MIXED BOUNDARY VALUE PROBLEM FOR A QUARTER-PLANE WITH A ROBIN CONDITION

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1 Jornal o Science, Ilamic Repblic o Iran 3: Naional Cener For Scieniic Reearch, ISSN 6-4 MIXED OUNDRY VLUE PROLEM FOR QURTER-PLNE WITH ROIN CONDITION ghili * Deparmen o Mahemaic, Facl o Science, Gilan Unieri, Rah, PO o 84, Ilamic Repblic o Iran brac We conier a mie bonar ale problem or a qarer-plane wih a Robin coniion on one ege We hae eelope wo procere, one bae on he aance heor o al inegral eqaion an he oher, in or opinion impler echniqe, reling on conormal mapping oh o he procere are o inere, becae he ormer ma be eaier o aap o oher bonar ale problem Kewor: Mie bonar ale problem; Dal inegral eqaion; Freholm inegral eqaion o econ kin Inrocion Problem Fin he olion o he ollowing mie bonar ale problem PDE: Φ, >, > C Φ, h Φ, q, > Φ,, < < 3 Φ,, > Noe Or raeg i o creae a eqence o racable bonar ale problem leaing o anar inegral eqaion which ma be ole nmericall Solion We ir in he olion o or ailiar problem b replacing he Robin coniion b Φ,, where i an amiible bonar ncion Φ, be he eire ncion, hen Φ, ha he ollowing inegral repreenaion: Φ, ep in ep co Φ, in * agili@cgacir 65

2 Vol 3 No Winer ghili J Sci I R Iran From Forier inerion-ormla or he ine ranorm in 3 From 3 we obain he ollowing eqaion: ep 4 The coniion on lea o he pair o al inegral eqaion co ep, <<, 5 co, >, 6 The eqaion 6 i aiie b [,3,4]: J g, where g i eermine o ai 5 Th 5 ma be rewrien a in, << 7 Following Sneon elemenar olion [4], we hae g, 8 Φ ep in,, 9 rom 3 an [] ln, Φ g Now we rn o he Robin coniion, which gie he eqaion ] ln [ g h q Frher impliicaion i poible We hae ] [ln ] [ g Th, i eermine b he Freholm inegral eqaion o he econ kin:, ln < < F h 66

3 J Sci I R Iran ghili Vol 3 No Winer where F q h ln ln Ping he piece ogeher, eermine g which in rn allow o compe b Φ, Noe In he aboe inegral eqaion we e he ollowing Inegral: a { }, b he inner inegral o righ ie can be wrien aer changing in ariable: co α in α ln an inα coα α Ne we how how he conormal mapping w in - z ma be gainll e o ole he problem w in - i z, i in implie ha, coα α in α Uing anoher change o ariable o he orm inα one ge, in coh, Φ, Ψ, co inh 67

4 Vol 3 No Winer ghili J Sci I R Iran The bonar coniion change o: or, ΔΨ, - Ψ, ec h hinh Ψ, qinh, - Ψ, ec h inh co, < <, 3- Ψ,, >, We ormlae an ailiar problem or Laplace eqaion ΔΨ, - Ψ,, >, - Ψ, inh co, < <, 3- Ψ,, > Since we hae a non-homogeno bonar coniion, Ψ, ha he ollowing inegral repreenaion: Ψ, n an ep[ n ]co n c inh co n an in co co n 5 From he aboe relaion one ge 4 a n in co co n n 6 Then Ψ, ν an, n Ψ, an ep[ n ] c inh co, 7 c coh co 8 We nee o epre Ψ, in erm o an Thereore rom 8 we ge c coh co 9 The aboe relaion iel: From bonar coniion an we obain c inh co, 3 n n a co n n inh co, 4 c inh co ν co coinh coh The inner inegral a righ ie can be wrien a 68

5 J Sci I R Iran ghili Vol 3 No Winer ollow []: anh [co co ] log[coh coh ] 4 4 in preio cae, he Robin coniion will gie a Freholm inegral eqaion o he econ kin wih a complicae kernel ha can be ole c inh co [log coh coh ], a n in co co n n ec h hinh Ψ, qinh wih, Ψ, ν n an ep[ n ] c inh co, Reerence ghili Docoral Dieraion, Sae Unieri o New York, Son rook, New York 999 Ereli, Magn W, Oberheinger F an Tricomi FG Table o Inegral Tranorm Vol I, McGraw Hill ook Compan Inc Melroe G Triple rigonomeric erie an heir applicaion o Mie onar ale problem PhD Thei, Ol Dominion Unieri, Norolk, Virginia Sneon Ian N The elemenar olion o Dal Inegral Eqaion Proceeing o he Glagow Mahemaical ociaion, Vol IV, par III, Janar 96 69