SINGULARLY PERTURBED BOUNDARY VALUE. We consider the following singularly perturbed boundary value problem
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1 Cater PARAETRIC QUITIC SPLIE SOLUTIO OR SIGULARLY PERTURBED BOUDARY VALUE PROBLES Introcton We conser te ollong snglarly ertrbe bonary vale roblem " L r r > an ere s a small ostve arameter st << r an are scently smoot nctons st r r > an < < We ave mose te conton r r > to ensre te estence an neness o te solton 8 We se arametrc ntc slne or ts solton Sc n o roblems arses n varos els o engneerng an scence or eamle elastcty l ynamcs otmal control teory yroynamcs etc any ators ave scsse te nmercal solton o snglarly ertrbe bonary vale roblems n ter or In ts cater e evelo a amly o metos base on arametrc ntc slne In secton e ave gven te bre ntrocton o arametrc ntc slne an erve mortant slne relatons In secton e ale te arametrc ntc slne tecne on snglarly ertrbe bonary vale roblem an obtane te
2 Parametrc Qntc Slne Solton sceme Secton ncles te eveloment o bonary contons o orer or an s an sbsecton enes te erent class o metos base on ter orer In secton convergence analyss o te metos s scsse an secton contans nmercal llstratons o te roose metos Parametrc Qntc Slne an ts Relatons In ts secton e ave resente te ormlaton o te arametrc ntc slne nterolant S τ C a b an erve te slne relatons or te ormlatons o or metos b a Conser a norm mes { } A ncton τ S o class C a b c nterolates at te mes onts eens on te arameterτ reces to ntc slne S n a b as τ s terme as arametrc ntc slne ncton Snce te arameter τ can occr n τ n many ays sc a slne s not ne S I S τ S s a arametrc ntc slne satsyng te ollong erental eaton n te nterval - S τ S τ τ Q Q ere Q τ S S an τ > ten t s terme arametrc ntc slne I 8
3 Parametrc Qntc Slne Solton Solvng an etermnng te or constants o ntegraton rom te nterolatng contons at an - e obtan!!! S ere sn / sn τ ± ± Smlarly n te nterval e get!!! S Usng te contnty contons o rst an secon orer ervatves e obtan rom an e obtan: ltlyng by an by an ang e get
4 Parametrc Qntc Slne Solton Usng e ave te ollong relatons s 7 ere s cos ec cot τ Remar: 7 8 As τ e ten an 8 s o 8 teslneene by reces to ntc slne8 Te eto At te mes onts a te erental eaton can be screte by sng te slne relaton 7 to obtan
5 Parametrc Qntc Slne Solton 7 s r r rs r r Remar: or an s or meto reces to ntc slne meto Develoment o Bonary Eatons We evelo t orer bonary contons at as s T Relacng te cttos vales - an - sng Taylor s teorem e get s T ere r s r r an ence
6 Parametrc Qntc Slne Solton 8 " " " s T Usng Taylor s seres an comarng te coecents o an e get s an Pttng tese vales n e get te ort orer bonary eaton as O r r r Smlarly at te oter en te bonary eaton s gven by O r r r rove rter n orer to obtan te st orer meto e evelo te bonary contons as ollos: t b b b b t b b b b By Taylor s eanson e obtan te vales o te nnons as ollos:
7 Parametrc Qntc Slne Solton 9 b b b b Te local trncaton error t assocate t te sceme s gven by t " " s " " " Class o etos Eatons to gve rse to a class o metos o erent orer as ollos: Secon orer meto or an s or sceme s o secon orer an te local trncaton error s t t / t 77 / 77 / ort orer meto 7 or / / s / or sceme s o ort orer an te local trncaton error s t t / 7 t 77 / 77 / 8 an or sceme reces to A an Kan s meto
8 Parametrc Qntc Slne Solton St orer meto or 7 / 7 9 / 9 s 7 / an b b b b Or meto s st orer an local trncaton error s 8 t 79 / 8 t 8 8 t / / v St orer at nteror noes an ort orer at en onts or 7 / 7 9 / 9 s 7 / or sceme s ort orer at te en onts an st orer at nteror noes an local trncaton error s t t 77 / 8 8 t / 8 77 / Convergence o te eto Pttng te enta agonal system n te matr e ave AU D C ere A a j s a enta agonal matr o orer - t coecent o ± n eaton a ± - - T U - - T
9 Parametrc Qntc Slne Solton D s s s an C c c c - c - T ere r c r c c r c r c or St orer meto D s s s
10 Parametrc Qntc Slne Solton an C c c c - c - T ere c r b c r c c r c r b Also e ave A U D T C T ere U s eact solton an T T t t t s te local trncaton error vector rom eaton an e ave A U U AE T ere E U U T e e e Clearly te ro sms S a j r s j j S a j r s j S a r s B j -
11 Parametrc Qntc Slne Solton S S j a j r s j j a r s We can coose scently small so tat te matr A s rrecble an monotone It ollos tat A - ests an ts elements are nonnegatve Hence rom eaton e ave E A T Also rom te teory o matrces e ave a S - ere a s element o matr A - Tereore a mn S B B ere B / mn S or some beteen an - > rom -9 an e ave e j a T j - j j an tereore or Secon Orer eto K e j j - 7 B ere K s neenent o
12 Parametrc Qntc Slne Solton Tereore t ollos tat E O or ort Orer eto ere K s neenent o Tereore t ollos tat E O or St Orer eto K e j j - 8 B K e j j - 9 B ere K s neenent o Tereore at te noal onts t ollos tat E O We smmare te above reslts n te ollong teorem: Teorem Te meto gven by along t te bonary contons o erent orers or solvng bonary vale roblem t r an scently small gves te ollong class o convergent soltons: E O s a secon orer convergent gven by 7 E O s a ort orer convergent gven by 8 E O s a st orer convergent gven by 9
13 Parametrc Qntc Slne Solton mercal Illstratons an Dscssons We conser nmercal eamles to llstrate comaratve erormance o or metos Te mamm absolte errors are tablate n tables - or erent vales o te arameters an an comare or reslts t te oter non estng metos All calclatons are mlemente by ATLAB7 Eamle Doolan et al8 " cos π π cosπ Te eact solton s gven by e / e / / e / cos π Snce > te bonary layer ests at bot ens Te mamm error at te noal onts ma s tablate n tables - Table : amm absolte errors or Secon orer meto 8 / 7E- 7E- 97E- 87E- 8E- / E- 7E- 7E- 8E- 7E- / E- 7E- 9E- 7E- 9E-
14 Parametrc Qntc Slne Solton Table : amm absolte errors or ort orer meto 8 / 879E-7 E-8 E-9 9E- 8E- / E-7 E-8 E-9 E- E- / 87E-7 E-8 9E-9 78E- 779E- Table : amm absolte errors or St orer meto at nteror noes 8 / 98E-8 E-9 9E- E- 7E- / E-7 7E-9 8E- 7E- E- / 9E-7 E-8 97E- E- 7E- Table : amm absolte errors or St orer meto 8 / 8E-9 78E- 9E- 7E- 79E- / E-9 E- 9E- 9E- 7E- / 79E-8 E- E- E- 7E-
15 Parametrc Qntc Slne Solton 7 Table : amm absolte errors 8 / E- 8 E- 7 E- 7 E- / 79 E- 8 E- E- 8 E- / E- E- E- E- Table : amm absolte errors 8 / E- E- E- 8 E- / 9 E- E- E- E- / 8 E- 8 E- E- E- Table 7: amm absolte errors 8 / 77 E- 7 E-7 9 E-8 9 E-9 / 8 E- E-7 E-8 99 E-9 / E- E-7 9 E-8 E-9
16 Parametrc Qntc Slne Solton 8 Table 8: amm absolte errors 8 / 77 E- E- E- 78 E- / E- E- 889 E- E- / E- E- E- 8 E- Table 9: amm absolte errors 8 / E- 8 E E-9 7 E- / E- 77 E-8 8 E-9 E- / E- E-7 E-8 89 E- Table : amm absolte errors 9 8 / 8 E-9 7 E- 7 E- 888 E- / 8 E-8 7 E- 9 E- 9 E- / 8 E-7 E-9 8 E- 97 E-
17 Parametrc Qntc Slne Solton 9 Eamle " e / e / Te eact solton s gven by e / e / Te mamm error at te noal onts ma s tablate n Tables - Table : amm absolte errors or Secon orer meto 8 / E- E- E- E- E- / 8 E- 87 E- 8 E- 7 E- 98 E- / 7 E- 97 E- 8 E- E- E- Table : amm absolte errors or ort orer meto 8 / 8 E-7 E-8 7 E- E- E- / E-7 7 E-8 97 E-9 E- 778 E- / E- 9 E-8 7 E-9 98 E- E-
18 Parametrc Qntc Slne Solton Table : amm absolte errors or St orer meto at nteror noes 8 / E-8 E- 7 E- 99 E- 9 E- / 9 E-7 7 E-9 E- 77 E- E- / 89 E-7 7 E-8 8 E- 7 E- E- Table : amm absolte errors or St orer meto 8 / 7 E- 78 E- 7 E- E- E- / E-9 9 E- 7 E- 87 E- E- / 7 E-8 8 E- 9 E- 88 E- E-
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