Logarithmic Sine and Cosine Transforms and Their Applications to Boundary-Value Problems Connected with Sectionally-Harmonic Functions

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pplied Mthemti 3 4 378-386 http://dx.doi.og/.436/m.3.458 Publihed Olie Febuy 3 (http://www.ip.

1 pplied Mthemti Publihed Olie Febuy 3 ( Logithmi Sie d Coie Tfom d Thei pplitio to Boudy-Vlue Poblem Coeted with Setiolly-Hmoi Futio Mitht Ideme Egieeig Fulty OKN Uiveity Itbul Tukey Emil: mideme@gmil.om eeived Septembe 3 ; evied Juy 8 3; epted Juy 5 3 BSTCT Let td fo the pol oodite i be give ott while u tifie the Lple equtio u i the wedge-hped domi D : j j o : j j j j D. Hee j j deote eti gle uh tht j j. It i kow tht if u tifie homogeeou boudy oditio o ll boudy lie j j dditio to o-homogeeou oe o the iul boudy the expliit expeio of i u i tem of eige-futio be foud though the lil method of eptio of vible. But whe the boudy oditio give o the iul boudy i homogeeou it i ot poible to defie diete et of eige-futio. I thi ppe oe how tht if the homogeeou oditio i quetio i of the Diihlet (o Neum) type the the logithmi ie tfom (o logithmi oie tfom) defied by f F i log d (o f Folog d ) my be effetive i olvig the poblem. The ivee of thee tfomtio e expeed though the me keel o o. Some popetie of thee tfom e lo give i fou theoem. illuttive exmple oeted with the het tfe i two-pt wedge domi how thei effetivee i gettig ext olutio. I the exmple i quetio the ltel boudie e umed to be o-odutig whih e expeed though Neum type boudy oditio. The pplitio of the method give lo the eey oditio fo the olvbility of the poblem (the ledy kow exitee oditio!). Thi kid of poblem ie i viou domi of pplitio uh eletotti mgeto-tti hydotti het tfe m tfe outi eltiity et. Keywod: Itegl Tfom; Hmoi Futio; Wedge Poblem; Boudy-Vlue Poblem; Logithmi Sie Tfom; Logithmi Coie Tfom. Itodutio Boudy-vlue poblem oeted with etiollyhmoi futio i wedge-hped domi d : j j D j D : j j j whee td fo the pol oodite i while > i give ott e impott fom both pue ietifi d egieeig poit of view. Figue epitomize imple e of D whih oepod to =. The ub-egio detemied by d model the egio filled with diffeet mteil hvig diffeet otitutive pmete. The field futio u tifie the bi equtio div gdu F () Copyight 3 Sie.

2 M. IDEMEN 379 L (u) = f () O y L(u) = f () L (u) = f () Figue. ompoite egio D filled with two diffeet mteil. i the ee of ditibutio ude the boudy oditio L u f L u f d Lu f how o the figue. Hee F td fo the deity of the exitig oue oetted o the itefe (if y) while Lu L ud L u e give lie (diffeetil) boudy opetio. The boudy oditio i quetio my lo ivolve eti tem epeetig the oue lolized o the boudy (if y). to the futio it h ott vlue d i the ub-egio i quetio. Thu o the itefe betwee the ub-egio two tmiio oditio of the followig fom e tified: u u (b) u u F. () i well-kow whe F f f oe defie et of othogol eige-futio whih pemit u to obti expliit expeio of u i tem of thee eige-futio. The oeffiiet i the eige-futio eie e detemied by uig the ohomogeeou boudy oditio give o the boudy = (i.e. though f ) togethe with the egulity oditio to be tted t =. Whe f t let oe of the futio F f d f mut be diffeet fom zeo i ode to hve o tivil olutio u. I thi e it i ot poible to defie et of diete eige-futio. To oveome uh kid of diffiulty i the midt of the lt etuy ome method whih e effetive whe the egio oit of D wee popoed. mog them we metio fo exmple the fiite Stum-Liouville tfom itodued by Eige [] d Chuhill [] d the fiite Melli tfom itodued by Nylo [34] (ee lo [5]). The fiite Stum-Liouville tfom e ot ppopite i the e oideed hee beue they e bed o the et of eige-futio whih ot be defied i the peet e. to the fiite Melli tfom they e defied follow (ee fo ex. [5 pp ]): x d () M f f d. (b) M f f Oe eily hek tht the fit o the eod tfom i ppopite to edue the Lple equtio witte i the iul pol oodite to odiy diffeetil equtio whe Diihlet o Neum type oditio e peibed epetively o the iul pt = of the boudy. The ivee tfom oit the of the lil Melli type itegl. The im of thi ote i to how tht the tfom of the fom d f Fi log d (3) f F olog d (3b) e effetive i gettig expliit expeio to the olutio of the poblem oeted with etiolly-hmoi futio defied i D d D metioed bove whe the boudy oditio o the = i homogeeou d of the Diihlet o Neum type. The impliity of thee tfom i tht thei ivee e lo give with the me keel (ee Setio B below). To lify the eetil popetie of the epeettio ((3) (b)) i wht follow we will oide without lo of geelity the e whee = (ee Figue d 3). To the bet of ou kowledge epeettio of the fom (3b) w fit oideed by Smythe (ee [6 pp. 7-7]) to fid the eletotti potetil due to lie oue loted pllel to dieleti wedge fo whih. Hi diuio i bed o moot phyil gumet d ome ptiul etitio. we will how lte o epeettio of the fom (3b) i ot uitble whe beue oly the dt kow i uffiiet to uiquely fo o L (u) = f () y L(u) = L (u) = f () O x Figue. wedge-hped egio ivolvig oly the igul poit =. D Copyight 3 Sie.

3 38 M. IDEMEN L(u) = y D D L (u) = f () L (u) = f () Figue 3. wedge-hped egio ivolvig oly the igul poit =. detemie F (i.e. the ivee tfom). Futhe- (3b f fo ll it give whih i ot om phyi poit of eptble f view. moe whe ) i ued to expe f f. Logithmi Sie d Coie Tfom Let > be give ott while F L i give futio. The oide the futio f d f defied though the oveget itegl tkle i (3) d (3b). Thee log td fo the pi- ig p ipl bh of the logithm futio. We will efe f d f to the logithmi ie tfom d logithmi oie tfom of F epetively. I wht follow we will lo deote th em by the ymbol SF d CF. Some iteetig d impott popetie of thee tfom e tted i the theoem give below. ) Limit Vlue fo d we will ee lte o (ee Setio B) the expeio of the futio f (o f ) kow oly i the itevl o i uffiiet to detemie the futio F uiquely. The futio f d f my be piee-wie otiuou i thee itevl. Tht me tht the limit vlue of the itegl tkig ple i (3) d (3b) ted to the ed poit o o my be diffeet fom the vlue obtied by eplig dietly o o i thoe itegl. Fom pplitio poit of view it i the limitig vlue tht e impott. Theefoe thee limit vlue mut be diued efully. The two theoem tht follow oe thi poit (fo thei poof ee ppedix). Theoem-. If F L the fom (3) oe get lim f f Theoem-. ) If oe get lim f f lim f f. F L x (4) the fom (3b) lim f f lim f f. b) If oe h lo F L lim. (4b) the f f (4) B) Ivee Tfom It i iteetig ft tht whe f (o f ) i kow fo ll o fo ll the the futio F be detemied ompletely. The theoem tht follow oe thi iveio poblem (fo thei poof ee ppedix). Notie tht whe f i piee-wie otiuou i wht follow f me f f. Theoem -3. ) Let f L be piee-wie otiuou i the itevl d f. The (3) yield F f i log d. (5) b) If f L i piee-wie otiuou i the itevl the (3b) yield F f o log d. (5b) Theoem-4. ) Let f L otiuou fo d f yield F be piee-wie. The (3) i log d. (6) f b) If f L the fom (3b) i piee-wie otiuou fo oe get F o log d. (6b) f The poof of thee theoem (exept Theoem ) eily be hieved by uig the ledy kow oe though imple tfomtio (See fo exmple [5] o [7]). Fo the ke of fluey of the ppe we pefe to potpoe the poof to the ppedix. I wht follow we will deote the ivee tfom give by (5) d (6) by S f. Similly the ivee tfom give by ( 5b) d (6b) will be deoted by C f. 3. pplitio to Boudy-Vlue Poblem Coeted with Setiolly-Hmoi Futio Whe oe h lo FL by ueive diffeetitio of (3) d (3b) oe get Copyight 3 Sie.

4 M. IDEMEN 38 d f F i log d (7) f Folog d (7b) whih how tht d f S f S (8) C f C f. Now oide futio i o j (8b) u whih i hmoi j j D : j : j j D d tifie homogeeou boudy oditio of the Diihlet o Neum type o the iul pt of the boudy mely: d o u u j j (9) u () u j j pplitio of the opeto d u. (b) S o C to (9) yield uˆ ˆ () d ((8) (b)) beig tke ito out. Hee uˆ td fo the logithmi ie o oie tf om of u. Fom () oe get j Bj j j j j uˆ ih oh () whee j d B j e the itegtio ott to be detemied though the boudy d tmiio oditio while j d j e two ott whih be hoe ppopitely to filitte the omputtio. They my lo be depedet o. Thu i the eto j j oe h the followig expeio o j u j ih Bj oh j i log d jih j u (3). (3b) Bj oh j o log d (3) i vlid fo the oditio () while (3b) i vlid fo the e of (b). 4. Illuttive pplitio To how the effetivee of the epeettio ((3) (b)) i wht follow we will give illuttive exmple whih oe the het odutio i two-pt ompoite egio how i Figue 4. poit oue of mout Q exit t the poit (b ) while the iul pt of the boudy i oted by iultig mteil. The phyil popetie of the ltel boudie (i.e. the boudy oditio o d ) will be defied lte o. Thu the field futio (tempetue) u tifie the followig field equtio ude the give boudy oditio: Q divgdu b b (4) u u f (5) u u f (5b) u u (5) u O. (5d) u/ = O y u/ = f () u/ = f () Q Q b Figue 4. wedge-hped egio with Neum type boudy oditio. x Copyight 3 Sie.

5 38 M. IDEMEN Hee h ott vlue d i the idited ub-egio. emk tht the poblem poed by (4)-(5d) h olutio if the followig eey oditio i tified by the boudy oditio : D o moe expliitly u d Q f d f dq (6) (6b) Thi i obtied by fit itegtig (4) o D the pplyig the Gee theoem. I wht follow we will ume tht (6b) i tified (it will be ued lte o!). I ode with the defiitio of d Q let u wite u u u. (7) Sie the field Equtio (4) i equivlet to the equtio d u D (8) u u (8b) u u bq b d oe wite (8) u oh Boh olog d (9 ) u Coh Doh olog d. (9b) d The oeffiiet B C D e detemied though the oditio (5) (5b) (8b) d (8) follow: C B ih ih oh ih Q o log b ih () (b) D Hee we put oh log b ih ih Q o f olog d f olog d. () () If we fit iet ()-() ito ((9) (9b)) d the ue the Eule fomul to wite the o futio though expoetil futio the we get d u u oh i oh e d ih ih oh Q oh b i o log e d ih oh i oh e d ih ih oh Q oh b i o log e d ih () (b) Copyight 3 Sie.

6 M. IDEMEN 383 whee i defied with log. () Now it i impott to obeve tht the poit whih i loted o the itegtio lie eem to be double pole of the itegd i () d (b). But beue of the eltio (6b) thee pole e emovble. Ideed the emovbility of thee pole equie the oditio Q (3) whih i equivlet to (6b) () beig tke ito out. Sie the expeio tkig ple i the bket i () d (b) e eve futio of (3) gutee the emovbility of the igulity t =. Thu o the bi of Jod lemm the itegl i (() (b)) be omputed though the eidue t the pole loted i the uppe J o lowe J hlf-ple. The eidue eie omig fom the pole whih ou t the zeo of ih() d oh() e oeted with the geomety of the wedge i quetio d hee oit of the eige-futio eie. But the tem omig fom the pole of d (if y!) e idepedet of the geomety of th e wedge d hve o oetio with the eige-futio. emk. It i wothwhile to otie hee tht the ovegee of the ivee tfom itegl i () d (b) equie the eltio (3). Tht me tht the el- tfomtio i quetio doe ot oly pemit u to tio tted by (3) i i ft oditio fo the exitee of the olutio. Oe eily hek tht thi i othig but (6b) (o (6)). Thi how tht the pplitio of the obti expliit expeio of the olutio but the how lo the eey oditio fo the exitee of olutio. 4.. Ptiul Ce. Poit Sik Loted o the Ltel Boudie ume moe ptiully tht f M f M whee (4) d M td fo two give ott uh tht (Cf. (3) o (6b)) M Q. (4b) I thi e the ltel boudie oit of iultig mteil d y ik t the poit d. By tightfowd omputtio oe get M o log (4) whih edue () to u i Q oh ih Q oh ih i log log e +e d log i log i b e +e d. (5) Sie i the peet e oe lwy h b whih yield oh exp exp ih exp oh exp exp ih exp i o i o by the Jod lemm the fit pt of thee itegl be omputed by oideig the eidue of the pole tkig ple i the uppe hlf-ple Jh. But depedig o the eltive poitio of d oe get o <. Theefoe the eod pt of the fit itegl ivolve the otibutio of the pole exitig i the hlf-ple Jh o Jh (imil itutio i lo vlid fo the eod pt of the eod itegl). Thu we hve to oide the followig fou e ep tely: Ce : b e : b Ce 3: b e 4: b. Copyight 3 Sie.

7 384 M. IDEMEN By tightfowd omputtio we get the followig eult: b Q u o Q b o b b Q u o b Q o b b Q u o b Q o b b Q u o b Q o b (6) (6b) (6) (6d) By ompig (b) with () oe obeve tht u u whee. It i lo iteetig to ompe ( 6)-(6d) with the eult petiet to the e of. Thu o e olude tht the peet two-pt wedge poblem i equivlet to the homogeewith otitutive pmete ou wedge poblem. 5. Coluio d Coludig emk Fom the lyi mde bove oe olude tht the logithmi tfomtio defied by (3) d (3b) my be ppopite i gettig expliit expeio of etiolly-hmoi futio whih tify the homogeeou Diihlet o Neum boudy oditio o the iul pt of the boudy of wedge hped domi. Thi kid of poblem ie i viou domi of ppli- tio uh eletotti mgeto-tti hydotti het tfe m tfe outi eltiity et. EFEENCES []. C. Eige The Fiite Stum-Liouville Tfom Qutely Joul of Mthemti Vol. 5 No. 954 pp. -9. doi:.93/qmth/5.. []. V. Chuhill Geelized Fiite Fouie Coie Tfom Mihig Mthemtil Joul Vol. 3 No. 955 pp doi:.37/mmj/3754 [3] D. Nylo O Melli Type Itegl Tfom Joul of Mthemti d Mehi Vol. No. 963 pp [4] D. Nylo O Itegl Tfom of the Melli Type Joul of Egieeig Mthemti Vol. 4 No. 98 pp doi:.7/bf3769 [5] I. N. Seddo The Ue of Itegl Tfom MGw- Hill Co. New Yok 97. [6] W.. Smythe Stti d Dymi Eletiity MGw- Hill Co. New Yok 95. [7] E. C. Tithmh Itodutio to the Theoy of Fouie Itegl Oxfod Uiveity Pe 948. Copyight 3 Sie.

8 M. IDEMEN 385 ppedix. Poof of Theoem. Poof fo Theoem- f i quite obviou. To fid the limit of f oide fit the e whe d fo ume tht bitily fixed (mll) i give. Sie F L we fid uh tht Thu fom (3) oe get F d. (7) 4 f f F i log i log d F ilog ilog d whih yield f f log F i d. Now by tkig ito out the ue ). d (7b) iequlitie (ee Fig- log log i x x x we hooe o mll tht F i log d F d F d. (7) (7d) Fom (7b) d (7d) we olude tht fo evey howeve it i mll we fid uh tht whih pemit u to wite f f f f. Fo thi give the fit equti o i (4). To pove the me equlity fo the e of we hooe bity umbe d epet the gumet mde bove by eplig by. ll the lie exept (7b) d ( 7) emi uhged while the ltte beome ow follow: d - O y y = x y = log(+x) y = log(- x) y = - x y = - mx m> Figue. Logithm futio. f f F ilog d log log m. Hee m x (8) (8b) td fo uitble umbe whih doe ot deped o (Fo detil ee Figue ). Thu by hooig uffiietly mll we gutee whih yield d m F d (8) f f f f. Fo = the ltte edue to the fit equlity i (4) whe. To ee the limit of f oide bitily give (mll) d hooe uh tht the eod pt i f F i log d F i log d meet the followig iequlity: (8d) (9) Fi log d F d. (9b) Copyight 3 Sie.

9 386 M. IDEMEN fte hvig fixed let u mke. By vitue of the well-kow iem-lebegue lemm [5 p. 3] the fit pt i (9) ted to zeo whe log. Theefoe fo uffiietly mll oe h lo F i log d. (9) Fom (9)-(9) oe olude tht fo uffiietly mll oe h f fo evey. Thi pove the eod equlity i (4). (9)-(9) e lo vlid fo whih how the lt equlity i (4). Poof fo Theoem- The equlitie give i (4b) be how by epetig the eoig mde i povig Theoem-. to the equlity give i (4) owig to the umptio d FL f Fi log d (3) it i edued to the fit equlity i (4). 3. Poof of Theoem-3 d 4 Poof of thee theoem e bed o the followig wellkow lemm (ee [5 p. 34]). Lemm (Fouie itegl theoem). If f t i piee-wie otiuou d bolutely itegble i x oe h the fo ll d o f t x t dt (3) f x f x. Let u iet (5) ito the ight-hd ide of (3) d mke the ubtitutio whih yield d log log (3) d e e d F i log d f e i di d. (3b) (33) Now let u defie the odd futio e L f follow (otie tht f e f ): f e f e. The (33) lo be witte o F i log d f e i di d F i log d d f e o d. Sie the futio f e metioed i the lemm fom get (33b) (33) (33d) meet the equiemet the lt expeio oe F i log d e e f f f f. (34) Thi pove (5). To pove (5b) oe tt fom (3b) d epet (3)- (34) with the oly exeptio tht (33b) i epled ow by the eve futio f e f f Poof of theoem-4 i quite imil to tht of Theoem-3. The oly diffeee i tht d defied i (3) e epled ow by d e e. log log. Copyight 3 Sie.

Generalized Functions in Minkowski Space

Generalized Functions in Minkowski Space Geelize Ftio i Miowi Spe Chiw Ch Agt t, Mthemti Deptmet, The Uiveit of Aizo Cl DeVito Mthemti Deptmet, The Uiveit of Aizo PDF ete with FiePit pffto til veio http://www.fiepit.om . Itoio Peioi ftio e i