =,..., of te dscrete mmum rcle. Ts cotradcto arose sce we assumed tat π k <. Terefore our assumto s wrog. Hece, π k. But k {,,,...,} s a arbtrar ostve teger, so π for all =,,...,. Teorem : If A ( t M > ad B ( t θ <, were M ad θ are ostve costats, te te soluto of te dscrete roblem (3 wt te boudar codtos (4 ests, s uque ad satsfes Y θ F + K( Φ + γ, (3, were K s a ostve costat. Here,. s te dscrete l, orm defed b t ma t = Proof: Suose { u } = ad { v } = be two solutos to te dscrete roblem (3, (4. Te, z = u v s a mes fucto satsfg z =, z = for, we ave L (z = L (u L (v. Sce u ad v satsf (5, terefore L (z =,. Tus, te mes fucto z satsfes te otess of te dscrete mmum rcle ad so b te alcato of t to te mes fucto z we get z = u v, (3 Aga let z = (u v, te z satsfes z = = z ad roceedg as above we get L (z =,.. Tus, te dscrete mmum rcle ca be aled for te mes fucto z wc gves z = (u v..e., u v,. (3 Hece, from (3 & (3 we get u v =, wc rove te uqueess of te soluto to te dscrete roblem (3-(4 ad for lear equatos, te estece s mled b uqueess. Now to rove te boud o { Y } = we cosder two barrer fuctos ψ defed b ψ = θ F + K( Φ + γ Y,, were K s a arbtrar ostve costat. Te, we ave ψ = θ F + K( Φ + γ Y, = θ F + ( K Φ Φ + K γ, sce Y = Φ, s ce Φ Φ ad K ad ψ = θ F + K( Φ + γ Y, = θ F + K Φ + K γ γ, sce Y = γ γ γ ad K, sce For m, we ave L (ψ D D ( = + ψ + B (t ψ = B (t θ F + K( Φ + γ L ( (33, Y Also for m, we ave L (Y Y B Y = D+ D + (t (Φ Φ F(t A (t m+ = m Te from (33 we get B L (ψ = (t θ F + K( Φ + γ (Φ Φ F(t A (t m+ m Sce, B (t θ <, tat s B (t θ, we get L B (ψ F + (t K( Φ + γ (Φ Φ F(t A (t m+ m F F(t B + (t K( Φ + γ (34 (Φ Φ A (t m+ m Sce, te above equalt (34 te frst ad secod terms are egatve, so we coose te costat K suc tat te sum of te modul of te frst ad secod terms domates te modulus of te trd term te above equalt. We te obta

6 6 R. Nageswar Rao et al.: A Numercal Patcg Tecque for Sgularl Perturbed Nolear Dfferetal-Dfferece Equatos wt a Negatve Sft L (ψ, m. (35 For m,we ave L (ψ = D+ D ψ + A (td+ ψ m + B (t ψ = B (t θ F K( Φ γ L (Y + + (36 We kow tat for =m, m+,..,-, L (Y D D Y A (t D Y B (t Y = m + = F(t Te from (36 we get L (ψ B (t θ F K( Φ γ = + + F(t Sce, B (t θ <, tat s B (t θ, we get L (ψ F B (t K( Φ γ + + F(t F F(t B + (t K( Φ + γ (37 Hece, L (ψ, m (38 Combg te results of (35 ad (38 we get L (ψ,. Tus a alcato of Lemma to te mes fucto ψ gves ψ θ F K( Φ γ Y = + +, wc roves te requred boud o te dscrete soluto, were θ s a ostve co- { Y } =. Case : We B (t θ > stat. Lemma : (Dscrete mamum rcle Let π be a mes fucto tat satsfes π ad π ad L (π te π for all =,,,,. Proof: Let k {,,,...,} be suc tat πk = ma π ad assume tat π k <. Clearl k {,}. Te, for k m we ave L (π D D π + B (t π k = + k k k πk πk + π = k + + B (t k πk (πk πk (πk πk = + + B (t k πk ad π <, s ce π π k +, B (t π > k k k Tus, we ave L (πk < for k m wc cotradcts te otess tat L (πk for =,,,- of te dscrete mamum rcle. Ts cotradcto arose sce we assumed tat π k <. Terefore our assumto s wrog. Hece, π k. But k {,,,...,} s a arbtrar ostve teger, so π for all =,,,. Teorem : If A (t M > ad B (t θ >, were M ad θ are ostve costats, te te soluto of te dscrete k roblem (3 wt te boudar codtos (4 ests, s uque ad satsfes Y θ F + K( Φ + γ, (39, were K s a ostve costat. Proof: Let { u } = ad { v } = be two solutos to te dscrete roblem (3, (4. Te, z = u v s a mes fucto satsfg z =, z, we ave Sce = L (z = L (u L (v, u ad v satsf (3, terefore L (z =,. Tus, te mes fucto z satsfes te otess of te dscrete mamum rcle ad so b te alcato of t to te mes fucto z we get z = u v, (4 Aga we cosder z = (u v, te z s a mes fucto satsfg z = = z ad roceedg as above we get L (z =,. Tus, te dscrete mamum rcle s aled o te mes fucto z wc gves z = (u v..e., u v,. (4 Hece, from te (4, (4 we get u v =, for wc roves te uqueess of te soluto to te dscrete roblem (3-(4, ad for lear equatos te estece s mled b uqueess. Now to rove te boud o { Y } = we cosder two barrer fuctos ψ defed b ψ = θ F + K Φ + γ Y,, ( were K s a arbtrar ostve costat. Te, we ave ψ = θ F + K Φ + γ Y ( = θ F ( K Φ Φ + + K γ, sce Y = Φ, sce Φ Φ ad K ad ψ = θ F + K( Φ + γ Y, = θ F + K Φ + ( K γ γ, sce Y γ =, sce γ γ ad K. ad for m, we ave L (ψ = D+ D ψ + B (t ψ = B ( t θ F K( Φ γ L (Y + + (4 We kow tat for =,,..m-,

7 Aled Matematcs, (: - 7 L (Y Y B Y = D+ D + (t = (Φ Φ F(t A (t m+ m Te from (4 we get B L (ψ = (t θ F + K( Φ + γ F(t (Φ Φ A (t m+ m Sce, B (t θ >, tat s B (t θ, we get L (ψ F + B t K Φ + γ F F(t ( ( (Φ A (t m+ Φ F(t B + (tk (Φ Φ A m+ (t m ( Φ + γ m (43 Sce, te above equalt (43 te frst ad secod terms are ostve, so we coose te costat K so tat te sum of te modul of te frst ad secod terms domates te modulus of te trd term te above equalt. We te obta For m, we ave L (ψ L (ψ, m. (44 = D+ D ψ + A (td + ψ m + B (t ψ + + = B (t θ F K( Φ γ L We kow tat for =m, m+,.., -, L (Y D D Y + A (t D Y + B (t Y = + + m = (Y F(t Te from (45 we get L (ψ B (t θ F K Φ γ = + + F(t Sce, B (t θ >, tat s B (t θ, we get L (ψ F + B (t K Φ + γ F(t ( ( K( Φ + γ (45 F F(t B + (t (46 Hece, L (ψ, m (47 Combg te results of (44 ad (47 we get L (ψ,. Tus a alcato of Lemma to te mes fucto gves ψ = θ F + K Φ + γ Y, ( ψ wc roves te requred boud o te dscrete soluto { Y } =. Tus, Teorems ( ad ( ml tat te soluto to te dscrete roblem (3, (4 s uforml bouded, deedetl of te mes arameter ad te arameter, wc roves tat te dfferece sceme s stable for all mes szes. Corollar: Uder te assumto tat A (t M >, te error e = Y (t Y betwee te soluto Y (t of te cotuous roblem (, ( ad te soluto Y of te dscrete roblem (3,(4 satsfes te estmate e θ T, were T,, satsfes ( Y 4 ma (t + 6 t t t+ T ( A Y + Y (t ma "(t (t + Y 4 (t t m t t m+ 3 Proof: Te trucato error T s gve b Y Y + Y + T = Y (t + Y + Y A m m (t Y (t m Now usg Talor s seres ad after smlfcatos, we obta Y (4 ma (t + 6 t t t + T A Y + Y (t ma "(t (t + Y (4 (t t m t t m+ 3 We ave L ( ( Y L ( Y e(t = L (t = T, =,,,- ad e =e =. Te b usg Teorems ( ad ( we obta te requred error estmate. 4. Numercal Results We ave aled te reset metod o two olear sgularl erturbed dfferetal-dfferece equatos wt small egatve sft. Te olear roblems are frst coverted to sequece lear sgularl erturbed dfferetal-dfferece equatos b usg quaslearzato metod. Te soluto of te reduced roblem s take as tal aromato. ( Eamle [,.593]: "( + ' ( δ e =, wt te terval ad boudar codtos ( = ; δ, ( = Te soluto at te cuttg ot s gve b te reduced roblem soluto ( = ( + ( were ( = loge 3 τ 3 ad ( = loge (3 ad ece = τ ( loge 3 3

8 8 R. Nageswar Rao et al.: A Numercal Patcg Tecque for Sgularl Perturbed Nolear Dfferetal-Dfferece Equatos wt a Negatve Sft Te learzed form of te gve olear equato b quaslearzato metod s "( + ' ( δ ( = log e wt te codtos ( = ; δ ad ( =. Te modfed er rego roblem s gve b Y"(t + Y' (t τ Y(t = log 3 t, t t e 3 t 3 t uder te codtos Y ( =, Y(t = γ. Te outer rego roblem s gve b "( + '( δ ( = loge 3 3 3, uder te codtos ( = γ, ( = Te umercal results are gve tables & for τ=.5 ad = 3, = 4 resectvel. For τ=.5 adτ=.5, te er soluto s lotted gras ad sow fg. to fg 4 for = 3, = 4 resectvel. Numercal Soluto Table. Numercal results of Eamle for = 3, τ=.5. (t=3 (t=6 (t= Fgure. Ier soluto of Eamle for = 3, τ=.5. Numercal Soluto Numercal Soluto Numercal Soluto Fgure. Ier soluto of Eamle for = 3, τ= Fgure 3. Ier soluto of Eamle for = 4, τ= Fgure 4. Ier soluto of Eamle for = 4, τ=.5. Table. Numercal results of Eamle for = 4, τ=.5. (t=3 (t=6 (t= Eamle [,.e9]: "( + (.'( δ ( = wt te terval ad boudar codtos

9 Aled Matematcs, (: - 9 ( = ; δ, ( = Te soluto at te cuttg ot s gve b te reduced roblem soluto ( = ( + ( were ( = ad ( = ad ece ( = Te learzed form of te gve olear equato b quaslearzato metod s "( + '( δ = wt te codtos ( = ; δ, ( = Te er rego roblem s gve b Y "(t + ty'(t τ = t, t t uder te codtos Y ( =, Y(t = γ Te outer rego roblem s gve b "( + '( δ =, uder te codtos ( = γ, ( = Te umercal results are gve tables 3 & 4 for τ=.5 ad = 3, = 4 resectvel. For τ=.5 ad τ=.5, te umercal soluto s lotted gras ad sow fg. 5 to fg. 8 for = 3, = 4 resectvel. Table 3. Numercal results of Eamle for = -3, τ=.5. Numercal Soluto Numercal Soluto Fgure 6. Numercal soluto of Eamle for = 3, τ= Fgure 7. Numercal soluto of Eamle for = 4, τ=.5..8 Numercal Soluto (t=3 (t=6 (t= Fgure 5. Numercal soluto of Eamle for = 3, τ=.5. Numercal Soluto Fgure 8. Numercal soluto of Eamle for = 4, τ=.5. Table 4. Numercal results of Eamle for = -4, τ=.5. (t=3 (t=6 (t=

10 R. Nageswar Rao et al.: A Numercal Patcg Tecque for Sgularl Perturbed Nolear Dfferetal-Dfferece Equatos wt a Negatve Sft 5. Coclusos I order to kow te beavor of te soluto of te sgularl erturbed dfferetal-dfferece equatos te boudar laer rego, t s alwas suggestve to dvde te orgal roblem to two roblems amel te er rego roblem ad te outer rego roblem ad solve tem searatel. We ave reseted a umercal atcg tecque for solvg sgularl erturbed olear dfferetal-dfferece equatos wt te boudar laer at oe ed ot. Te orgal olear boudar value roblem s learzed usg quaslearzato. Te t s dvded to two roblems amel er rego roblem ad outer rego roblem. Te boudar codto at te cuttg ot s obtaed from te teor of sgular erturbatos. A ew er rego roblem s costructed ad solved b usg uwd fte dfferece sceme. Te outer rego roblem s solved b Talor olomal aroac. We ave mlemeted te reset metod o two olear eamles ebtg a left ed boudar laer. Te roosed metod s teratve o te cuttg ot. Te rocess s to be reeated for varous coces of te cuttg ot, utl te soluto rofles do ot dffer materall from terato to terato. Numercal results are reseted tables. It ca be observed from te tables tat te reset metod aromates te soluto avalable te lterature as well. ACKNOWLEDGMENTS Te autors ws to tak te Deartmet of Scece & Tecolog, Govermet of Ida, for ter facal suort uder te roject No. SR/S4/MS: 598/9. (5 e99-e94 [] Moa K. Kadalbajoo ad Devedra Kumar, A comutatoal metod for sgularl erturbed olear dfferetal-dfferece equatos wt small sft, Aled Matematcal Modellg 34 ( [3] Moa K. Kadalbajoo ad Kal K. Sarma, A umercal metod based o fte dfferece for boudar value roblems for sgularl erturbed dela dfferetal equatos, Aled Matematcs ad Comutato 97 ( [4] C.G. Lage ad R.M. Mura, Sgular erturbato aalss of boudar-value roblems for dfferetal dfferece equatos, IV, a olear eamle wt laer beavor, Stud. Al. Mat. 84 ( [5] C.G. Lage ad R.M. Mura, Sgular erturbato aalss of boudar-value roblems for dfferetal dfferece equatos. V. Small sfts wt laer beavor, SIAM Joural o Aled Matematcs 54 ( [6] Mustafa Gulsu ad Memet Sezer, A Talor olomal aroac for solvg dfferetal- dfferece equatos, Joural of Comutatoal ad Aled Matematcs 86 ( [7] E.P. Doola, J.J.H. Mller, W.H.A. Sclders, Uform Numercal Metods for Problems wt Ital ad Boudar Laers, Boole Press, Dubl, 98 [8] R.E. Bellma, R.E. Kalaba, Quaslearzato ad Nolear Boudar Value Problems, Amerca Elsever, New York, 965 [9] F.A. Howes, Sgular Perturbatos ad Dfferetal Iequaltes, vol. 68, Memors of te Amerca Matematcal Socet, Provdece, RI, 976 [] Y. Kuag, Dela Dfferetal equatos wt alcatos oulato damcs, Academc Press, 993 [] R.B. Ste, A teoretcal aalss of euroal varablt, Boscal Joural 5 ( REFERENCES [] Moa K. Kadalbajoo ad Kal K. Sarma, Numercal treatmet for sgularl erturbed olear dfferetal dfferece equatos wt egatve sft, Nolear Aalss 63 [] L.E. El sgol ts, Qualtatve Metods Matematcal Aalses, Traslatos of Matematcal Moogras, Amerca matematcal socet, Provdece, RI, 964