A CLASS OF SINGULAR PERTURBATED BILOCAL LINEAR PROBLEMS
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1 Proeegs of the Iteratoal Coferee o Theor a Applatos of Matheats a Iforats ICTAMI 3 Alba Iula A CLASS OF SINGULAR PERTURBATED BILOCAL LINEAR PROBLEMS b Mhaela Jaraat a Teoor Groşa Abstrat. Ths paper presets the algorths for solvg a bouar laer bloal sgular perturbate proble whe the bouar laer s stuate org. The solvg of ths proble has ver goo results whe approatve ethos are use. For ths proble a ufor frst orer aso was obtae. e wor: bouar laer ufor aso outer a er aso ahg. As t s kow fro lterature there are a ethos to etere the ufor soluto for the sgular perturbate probles attahe to sae bloal probles. The stu of ths k of sgular perturbate probles attahe to soe orar fferetal equato of hgh orer has goo results f we use approatve ethos. Thus for these probles we wll searh asptot solutos usg the ahg asptot aso ethos. Let s oser the followg bloal lear proble: (a) (b) for whh we obta a frst orer ufor aso oserg the bouar laer stuate org. We suppose that the bouar laer s stuate org. Thus the outer aso ust satsf the rght lt oto a the er aso ust satsf the left lt oto a f there s ol oe stt lt the the er a outer aso ust be ahe. Fro equato (a) we a see that the sall paraeter ultpl the seo orer ervatve thus the bouar laer posto epe b the oeffet a beause the value of epe b the lts a the posto of the bouar laer wll epe also b ths paraeters. The outer aso. We hoose the frst orer outer aso the for: 57
2 Proeegs of the Iteratoal Coferee o Theor a Applatos of Matheats a Iforats ICTAMI 3 Alba Iula 58 ) ( ) ( ) ( () a usg equato (a) a the rghts lt oto () after equalzato of the oeffets of we have: () (3) whh have the soluto: f f (4ab) To etere the er aso we troue the strethg trasforato : > equato (a) a we obta (5) Wth the stt lts of equato (5) are: a f () orrespog to () < a f ()
3 Proeegs of the Iteratoal Coferee o Theor a Applatos of Matheats a Iforats ICTAMI 3 Alba Iula orrespog to () a respetvel f > a () orrespog to (). We a see that the lts are ot stt a the bouar laer org o t est whe - a - - a > - or - a > -. I ase () the stt lt s the sae lke the orgal proble a o splfato was ae; thus for ase t s eessar to solve the orgal proble. I ase () the geeral soluto of the proble s: (6) for -. For < - the tegral o to ot est a for () () fte. Thus the seo ase o ot est bouar laer org. I the frst ase beause the er evelop ust satsf the lt oto org a for orrespo to () a we obta. (7) To obta we ah (4b) a (7). We have: ( ) ( ) (8a) (8b) Takg t we have t fro where we have: 59
4 Proeegs of the Iteratoal Coferee o Theor a Applatos of Matheats a Iforats ICTAMI 3 Alba Iula 6 [ ] t e t t Usg (8a) a (8b) we obta: [ ] (9a) a [ ] (9b) Ag (4b) a (9b) a subtratg (8a) we obta the ufor evelop: [ ] () the ase - < a -. For () the soluto of the proble s: I () where ~ ~ π π for e e I Mahg a we wll obta. Beause the er evelop ust satsf the lt oto org a beause orrespo to (). More beause
5 Proeegs of the Iteratoal Coferee o Theor a Applatos of Matheats a Iforats ICTAMI 3 Alba Iula 6 ~ z for z z we have ~ Thus posg the oto that () we obta Therefore () Beause ( ) a ( ) t s possble to ah the er evelop a the outer evelop. Ag (4b) a () we obta the ufor evelop: () the ase (-) > a -. Thus we etere a frst orer ufor aso oserg the bouar laer stuate org. Referees []Va Dke M. Perturbato ethos flu ehas. Aa. Press New York 964; the parabol Press Stafor Calfora 975. []Georgesu A. Aproaţ asptote Etura tehă Buureşt 989. [3]Gheorghu C.I. Curs e hroaă ş trasfer e ălură petru stu aprofuate Cluj Napoa 995 ltografat
6 Proeegs of the Iteratoal Coferee o Theor a Applatos of Matheats a Iforats ICTAMI 3 Alba Iula [4]Nafe A. H. Probles perturbato. Joh & Sos New York Chhester Brsbae Toroto Sgapore 985 Authors: Jaraat Mhaela Boga Voă Uverst Fault of Eoo Sees Cluj Napoa. Groşa Teoor Babeş-Bola Uverst Fault of Matheats a Iforats Cluj-Napoa. 6
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