On a Volterra equation of the second kind with incompressible kernel

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1 Jenliyev e l. Advnces in Difference Equions 25 25:7 DOI.86/s R E S E A R C H Open Access On Volerr equion of he second kind wih incompressible kernel Muvshrkhn Jenliyev * Meirmkul Amngliyev MinzilyKosmkov 2 nd Mur Rmznov 3 * Correspondence: muvshrkhn@gmil.com Insiue of Mhemics nd Mhemicl Modeling Pushkin 25 Almy 5 Kzkhsn Full lis of uhor informion is vilble he end of he ricle Absrc Solving he boundry vlue problems of he he equion in noncylindricl domins degenering he iniil momen leds o he necessiy of reserch of he singulr Volerr inegrl equions of he second kind when he norm of he inegrl operor is equl o. The pper dels wih he singulr Volerr inegrl equion of he second kind o which by virue of he incompressibiliy of he kernel he clssicl mehod of successive pproximions is no pplicble. I is shown h he corresponding homogeneous equion when λ > hs coninuous specrum nd he mulipliciy of he chrcerisic numbers increses depending on he growh of he modulus of he specrl prmeer λ. By he Crlemn-Veku regulrizion mehod Veku in Generlized Anlyic Funcions 988 he iniil equion is reduced o he Abel equion. The eigenfuncions of he equion re found licily. Similr inegrl equions lso rise in he sudy of specrl-loded he equions Amngliyev e l. in Differ. Equ. 472: MSC: Primry 45D5; 45C5; secondry 45E Keywords: Volerr inegrl equion; Abel equion; specrum; nonrivil soluion; chrcerisic equion; Crlemn-Veku regulrizion mehod Inroducion Invesigion of boundry vlue problems for he he equion in noncylindricl domins hs wide prcicl pplicion [ 3]. For exmple in he sudy of herml regimes of he vrious elecricl concs here is he necessiy o sudy he processes of he nd mss rnsfer king plce beween he elecrodes. Afer chieving he meling emperure he conc surfce of elecrodes here is liquid mel bridge beween hese elecrodes. When he concs open his bridge is divided ino wo prs i.e. he conc meril is rnsferred from one elecrode o noher nd his leds o he bridging erosion. Ulimely he smooh surfce of concs is desroyed which mens h heir proper operion is violed. The mhemicl descripion of he herml processes which go wih he bridging erosion leds o solving he boundry vlue problems for he he equion in domins wih moving boundry nmely in he domins which degenere ino poin he iniil momen. Using he pprus of he poenils solving he problems under considerion is reduced o he sudy of singulr Volerr inegrl equions of he second kind when he norm of he inegrl operor is equl o. A feure of hese equions is he incompressibiliy of he kernel nd his is ressed in he fc h he corresponding nonhomogeneous equion cnno be solved by clssicl mehods. 25 Jenliyev e l.; licensee Springer. This is n Open Access ricle disribued under he erms of he Creive Commons Aribuion License hp://creivecommons.org/licenses/by/4. which permis unresriced use disribuion nd reproducion in ny medium provided he originl work is properly credied.

2 Jenliyevel. Advnces in Difference Equions 25 25:7 Pge 2 of 4 For he problem of he solvbiliy of he Volerr inegrl equion of he second kind wih specil kernel sed in Secion 2 fer some rnsformion in Secion 3 we obin he corresponding chrcerisic inegrl equion. An imporn momen of our reserch is fc h using Crlemn-Vekuregulrizion mehod [4] we reduce he iniil problem o solving he Abel inegrl equion of he second kind. The soluion of he ls equion provides finding ll soluions of he iniil inegrl equion from Secion 2. These resuls re sed in Secions 4-6. The min resul bou solvbiliy of he inegrl equion in clss of essenilly bounded funcions is formuled in he form of he heorem in Secion 6. 2 Semen of he problem When solving model problems for prbolic equions in domins wih moving boundry he singulr inegrl equions of he following form rise: ϕ λ K τ= K τϕτ dτ = f > + τ π τ 3/2 + τ2 τ + τ. τ /2 The kernel K τ hs he following properies: K τ nd coninuously <τ < <+ ; 2 lim K τ dτ = ε >; 3 lim K τ dτ = lim + K τ dτ =. To verify propery 3 we mke he subsiuion x = τ.wehve K τ dτ = 2 2 π π 2 2 = erfc x2 3 d + erf x + x x 2 d x + x. 2 From 2 he vlidiy of propery 3 direcly follows. Moreover i lso follows h he norm of he inegrl operor in cing in he clss of essenilly bounded funcions is equl o. Also he kernel K τ is summble wih weigh funcion /2. Indeed K τ dτ = 2/2 τ π + π + π =I +I 2 +I 3. 2 x 2 x 2 x 2 x2 x 2 x + x x x dx 2 dx 2 dx

3 Jenliyevel. Advnces in Difference Equions 25 25:7 Pge 3 of 4 For he firs inegrl fer inroducing he replcemen y =/x we hve he esime I 2 / 2 π + /2 2 y 2 dy. 2 y 2 For smll vlues he ls inegrl is bounded. For lrge vlues wehvehefollowing esime: 2 / 2 π + 2 y 2 dy = cons. 2 y 2 Thus we hve esblished he boundedness of he firs inegrl I. We esime he inegrls I 2 ndi 3 using he following inegrl: π dx = x 2 π dy π = y y. Problem To find he soluion ϕ ofinegrlequion sisfying he condiion ϕ L for ny given funcion f L ndechgivencomplexspecrl prmeer λ C. We noe h he inegrl equions of he form rise in he sudy of boundry vlue problems of he conducion in n infinie ngulr domin which degeneres he iniil momen. Such equions re clled by us Volerr inegrl equions wih incompressible kernel. The feure of he equion in quesion consiss in propery 3 of he kernel K τ nd is ressed in he fc h he corresponding nonhomogeneous equion cnno be solved by he mehod of successive pproximions for λ >.Obviouslyif λ < hen hs unique soluion which cn be found by he mehod of successive pproximions. The cse λ = ws considered in [5] i is shown h hs only one nonrivil soluion f wihin consn fcor. Furher in his pper we ssume h λ >. The equions of he form werefirsconsideredbysnkhrin:hesympoicsof inegrls of he double lyer poenils were sudied nd pproxime soluions of some pplied problems were consruced [6 7]. Subsequenly such inegrl equions were he subjec of invesigion by mny uhors. I should be noed h he boundry vlue problems for specrlly loded prbolic equions lso re reduced o he singulr inegrl equions under considerion when helodlinemovesbyhelwx = α[8 9]. 3 Trnsforming he inegrl equion We will use Crlemn-Veku regulrizion mehod [4]. To do his we rnsform. By mens of he relions + τ =2 τ + τ 2 τ = τ 2 τ + τ 4 2

4 Jenliyevel. Advnces in Difference Equions 25 25:7 Pge 4 of 4 we reduce oheform ϕ 2 π τ + τ /2 3/2 τ τ τ 2 τ τ /2 2 τ ϕτ dτ = f. 3 From [] p.83 i follows h i suffices o find soluion o he simplified equion ϕ λ k τ ϕτ dτ = f 4 ϕ= / ϕ f = / f k τ= 2 τ τ + π τ 3/2 2 τ τ /2 2 τ / ϕ L /4 2 f L 5 τ/ k τ L. To invesige he full equion 4 we will exrc is chrcerisic pr nmely ϕ λ k τ= k τ ϕτ dτ = f 6 π τ k h τ= π τ /2 f = f +λ 3/2 τ 2 τ τ 2 τ k h τ ϕτ dτ. 7 Equion 6 is chrcerisic for4 since lim k τ dτ =; lim k h τ dτ =. 4 Solving he chrcerisic inegrl equion Considering he righ side of 6 s known we find is soluion i.e. he soluion of he chrcerisic equion 6. Anlogously [] p.74 inegrl equion 6 isreducedo n equion wih difference kernel.todohiswemkeinireplcemens: = y ; τ = x ; ψy= y ϕ y ; f 2 y= f. 8 y y

5 Jenliyevel. Advnces in Difference Equions 25 25:7 Pge 5 of 4 Then we obin he equion of he form ψy λ y πx y 3/2 2 x y ψx dx = f 2 y y > 9 / y ψy L / y f 2 y L. The soluion of 9 cn be found eiher by he operionl mehod or by is reducion o Riemnn boundry vlue problem [ ]. If we denoe L[ψy] = ψp s he Lplce rnsformion of he funcion ψy hen he following formul holds for he convoluion: As [ ] L Ky xψx dx = K pψp y K p= [ b L 2 π3/2 K p d. b2 4 ] = b p b =cons hen by virue of 9isrnsformedo ψp λ 2 p = f 2 p. 2 The corresponding homogeneous equion hs he form ψp λ 2 p =. 3 In he cse when λ 2 p = 4 he nonzero soluions of 3re ψ k p=c k δp p k δx is he del-funcion C k =consndp k k =± ±2... re roos of 4. Applying o he ls equliy he inverse Lplce rnsformion we obin ψy= 2πi σ +i σ +i δp p k py dp = p k y

6 Jenliyevel. Advnces in Difference Equions 25 25:7 Pge 6 of 4 he inegrl is ken long ny srigh line Re p = σ nd undersood in he sense of he principl vlue. Therefore if p = p k re roos of 4 hen he eigenfuncions of 9 willhveheform [] ψ k y=c k p k y C k = cons. 5 We shll find he roos of 4. When λ wehve 2 p = λ []. Tking he logrihm we obin 2 p = ln λ + irg λ +2kπ; k =2... p k = 2 ln 2 λ rg λ +2kπ 2 + i ln λ 2 rg λ +2kπ. 6 For he boundedness of funcions 5 infiniy i is necessry h Re p k i.e. ln 2 λ rg λ +2kπ 2 or ln λ rg λ +2kπ ln λ. Hence N k N 2 [ ] ln λ + rg λ N = N 2 = 2π [ ln λ rg λ 2π N + N 2 + is number of eigenfuncions of 5nd[] is he ineger pr of.obviously he lrger λ he greer he mulipliciy of he eigenfuncions. Thereby λ λ wehve ψ hom y= k= N C k p k y. Using he replcemens which re inverse o 8 we obin he soluion of he homogeneous equion 6 ϕ hom = k= N C k Re p k byvirueof6. We noe h if λ =henp =. This cse is considered in deil in [5 2]. We rewrie he nonhomogeneous operor equion in he form ψp=f 2 p+ λ 2 p λ 2 f p 2 p Re p. Inroducing he noion 2 r λ p= p λ 2 p ]

7 Jenliyevel. Advnces in Difference Equions 25 25:7 Pge 7 of 4 we will find he originl of his imge: r λ y= i 2πi i 2 p λ 2 dp r λ y if y >. p In he ls inegrl we hve crried ou he inegrion long he conour voiding he poins p k deerminedby6 on he lef. The inegrl is undersood in he sense of he Cuchy principl vlue. Since we consider y we close on he righ cuing he hlfplne sli is long he posiive rel semixis. The zeros of he denominor of he funcion 2 Ap= p λ 2 p re numbers p k k =± ±2... which we need o circumven wice in opposie direcions. Therefore ccording o [] pp we hve r λ y= res Ap= p=p n π y 3/2 n= n= n λ n2. n 2 y Thus he soluion of he nonhomogeneous equion 9hsheform[] pp.86-87: ψy=f 2 y+λ y r λ y xf 2 x dx + k= N C k e y C k cons 7 he resolven r λ yisdefinedbove. Performing he reverse replcemens o 8 ino7 we obin he soluion of he nonhomogeneous equion 6 ϕ=f +λ r τf τ dτ + k= N C k e p k 8 r τ= π τ 3/2 n= n λ n 2 τ. 9 n 2 τ 5 Reducing he inegrl equion 4 o Abelequion We shll now ge o solving 4 i.e. he simplified vrin of he iniil equion. Using heformulforhesoluionofhechrcerisicequion 8 king ino ccoun 7forhefuncionf we obin ϕ= f +λ f τ+λ + k= N C k π τ τ πτ τ. ϕτ dτ + λ τ 2 τ ττ 2 τ τ r τ ϕτ dτ dτ

8 Jenliyevel. Advnces in Difference Equions 25 25:7 Pge 8 of 4 Chnging he order of inegrion in he righ-hnd side of his equion nd inerchnging he roles of τ nd τ wehve ϕ=λ τ π τ 2 τ + λ r τ τ τ τ dτ πτ τ 2 ϕτ dτ τ τ + f +λ r τ f τ dτ + C k. 2 k= N We compue he inner inegrl in 2 J τ; λ= r τ τ πτ τ = 2 π n= τ τ τ 2 τ τ n [ = I 2 π λ n n n= n λ n τ 3/2 τ τ dτ I n τ= τ 3/2 τ τ n 2 τ τ I n 2 τ= τ τ 3/2 τ τ τ τ 2 τ τ dτ n 2 τ 2 τ τ I2 n τ] 2 2 τ n 2 τ 2 τ dτ τ τ dτ 2. τ τ Using he subsiuion z = τ τ/ τ we compue he inegrls I n τndi n 2 τ. We will hve I n 2 τ= τ n2 τ 2 τ = π n τ n2 τ I n 2 2 τ= τ = π n τ 2 τ n2 +τ 2 τ n +2 τ. 2 τ n2 2 z 2 dz 2 τ n2 2 z 2 2 τ τ 2 dz 2 τz 2 When clculing he inegrl I n 2 τ he following formul ws used [3] p.32 formul 3.325: μx 2 ηx π = 2 2 μ 2 μη μ >η >.

9 Jenliyevel. Advnces in Difference Equions 25 25:7 Pge 9 of 4 Thus for he difference I n τ I n 2 τweobin I n τ I2 n τ= π n n2 τ n +2 τ. τ 2 τ 2 τ Subsiuing he ls ression ino 2 we obin J τ; λ= n2 τ π τ λ n 2 τ n= = λ π τ Then 2cnberewriens ϕ=λ π τ τ 2 τ + k= N C k τ 2 τ. τ 2 τ ϕτ dτ + f +λ. Finlly fer inroducing he noion f 2 = f +λ r τ isdefined by9 we obin ϕ λ π n +2 τ 2 τ + π τ r τ f τ dτ r τ f τ dτ 22 ϕτ τ dτ = f 2 + k= N C k 23 he soluion nd he righ side of he inegrl equion 23belongoclsses5. Thus he iniil simplified inegrl equion 4 is reduced o 23 h is n Abel inegrl equion of he second kind. 6 Soluion of Abel inegrl equion. The min resul According o [] p.7 soluion of he Abel equion of he second kind x y yx+μ d = gx x hs he form x yx=gx+πμ 2 [ πμ 2 x ] G d 24 x g Gx=gx μ d. x

10 Jenliyevel.Advnces in Difference Equions 25 25:7 Pge of 4 We find he soluion of he Abel equion 23 for f 2 =hiswewillfindsoluion of he corresponding homogeneous equion 4 forechk N k N 2 eigenfuncions. Under his condiion 23forechk N k N 2 hsheform ϕ k λ ϕ k τ dτ = π τ The soluion of his equion cn be wrien s see 24 ϕ k =G k + λ2 λ 2 τ G k τ dτ G k = = + λ π + λ π erfc. p k τ τ dτ. In clculing he ls inegrl we use he formuls [3] p.367 formul ; p.25 formul Indeed ccording o hese formuls we obin λ π W /4/4 p k τ τ dτ = λ = π /4 /4 2 erfc W /4/4 2 W αβ z is he Whiker funcion. The funcion G k isboundedfor [; + + nd G k =. Thus he eigenfuncions of 4hve he form or ϕ k = + λ2 + λ π λ 2 τ erfc τ We rewrie he previous funcion in he form ϕ k = + λ π erfc + λ2 λ 2 4 λ τ + p k 2 τ + λ π erfc τ 4 τ 2 ϕ k = + λ2 + λ π λ 2 erfc τ + λ π erfc dτ. τ τ dτ dτ I k ; λ+ λ π I 2k; λ 25

11 Jenliyevel.Advnces in Difference Equions 25 25:7 Pge of 4 I k ; λ= I 2k ; λ= 4 τ + p k 2 τ erfc τ τ dτ τ dτ. Afer he replcemen z = τ he inegrl I k ; λcnbewriens I k ; λ=2 z2 + p k dz. z 2 We compue he inegrl I 2k ; λ inegring by prs: u = erfc p k τ ; dv = τ dτ du = p k πτ 3/2 p k τ dτ; v = 42 τ. 26 λ 2 Then using 26 we hve I 2k ; λ= 42 λ 2 λ2 4 erfc p k λ 2 π Afer replcing z = τ we obin I 2k ; λ= 42 λ 2 4 τ + p k dτ. 2 τ τ 3/2 λ2 4 erfc p k λ 2 π z2 + p k z 2 z dz. 2 Afer subsiuing he ressions obined for I k ; λndi 2k ; λino25wehve ϕ k = + λ2 + λ π + 82 p k λ 2 π + λ π λ 2 2 [ 42 λ 2 erfc z2 + p k z 2 4 erfc 2 z2 + p k z 2 Afer some simple rnsformions we obin ϕ k = + λ2 λ 2 2 z2 + p k z 2 dz ] z dz. 2 + p k λz 2 dz

12 Jenliyevel.Advnces in Difference Equions 25 25:7 Pge 2 of 4 = + λ λ 2 z2 + p k λ z 2 + z 2 = λ λ 2 Afer he inroducion of he replcemen ξ = p k z ϕ k = = IRP = lim p k λ + s [3] p.89 formul λ λ 2 4 λ 2 + λ π λ 2 4 λ 2 dz z λ2 2 z2 λ z we obin IRP p k λ erfc λ erfc def = 2 IRP π ξ 2 dξ p k λ d λ. z z ξ 2 dξ λ C is infiniely remoe poin nd he ression denoed is n inegrl over n open ended conour from he sring poin p k λ remoe poin IRP. Thus he funcion ϕ k = + λ π λ 2 erfc λ o he infiniely is n eigenfuncion of he simplified equion 4forechk; N k N 2 [ ] ln λ + rg λ N = N 2 = 2π nd [] is he ineger pr of. Then he funcion ϕ= [ ln λ rg λ 2π ] 27 k= N C k ϕ k 28 is soluion of he Abel equion 23 for f 2 = h is soluion of he simplified homogeneous equion 4 nd he funcions ϕ k ndvluesp k re deermined by 27 nd 6 respecively. We noe h fer muliplying equliy 28by / we obin he soluion of he homogeneous equion corresponding o he originl equion : ϕ= k= N C k + λ π λ 2 4 λ 2 erfc λ. 29

13 Jenliyevel.Advnces in Difference Equions 25 25:7 Pge 3 of 4 The funcion ϕ belongs o he spce L.Indeedwehveforhefirserms of he sum 29 L. For he second erms in he sum 29 he following inclusions re lso vlid: λ π λ 2 4 λ 2 erfc λ L. Here i is sufficien o ke ino ccoun h he numbers p k k [ N N 2 ] re he roos of 4 for ech fixed complex specrl prmeer λ C nd o use he sympoic form of he funcion erfcz for lrge vlues of z [3] p.89 formul ;[4] p.758. Obviously here is limi relion z = p k λ Thus he following heorem holds. IRP nd for ech λ >. Theorem The nonhomogeneous inegrl equion is solvble in he clss ϕ L for ny righ-hnd side f L ; nd for ech λ >.The corresponding homogeneous equion hs N + N 2 +eigenfuncions ϕ k = + λ π λ 2 4 λ 2 erfc λ nd he generl soluion of inegrl equion cn be wrien s ϕ=f+ λ2 F= f 2 λ 2 τ λ f 2 τ dτ π τ F dτ + k= N C k ϕ k nd he funcion / f 2 L is defined by Conclusion We sudied he problems of resolvbiliy of singulr Volerr inegrl equions of he second kind in he spce of essenilly bounded funcions. I is proved h λ >hehomogeneous equion which corresponds o hs coninuous specrum nd he mulipliciy of he chrcerisic numbers increses depending on he growh of he modulus of he specrl prmeer λ. The iniil equion is reducedo he Abel inegrl equion 23 by he regulrizion mehod of Crlemn-Veku [4] which ws developed for

14 Jenliyevel.Advnces in Difference Equions 25 25:7 Pge 4 of 4 solving singulr inegrl equions. The eigenfuncions of re found licily nd heir mulipliciy depending on he modulus of he chrcerisic number λ is found. Compeing ineress The uhors declre h hey hve no compeing ineress. Auhors conribuions All uhors conribued eqully o he wriing of his pper. All uhors red nd pproved he finl mnuscrip. Auhor deils Insiue of Mhemics nd Mhemicl Modeling Pushkin 25 Almy 5 Kzkhsn. 2 Al-Frbi Kzkh Nionl Universiy Al-Frbi 7 Almy 54 Kzkhsn. 3 E.A. Bukeov Krgnd Se Universiy Universiesky 28 Krgnd 28 Kzkhsn. Acknowledgemens This sudy ws finncilly suppored by Commiee of Science of he Minisry of Educion nd Sciences Grn 2 RK 69/GF on prioriy Inellecul poenil of he counry. Received: 8 November 24 Acceped: 7 Februry 25 References. Kim EI Omel chenko VT Khrin SN: Mhemicl Models of Therml Processes in Elecricl Concs in Russin [Memicheskie modeli eplovyh processov v elekricheskih konkh]. Akdemy nuk Kzkh SSR Almy Krshov EM: Mehod of Green funcions for solving he boundry vlue problems of prbolic equion in noncylindricl domins in Russin [Meod funcsii Grin pri reshenii krevyh zdch dly urvnenii prbolicheskogo ip v necilindricheskih oblsyh]. Dokl. Mh. [Dokldy Akdemii Nuk SSSR] Orynbsrov MO: On solvbiliy of boundry vlue problems for prbolic nd poliprbolic equions in noncylindricl domin wih nonsmooh lerl boundries in Russin [O rzreshimosi krevyh zdch dly prbolicheskogo i poliprbolicheskogo urvnenii v necilindricheskoi oblsi s negldkimi bokovymi grnismi]. Differ. Equ. [Differeilniye urvneniy] Veku IN: Generlized Anlyic Funcions in Russin [Obobshhennye nliicheskie funkcii]. Glvny redksiy fiz.-m.li. Moscow Akhmnov DM Kosmkov MT Rmznov MI Tuimebyev AE: On he soluions of he homogeneous muully conjuged Volerr inegrl equions. Bull. Univ. Krgnd ser. Mh. in Russin [Vesnik Krgndinskogo universie. Ser. Memik] Khrin SN: Therml Processes in he Elecricl Concs nd Reled Singulr Inegrl Equions in Russin [Teplovye processy v jelekricheskih konkh i svjznnyh singuljrnyh inegrlnyh urvnenij]. Disserion for he degree of c.ph.-m.sc...2 Insiue of Mhemics nd Mechnics AS Almy Khrin SN: The nlyicl soluion of he wo-phse Sefn problem wih boundry flux condiion. Mh. J. in Russin [Memicheskii zhurnl] Amngliyev MM Akhmnov DM Jenliyev MT Rmznov MI: Boundry vlue problems for specrlly loded he operor wih lod line pproching he ime xis zero or infiniy. Differ. Equ. in Russin [Differeilniye urvneniy] Akhmnov DM Jenliyev MT Rmznov MI: On priculr second kind Volerr inegrl equion wih specrl prmeer in Russin [Ob osobom inegrlnom urvnenii Volerr vorogo rod so spekrlnym prmerom]. Sib. Mh. J. [Sib. m. zhurnl] Polynin AD Mnzhirov AV: Hndbook on Inegrl Equions in Russin [Sprvochnik po inegrl nym urvnenijm]. FIZMATLIT Moscow 23. Jenliyev MT Rmznov MI: The Loded Equions s Perurbions of Differenil Equions in Russin [Ngruzhennye urvnenij kk vozmushhenij differencilnyh urvnenij]. Gylym Almy 2 2. Akhmnov DM Jenliyev MT Kosmkov MT Rmznov MI: On singulr inegrl equion of Volerr nd is djoin one. Bull. Univ. Krgnd ser. Mh. in Russin [Vesnik Krgndinskogo universie. Ser. Memik] Grdshejn IS Ryzhik IM: Tble of Inegrls Series nd Producs 7h edn. Acdemic Press New York Tikhonov AN Smrskii AA: Equions of he Mhemicl Physics in Russin [Urvnenij memicheskoj fiziki]. MGU Moscow 999