A UNIFIED APPROACH TO SINGULAR PROBLEMS ARISING IN THE MEMBRANE THEORY*

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1 55(2) APPLICATIONS OF MATHEMATICS No., A UNIFIED APPROACH TO SINGULAR PROBLEMS ARISING IN THE MEMBRANE THEORY* Irna Rachůnková, Olomouc, Grno Pulvrr, Win, Ewa B. Winmüllr, Win (Rcivd January 4, 28, in rvid vrion Augu 9, 28) Abrac. W conidr h ingular boundary valu problm ( n u ()) + n f(, u())=, + n u ()=, a u()+a u ( )=A, whr f(, x)iagivnconinuoufunciondfindonh(,] (, )whichcanhav aimingulariya =andapacingulariya x=. Morovr, n Æ, n 2,and a, a, Aarralconanuchha a (, ),whra a, A [, ). Thmainaim ofhipapriodicuhxincofoluionohabovproblmandapplyh gnral rul o covr crain cla of ingular problm ariing in h hory of hallow mmbran cap, whr w ar pcially inrd in characrizing poiiv oluion. W illura h analyical finding by numrical imulaion bad on polynomial collocaion. Kyword: ingular mixd boundary valu problm, poiiv oluion, hallow mmbran, collocaion mhod, lowr and uppr funcion MSC2:34B6,34B8. Inroducion W inviga h olvabiliy of h ingular mixd boundary valu problm (.a) (.b) ( n u ()) + n f(, u()) =, < <, + n u () =, a u() + a u ( ) = A, *ThfirauhorwauppordbyhgranNo.A973ofhGranAgncyofh Acadmy of Scinc of h Czch Rpublic and by h Council of Czch Govrnmn MSM ; h cond and h hird auhor wr uppord by h Aurian Scinc Fund Projc P

2 whr n N, n 2, a (, ), a, A [, )andwdno u ()by u ( ). For h givn funcion f(, x) w mak h following aumpion: A:Thdaafuncion f(, x)iconinuouon (, ] (, )andcanhavaim ingulariya = andapacingulariya x =. Dfiniion.. Afuncion f(, x)haaimingulariya =,ifhrxi x (, )uchha ε f(, x) d =, ε (, ). Afuncion f(, x)haapacingulariya x =,if up f(, x) =, (, ). x + W focu our anion on h xinc of poiiv oluion of problm(.) which ar characrizd in h following dfiniion. Dfiniion.2. A funcion u i calld a poiiv oluion of problm(.) if u aifi h following condiion: (i) u C[, ] C 2 (, ), (ii) u() > for (, ), (iii) u aifi quaion(.a) and boundary condiion(.b). W wan o prov a gnral xinc horm for problm(.) which will nabl a unifid approach o h xinc and localizaion of poiiv oluion for crain cla of ingular problm, uch a (.2a) (.2b) ( ( 3 u ()) + 3 8u 2 () µ ) u() λ2 2 2γ 4 =, + 3 u () =, a u() + a u ( ) = A. Wih µ, λ >, γ > problm(.2)iapcialcaof(.). Boundaryvalu problm(.2) ari in h hory of hallow mmbran cap and ar invigad in[4],[5],[6],and[2].equaion (.3) u () + 3 u () + q() u 2 () =, whr qiconinuouon [, ]andpoiivon (, ),augmndbyboundarycondiion(.b) wa udid in[2]. I dcrib h bhavior of ymmric circular 48

3 mmbran and can b aily ranformd o h pcial ca of(.). Finally, h problm pod on a mi-infini inrval, (.4a) (.4b) z () + ( λ γ 2 32z 2 () + µ ) =, < <, 4 <, b z() b z ( ) = A, aloariinhmmbranhoryandfor A > iwadicudin[]and[8].i canbwrininhform(.2),whr a = b, a = 2b,byuinghubiuion (.5) = 2, = z ( 2 ) =: u(). 2. Exinc horm for problm(.) Our analyical approach i bad on h lowr and uppr funcion mhod which ihrxnddohgnralingularproblmofhform(.).inhqul,w hall u h following dfiniion: Dfiniion 2.. A funcion σ i calld a lowr funcion of quaion(.a), if σ aifi h following rquirmn: (i) σ C[, ] C 2 (, ), (ii) ( n σ ()) + n f(, σ()), (, ). Ifhinqualiyin(ii)irvrd, σicalldanupprfuncionofquaion(.a). If σaifi(i),(ii)and (iii) + n σ (), a σ() + a σ ( ) A, hn σicalldalowrfuncionofhboundaryvaluproblm(.).ifhinqualiiin(ii)and(iii)arrvrd,hn σicalldanupprfuncionofhboundary valu problm(.). Ingnral, σ ()canbcomunbounddahndpoinofhingraioninrval, = and =.Formorgnraldfiniionoflowrandupprfuncion,.g.[2],[7]or[22]. For h nx wo horm w nd h following aumpion: A2.: σ and σ 2 aralowrandanupprfuncionofproblm(.),rpcivly. A2.2: < σ () σ 2 ()for (, ). A2.3:Thrxi p < 2uchha + p h() <,whr h() = up{ f(, x) : σ () x σ 2 ()}. 49

4 Noha σ and σ 2 canvaniha = and =. Sinc f(, x)mayxhibi ingulariia = and x =,wailyha hcanbcomunboundd,i.. (2.) up h() =, + up h() =. Thorm 2.2. Aum ha A and A2. A2.3 hold. (i)l hbbounddon [, ].Thnproblm(.)haapoiivoluion uuch ha u C [, ]and u () =.Morovr, (2.2) σ () u() σ 2 (), [, ]. (ii) L h aify(2.). Furhrmor, l u aum ha hr xi a conan δ (, )uchha (2.3) ( n σ ()), ( n σ 2()), (, δ ), σ () = σ 2 (),andhrxi δ 2 (, ), K Ruchha (2.4) ( n σ ()) K, ( n σ 2 ()) K, ( δ 2, ). Thn problm(.) wih A = in(.b) ha a poiiv oluion u aifying(2.2). Proof. (i)for hbounddon [, ],(i)followbyarguingainhrgularca, whr ficoninuouoraifihcarahéodorycondiionon [, ] [, ),.g. Thorm 2.3 in[2]. (ii)l haify(2.)andl(2.3),(2.4),and σ () = σ 2 ()hold.nowhproof icarridouinfivp. Sp. Wfirhowha A = : Thcondiion up h() = anda imply σ () =. From σ () = σ 2 ()alo σ 2 () = follow. If a =,hn Dfiniion2.(iii)yild = a σ () Aand = a σ 2 () A. Thrfor, A =. If a >,Dfiniion2.(iii)yild σ 2 ( ) A/a. DuoA2.2, σ 2 () > for (, )andhnc, σ 2 ( ).Thrfor, A =. Sp2.Approximaoluion u k :Choo k N, /k min{δ, δ 2 },anddfin [,, ), k [ f k (, x) := f(, x), k k],, 5 K n, ( k, ].

5 Conidr h quaion (2.5) ( n u ()) + n f k (, u()) =. Wha σ and σ 2 arlowrandupprfuncion ofquaion(2.5)ubjc o(.b) and h k () := up{ f k (, x) : σ () x σ 2 ()} ibounddon [, ]. ByPar(i)ofhproof,problm(2.5),(.b)haaoluion u k C [, ] C 2 (, )aifying u k () = and (2.6) σ () u k () σ 2 (), [, ]. Sp3. Propriofhfuncion h: Wnowdrivomufulpropriof h whichwillbrquirdinhnxpofhproof.chooaninrval [, b] [, ). DuoAandA2.2,hfuncion n h()iconinuouon (, b].sinc p < 2 n,i follow from A2.3 ha + n h() = hold.thrfor, (2.7) b n h()d =: M b (, ). Thu, by d l Hopial rul and A2.3, + n p+ n h()d n h() = + (n p + ) n p = n p + + p h() =: c (, ). Thiyildhxincof ε (, )uchha n n h()d (c + ) p, [, ε]. Morovr, by(2.7), n n h()d ε n b n h()d = M b ε n for [ε, b]. Finally, imply h la wo inqualii (2.8) b n n h()d d <. 5

6 Sp4.Propriofhqunc {u k }:Conidrhquncofquaion(2.5) ubjco(.b)wih k N, /k min{δ, δ 2 },whr δ and δ 2 arpcifidby(2.3) and(2.4),rpcivly. FromSp2wobainhcorrpondingqunc {u k }of hir oluion which ar approximaion for u. L u fir dicu h convrgnc propriof {u k }. Chooaninrval [, b] [, ). Thnhrxianindx k N, /k min{δ, δ 2 },uchha [ [, b], ], k k. k Du o boundary condiion(.b) and quaion(2.5) w hav (2.9) n u k() + Th inqualiy (2.) f k (, u k ()) h(), condiion(2.7) and qualiy(2.9) yild (2.) n u k() n f k (, u k ())d =, [, b], k k. [, ], k k, k n h()d M b, [, b], k k. Accordingo(2.6)and(2.)hqunc {u k }and { n u k }arbounddon [, b]. Morovr,by(2.7)and(2.8),forach ε > hrxiaδ> uchhaforany, 2 [, b]wih 2 < δandany k k whav n u k ( ) n 2 u k ( 2 2) n h()d < ε and u k ( ) u k ( 2 ) 2 n n h()d d < ε hold. Hnc, hqunc {u k }and { n u k }arquiconinuouon [, b]. Th Arzlà-Acolihormnowimplihahrxiaubqunc {u l } {u k }uch ha u l = u, n u l = n u l l uniformlyon [, b].finally,byhdiagonalizaionprincipl,wfindaubqunc {u k }aifying (2.2) k u k = u, locally uniformly on [, ). k n u k = n u Forimpliciy,hprviounoaion {u k }forhiubqunciud. 52

7 Sp5.Propriofhfuncion u:wnowprovhahifuncion uia poiiv oluion of problm(.) aifying(2.2). Du o(2.6) and(2.2) w hav (2.3a) (2.3b) σ () u() σ 2 (), [, ), u C[, ), n u () C[, ), + n u () =. Choo (, ).Thnhrxi k k uchha andhnc,byaand(2.2), f(, u k ()) = f k (, u k ()), k k, f k(, u k ()) = f(, u k()) = f(, u()). k k Conqunly,hqunc {f k (, u k ())}ipoinwiconvrgingon (, ).Furhrmor,foranarbiraryinrval [, b] [, )whav,by(2.), n f k (, u k ()) n h(), [, b], k k. Thrfor, du o(2.7), w can u h Lbgu dominad convrgnc horm forhquncofqualii(2.9). Havinginmindha b (, )iarbiraryand ling k,wconcludha n u () + n f(, u())d =, [, ). Thu u C 2 (, )and uaifiquaion(.a)for (, ). BySp,whav σ () = σ 2 () = A = andconqunly,by(2.3a), u() = follow. For u() =,wcanha u C[, ]iapoiivoluionofproblm(.),which compl h proof. Thorm 2.3. Aum ha A and A2. A2.3 hold. (i)l hbboundda = andluaumha up h() = and condiion(2.4)hold.thnproblm(.)wih A = in(.b)haapoiivoluion u C [, )whichaifiima(2.2)and u () =. (ii)l hbboundda = andl up h() = andcondiion(2.3)hold. + Thnproblm(.)haapoiivoluion u C (, ]whichaifiima(2.2). Proof. WuargumnimilarohofromhproofofThorm

8 (i)sinc hiboundda =,wdfin f(, x), f k (, x) := [, ], k K n, ( k, ], whr k N, /k δ 2,and δ 2, Kargivnby(2.4).AinSp2 4,wconruc hqunc {u k }ofoluionofquaion(2.5)ubjco(.b)whichaify(2.6) and(2.2). BySp5,hifuncion uiapoiivoluionofproblm(.) aifying(2.2).sinc hiboundda =,whav and hrfor, u () n { [ up h() :, ]} =: M < 2 n h()d For u () =, u C [, )follow. (ii)sinc hiboundda =,w, f k (, x) := f(, x), M [, n +, ]. 2 [, ), k [ k, ], whr k N, /k δ,and δ ipcifidby(2.3). AinSp2wdrivh qunc {u k }ofoluionofquaion(2.5)ubjco(.b)andaifying(2.6). Morovr, imilarly o Sp 3, w obain n h()d <, andwdduc,ainsp4,ha n n h()d d <, u k = u, k k n u k = n u holduniformlyon [, ].Thrfor, u C[, ] C (, ]and uaifi(.b)and (2.2). By h Lbgu dominad convrgnc horm, a in Sp 5, w conclud ha u C 2 (, )aifiquaion(.a)for (, )andhrulfollow. No ha h xinc of nonngaiv oluion for mixd problm, whr f may bingularjua x =,waaloprovdin[3]. 54

9 3. Singular mmbran problm InhicionwuThorm2.2and2.3oprovholvabiliyofingular mmbran problm. W udy h boundary valu problm (3.a) (3.b) ( n u ()) + n( a u 2m () b u m () c2r) =, + n u () =, a u() + a u ( ) = A, whr a (, ), b, c [, ), r (, ), m, n N,and n 2. Problm(3.) covr h mmbran problm(.2) and, afr ubiuion(.5), alo h infini inrval problm(.4). InordrobablouilizhrulformuladinThorm2.2and2.3,i incaryohowhowofindproprlowrandupprfuncionofhabov problm. W bgin wih lowr and uppr funcion of quaion(3.a), h choic of whichdpndonhparamr a, b, c, r, n,and m. Lmma3.. Aumha c (, x /m ],whr x = 2 (b + b 2 + 4ac)/a.For [, ],wdfin { c, r, (3.2) σ () := c r/m, r (, ). Thn σ ialowrfuncionofquaion(3.a). Proof. Sinc c m x and x iapoiivoluionofhquaion ax 2 bx c =,whav (3.3) a c 2m b c m c. L r.thn σ () c and,by(3.3), ( n σ ()) + n( a σ 2m() b σ m() c2r) n( a c 2m b c m ) c, (, ). L r (, ).Thn σ () = c r/m and,by(3.3), ( n σ ()) + n( a σ 2m () b σ m() c2r) n+2r( a c 2m b c m ) c, (, ). Thiman σ aificondiion(i)and(ii)ofdfiniion2.. 55

10 Lmma 3.2. L u aum ha c 2 [x /m, ), whr x i dfind in Lmma3..For [, ]dfin c 2 + c ( ), r >, (3.4) σ 2 () := n c 2, r (, ]. Thn σ 2 ianupprfuncionofquaion(3.a). Proof. L r (, ].Thn σ 2 () c 2.Sinc < c m 2 x,whav ( n σ 2 ()) + n( a σ2 2m() b σ2 m() c2r) n( a 2 L r >.Thn σ 2 () = c 2 + c n ( )and whr c 2m b c m 2 ) c, (, ). ( n σ 2 ()) + n( a σ 2m 2 () b σ m 2 () c2r) n ( c + ψ()), (, ), a ψ() = [c 2 + c n ( b )]2m [c 2 + c. n ( )]m If ψ()ipoiivforom (, ),wcanconclud c + ψ() c + a c 2m b 2 c m 2 andhu,bydfiniion2.,hfuncion σ 2 ianupprfuncionof(3.a). Wnowpcifyh c and c 2 in σ and σ 2 fromlmma3.and3.2,rpcivly,inordroaifycondiiona2.2anddfiniion2.(iii). For σ 2 wak Dfiniion 2.(iii) wih h rvrd inqualii. Lmma3.3.L A > and x bainlmma3..s r := max{, r}and { c := min Am a m + a r, x /m } { (, c 2 := max a A + a c n ), x /m Thn σ and σ 2 givnby(3.2)and(3.4)ar,rpcivly,lowrandupprfuncion of problm(3.) and aify A2.2. Proof. ByLmma3.and3.2, σ and σ 2 arlowrandupprfuncionof quaion(3.a). W ha A2.2 hold and(3.2),(3.4) yild }. + n σ () =, + n σ 2() =. 56

11 Finally, a σ () + a σ ( ) = a σ 2 () + a σ 2( ) = a c A, r, ( r ) c a a A, r (, ), m a c 2 A, r (, ], c a c 2 a n A, r >. Lmma3.3dalwihhca A >.Inhnxwolmmawwilldicuh ca A =,whrconanlowrandupprfunciondonoxi. Lmma3.4. L A = and a >. S k := + a /(a a (r/m))andfor [, ]dfin ( ν a ), r, (3.5) σ () := a ν r/m (k ), r (, ), ( σ 2 () := β a ). a Thnhrxiconan ν, β (, )uchhaforach ν (, ν )and β β, hfuncion σ and σ 2 aralowrfuncionandanupprfuncionofproblm(3.) aifying A2.2. Proof. Bydirccalculaionwcanha σ and σ 2 aify + n σ i() =, a σ i () + a σ i( ) =, i =, 2. L r.thn σ () = ν( 2 + 2a /a )and whr ( n σ ()) + n( a σ 2m () b σ m () c2r) n ϕ (, ν), (, ), ϕ (, ν) = 2ν(n + ) + a [ν( 2 + 2a /a )] 2m b [ν( 2 + 2a /a )] m c. Sinc ν + ϕ (, ν) = uniformlyon [, ],wcanfind ν > uchhaforach ν (, ν ],hinqualiy ϕ (, ν) holdfor [, ]. L r (, ).Thn σ () = ν r/m (k )and ( n σ ()) + n( a σ 2m () b σ m () c2r) n r/m 2 ψ (, ν), (, ), 57

12 whr and ψ (, ν) = νl() + 2r+r/m+2 h(, ν), h(, ν) = (3.6) l() = rk m a [ν(k )] 2m b [ν(k )] m c, (n r ) ( m n r )( r ). m m Choo c > ainlmma3.andl ν (, c /k].thn,by(3.3), h(, ν) for ν (, ν ], [, ].Wnowdnohuniquzroof l()by andhav l() for [, ].Conqunly, Furhrmor, ψ (, ν), ν (, ν ], [, ]. ψ (, ν) = ν + uniformlyfor [, ]. Thrfor,wcanfind ν (, ν ]uchhaforach ν (, ν ],hinqualiy ψ (, ν) holdfor [, ]. Lunowconidr σ 2 () = β( 2 + 2a /a ).Whav whr ( n σ 2()) + n( a σ 2m 2 () b σ m 2 () c2r) n ϕ(β), (, ), ( ϕ(β) = 2(n + )β + a 2β a ) 2m. a Sinc ϕ(β) =,hrxi β β > ν uchhaforach β β whav ϕ(β) >. Lmma3.5.L A = and a =.For [, ]ludfin (3.7) σ () := { ν( 2 ), r, ν r/m ( ), r (, ), σ 2 () := β( 2 ) /2m. Thnhrxiconan ν, β (, )uchhaforach ν (, ν )and β β, hfuncion σ and σ 2 aralowrfuncionandanupprfuncionofproblm(3.) aifying A Proof. Wcanailychckha σ and σ 2 aify + n σ i() =, σ i () =, i =, 2.

13 L r.thn σ () = ν( 2 )and whr ( n σ ()) + n( a σ 2m () b σ m () c2r) n ϕ(, ν), (, ), ϕ(, ν) = 2ν(n + ) + a [ν( 2 )] 2m b [ν( 2 )] m c. Sinc ν + ϕ(, ν) = uniformlyon [, ],wcanfind ν > uchhaforach ν (, ν ],hinqualiy ϕ(, ν) holdfor [, ]. L r (, ). Thn σ () = ν r/m ( ) and imilarly o h proof of Lmma3.4wconcludhaforachufficinlymallpoiiv νhfuncion σ i a lowr funcion of problm(3.). Now,conidr σ 2 () = β( 2 ) /2m.Whav ( n σ 2()) + n( a σ 2m 2 () b σ m 2 () c2r) n ( 2 ) /2m 2 ϕ 2 (, β), (, ), whr ϕ 2 (, β) = 2β m ( ) + aβ 2m ( 2 ) /2m. 2m Sinc ϕ 2(, β) = uniformlyon [, ],wcanfindaβ > ν uchhafor β ach β β, ϕ 2 (, β) on (, )hold.thrfor, σ 2 ianupprfuncionof(3.) and A2.2 i aifid. Having drivd lowr and uppr funcion of problm(3.) for all valu of i paramr, w can prov h xinc of a poiiv oluion u o hi problm and dcribhow u bhavahingularpoin = and =. (3.8) Thorm 3.6. Problm(3.) ha a poiiv oluion u uch ha u() >, u (+) =, r > 2, u() >, u (+) = c n, r = 2, u(), u (+) =, r < 2, and (3.9) u ( ) R, A >, u ( ) R, A =, a >, u ( ) =, A =, a =. 59

14 Proof. Lowrandupprfuncion σ and σ 2 ofproblm(3.)aifyinga2.2 ar givn according o Lmma 3.3, 3.4, and 3.5. Th funcion f(, x) = a x 2m b x m c2r aifi A. Conidr h funcion h from A2.3. Thn w hav (3.) h() a σ 2m () + b σ m () + c2r, (, ). Ca.Waumha A > or A =, a >.Wfirfind c bylmma3.3, andhnchoo ν (, ν )in(3.5)uchha νk c. L r. Thn σ ipoiivon [, ]and(3.)impliha hiboundd on [, ]and h() =. Thu, haificondiiona2.3wih p = and,by + Thorm2.2(i),problm(3.)haapoiivoluion u C [, ]aifying u () = and(2.2).sinc σ () >,hinqualiy u() > follow. L r (, ).Thn(3.)yild (3.) h() 2r( a [ν(k )] 2m + b ) [ν(k )] m + c, (, ). Alo, h() a σ 2m () b σ m() c2r 2r( a c 2m b c m ) c >, (, ). By(3.)andhlainqualiywhav up h() =, + up h() <. Duo(3.),for p = 2rwcanhowA2.3,inc + p h() a [ν(k )] 2m + b [ν(k )] m + c <. Nowwprov(2.3). If A >, wulmma3.3andhav σ () = c r/m, σ 2 () c 2.Hnc, ( ( n σ ()) = c r )( n r ) n 2 r/m, ( n σ 2 m m ()) =, (, ). For A = and a > wulmma3.4andhav σ () = ν r/m (k ), σ 2 () = β( 2 + 2a /a ).Hnc, 6 ( n σ ()) = ν n 2 r/m l(), ( n σ 2 ()) = 2β(n + ) n, (, δ ),

15 whr l()igivnby(3.6)and δ = iiuniquzro.thrfor,condiion(2.3) hold. Conqunly, by Thorm 2.3(ii), problm(3.) ha a poiiv oluion u C (, ]aifying(2.2). Irmainoprov(3.8)for r (, ). Equaion(3.a)andcondiion(3.b) rul in (3.2) n u () = andconqunly,inc n 2and r >, (3.3) + n( a u 2m () b u m () c2r) d, (, ), n( b u m () a ) u 2m d =. () Aum u() =.Sinc σ () = and + σ () =,inqualiy(2.2)impli (3.4) + u () =. Onhohrhand,haumpion u() = guaranhxincof δ > uch ha u m () a/bfor [, δ].thn,by(3.2), u () = n n u 2m () (bum () a)d + c2r+ n + 2r + c2r+ n + 2r +, (, δ). If r [ 2, ),hn u () c/non (, δ),aconradiciono(3.4).thimanha w hav hown (3.5) r = u() >. 2 For r [ 2, ),uing(3.2),(3.3),(3.5)anddl Hopial rulwobain c 2r+ (3.6) + u () = + n + 2r + =, r ( 2, ), c n, r = 2. L r (, 2 ).If u() =,hn(3.4)hold.if u() >,hnby(3.2),(3.3), and d l Hopial rul w dduc a bfor, c 2r+ + u () = + n + 2r + =. Ca2.Now,wconidrhca A =, a =. 6

16 L r,hnbylmma3.5, σ () = ν( 2 ), σ 2 () = β( 2 ) /2m, whr < ν < βwihaufficinlymall νandaufficinlylarg β.for (, ) w hav < ( 2 ) 2m ( a ν 2m b ) ν m c h() ( 2 ) 2m ( a ν 2m + b ν m + c ) and conqunly, Hnc, A2.3 hold. Morovr, up h() <, + up h() =. σ () = σ 2 () = = A, ( n σ ()) = 2ν(n + ) n, and ( n σ 2()) = β m n ( 2 ) /2m 2( (n + )( 2 ) + 2 2( m )). Thimanhahrxi δ 2 (, )uchha(2.4)ivalidfor K = 2ν(n + ). Thrfor,byThorm2.3(i),problm(3.)haapoiivoluion u C [, ) aifying u () = and(2.2).sinc σ () >,whav u() >. L r (, ).ByLmma3.5, σ () = ν r/m ( ), σ 2 () = β( 2 ) /2m, whr < ν < βand νiufficinlymall,whil βiufficinlylarg. Thnfor (, ) < Conqunly, 2r ( ) 2m ( a ν 2m b ) ν m c h() up h() =, + 2r ( ) 2m ( a ν 2m + b ν m + c ). up h() =. For p = 2rwobain + p h() < andhnca2.3follow.morovr,whav 62 ( n σ ()) = ν n r/m 2( r (n r ) ( m m n r )( r ) ), (, ), m m ( n σ 2()) = β m n ( 2 ) /2m 2( (n + )( 2 ) + 2 2( )), (, ). m

17 Thu,wcanfind δ, δ 2 (, )whicharufficinlymalloguaran ( n σ ()), ( n σ 2()), (, δ ), ( n σ ()) K, ( n σ 2 ()) K, ( δ 2, ), whr K = ν(n r/m)( r/m).wcanha(2.3)and(2.4)holdanduing Thorm2.2(ii)wdduchaproblm(3.)haapoiivoluion u C (, ) aifying(2.2).for r (, ),propry(3.8)canbprovdinhamwayain Ca. Finally,whowhaif A = and a =,hn u ( ) =. Sinc u() =, hrxi ξ (, )uchha u m () a/2bfor [ξ, ].Morovr,whav ξ d u 2m () ξ d () σ 2m 2 2β 2m Thrfor, by ingraing(3.a), w obain ξ n u () = ξ n u n (ξ) + ξ u 2m () (bum () a)d + c ξ n u (ξ) + a ) d ( 2 ξn u 2m () ξ d = ln, (ξ, ). 2β2m ξ ξ n+2r d ξ n u (ξ) + aξn ln 4β2m ξ + c, (ξ, ). n + 2r + Hnc, n u () = u ( ) =. FromThorm3.6warnowablodrivhfollowingxincrulfor problm(.4). Thorm 3.7. Problm(.4) ha a poiiv oluion z uch ha >, γ z () = λ2 8γ, γ 3 2, (3.7), 3 z () =, γ < 3 2, and (3.8) z (+) R, A >, z (+) R, A =, b >, z (+) =, A =, b =. 63

18 Proof. Problm(3.)wih n = 3, a = 8, b = µ, c = λ2 /2, r = γ 2hah form(.2). ByLmma3.3,3.4,and3.5hrxilowrandupprfuncion σ and σ 2 ofproblm(.2)aifyinga2.2.bythorm3.6,hriapoiivoluion uof(.2)aifying(2.2),(3.8),and(3.9).l r 2 := max{ σ 2 () : [, ]}andl zbdfindby ( ) := z 2 = u(), (, ]. Thn < < r 2 for [, )and ziaoluionofproblm(.4).furhrmor, w hav 2 3 z () = u (). L γ 3 2.Thn,by(3.6), and λ 2 + u () = + 4γ 2γ 3, γ z () = γ 3/2 ( 3/2 z ()) = 3 2γ( ) + 2 u () = λ2 8γ. Conqunly, du o(3.8) and(3.9), z aifi(3.7) and(3.8). 4. Numrical approach Hr, w fir dcrib how w approxima oluion of wo-poin boundary valu problm for ym of ordinary diffrnial quaion of h form f(, u (), u()) =, [, ], g(u(), u()) =. W aum ha h analyical oluion u i approprialy mooh and amp o olv hi problm numrically uing h collocaion mhod implmnd in our Malabcod bvpui. IianwvrionofhgnralpurpoMalabcod bvp,cf.[4],[5],and[8],whichhaalradybnuccfullyapplidoavariyof problm, for xampl[9],[],[],[9], and[2]. Collocaion i a widly ud and wll-udid andard oluion mhod for wo-poin boundary valu problm, forxampl[23]andhrfrnchrin. Ialoprovdobrobuinh ca of ingular boundary valu problm. Th cod i dignd o olv ym of diffrnial quaion of arbirary ordr. For impliciy of noaion w formula blow a problm who ordr vari bwn 64

19 four and zro, which man ha algbraic conrain which do no involv drivaiv araloadmid.morovr,hproblmcanbgivninafullyimpliciform (4.a) (4.b) F(, u (4) (), u (3) (), u (), u (), u()) =, <, b(u (3) (), u (), u (), u(), u (3) (), u (), u (), u()) =. Thprogramcancopwihfrparamr, λ, λ 2,..., λ k,whichwillbcompud along wih h numrical approximaion for u, (4.2a) (4.2b) F(, u (4) (), u (3) (), u (), u (), u(), λ, λ 2,...,λ k ) =, <, b aug (u (3) (), u (), u (), u(), u (3) (), u (), u (), u()) =, providdhahboundarycondiion b aug includ kaddiionalrquirmnob aifid by u. Th numrical approximaion dfind by collocaion i compud a follow: On a mh := {τ i : i =,...,N}, = τ < τ... < τ N =, w approxima h analyical oluion by a collocaing funcion p() := p i (), [τ i, τ i+ ], i =,...,N, whrwrquir p C q [, ]inhcahahordrofhundrlyingdiffrnialquaioni q.hr p i arpolynomialofmaximaldgr m +qwhichaify h ym(4.a) a h collocaion poin { i,j = τ i + j(τ i+ τ i ), i =,...,N, j =,...,m}, < <... < m <, andhaociadboundarycondiion(4.b). For y R n, y = (y,...,y n ) T,w hav y := max k n y k. L y C[, ], y: [, ] R n.for [, ], and y() := max k n y k() y := max y(). Claical hory, cf.[23], prdic ha h convrgnc ordr for h global rror of hmhodiala O(h m ),whr hihmaximalpiz, h := max(τ i+ τ i ). i 65

20 Morprcily,forhglobalrrorof p, p u = O(h m )holduniformlyin. For crain choic of h collocaion poin h o-calld uprconvrgnc ordr can b obrvd. In h ca of Gauian poin hi man ha h approximaion i xcpionallyprciahmhpoin τ i, max τ i p(τ i) u(τ i ) = O(h 2m ). To mak h compuaion mor fficin, an adapiv mh lcion ragy bad on an a poriori ima for h global rror of h collocaion oluion may b uilizd. W u a claical rror ima bad on mh halving. In hi approach, wcompuhcollocaionoluion p ()onamh.subqunly,wchooa condmh 2 whrinvryinrval [τ i, τ i+ ]of winrwoubinrvalof quallngh. Onhinwmh,wcompuhnumricaloluionbadonh amcollocaionchmoobainhcollocaingfuncion p 2 (). Uinghwo quanii, w dfin (4.3) E() := 2m 2 m(p 2 () p ()) aanrrorimaforhapproximaion p (). Aumhahglobalrror δ() := p () u()ofhcollocaionoluioncanbxprdinrmofh principal rror funcion (), (4.4) δ() = () τ i+ τ i m + O( τ i+ τ i m+ ), [τ i, τ i+ ], whr () i indpndn of. Thn obviouly, h quaniy E() aifi E() δ() = O(h m+ )andhrrorimaiaympoicallycorrc. Ourmhadapaion i bad on h quidiribuion of h global rror of h numrical oluion. Thu,wdfinamoniorfuncion Θ() := m E()/h(),whr h() := τ i+ τ i for [τ i, τ i+ ].Now,hmhlcionragyaimahquidiribuionof τi+ τ i Θ()d onhmhconiingofhpoin τ i obdrmindaccordingly,whrah am im maur ar akn o nur ha h variaion of h piz i rricd and olranc rquirmn ar aifid wih mall compuaional ffor. Dail of hmhlcionalgorihmandaproofofhfachaourragyimpliha h global rror of h numrical oluion i aympoically quidiribud ar givn in[7]. W now dicu h numrical oluion of problm(3.) who analyical propri ar formulad in Thorm 3.6. For h numrical xprimn w pcify h following paramr ing: 66 n = 3, m =, a = λ2, b = µ =, c = 8 2 =, λ =, r = γ 2, 2

21 Thorm3.6. Inordrobabloformulahfirboundarycondiion in(3.b),winroducanwvariabl v() := 3 u ()andranformhcalarboundary valu problm(3.) o an aociad boundary valu problm for h following ym of wo implici diffrnial quaion of fir ordr: (4.5a) (4.5b) (4.5c) v () + 3( 8u 2 () µ u() λ2 2 2γ 4) =, v() 3 u () =, v() =, a u() + 2 a u () = A, wih [, ]. For numrical imulaion, problm(4.5) ha bn rarrangd o (4.6a) (4.6b) (4.6c) v ()u 2 () + 3( 8 µu() λ2 u 2 () 2γ 4) =, 2 v() 3 u () =, v() =, a u() + 2 a v() = A. 4.. Numrical rul In hi cion w illura h horical finding of Thorm 3.6 by appropria numrical xprimn which hav bn carrid ou uing collocaion a 4 Gauian collocaion poin. Th numrical oluion ha bn calculad on a fixd quidianmhwihpoin. Thrahrdngridwrncaryforagood viualizaion of approximaion whn ranforming hm from h andard inrval [, ]backohinfiniinrval [, ). Thrrorimaandhridualwr alo rcordd a indicaor for h accuracy of h numrical oluion. Th rror ima wa compud from(4.3) by coupling oluion rlad o mh wih and 2 mhpoin. Th ridual wa obaind by ubiuing h numrical oluion p ino h ym of diffrnial quaion(4.6a),(4.6b). Fir,w a =, a = and A =. AccordingoThorm3.6himan ha u ( ) R.Thcorrpondingnumricalrulforwodiffrnvaluof γ, bohcovringhca r > 2,canbfoundinFig.and Soluion Error Eima 2 Ridual.996 u() r Figur. Problm(4.6), γ = 2.5: Th numrical approximaion for h oluion componn u(), h rror ima and h ridual. 67

22 u() Soluion.955 Error Eima r Ridual 4 6 Figur 2. Problm(4.6), γ = 2: Th numrical approximaion for h oluion componn u(), h rror ima and h ridual. Inbohhfigur u() > and u (+) =,awaprdicdbythorm3.6. Morovr, boh h rror ima and h ridual ar vry mall hu indicaing anxcllnaccuracyofhapproximaion. InFig.3hrulfor r = 2 ar dpicd. Again, u() > iclarlyviibl. Hrwhav n = 3, c = 2 and hrfor, u (+) = c/n.67whichiinagoodagrmnwihthorm3.6. Soluion Error Eima Ridual u().92 r Figur 3. Problm(4.6), γ =.5: Th numrical approximaion for h oluion componn u(), h rror ima and h ridual. Finally,Fig.4and5howhlaca r < 2. Soluion Error Eima Ridual u().85 r Figur 4. Problm(4.6), γ =.3: Th numrical approximaion for h oluion componn u(), h rror ima and h ridual. 68

23 Soluion Error Eima Ridual u() r 2 4 Figur 5. Problm(4.6), γ =.2: Th numrical approximaion for h oluion componn u(), h rror ima and h ridual. Forbohing, u() and u (+) =. Wnow A = andlavallhohrparamrunchangd. Accordingo Thorm3.6hirulin u ( ) =. Fig.6ohowhcorrponding numricalrunfor γ = 2.5, 2,.5,.3,.2,rpcivly..3 Soluion 5 Error Eima Ridual.25 u().2.5 r Figur6. Problm(4.6), A=, γ=2.5: Thnumricalapproximaionforholuion componn u(), h rror ima and h ridual..25 Soluion 5 Error Eima Ridual u().2.5 r Figur7. Problm(4.6), A =, γ =2: Thnumricalapproximaionforholuion componn u(), h rror ima and h ridual. 69

24 .25 Soluion 5 Error Eima Ridual.2 u().5 r Figur8. Problm(4.6), A=, γ=.5: Thnumricalapproximaionforholuion componn u(), h rror ima and h ridual. Again, u (+) Soluion 5 Error Eima Ridual.5 5 u(). r.5 5 Figur9. Problm(4.6), A=, γ=.3: Thnumricalapproximaionforholuion componn u(), h rror ima and h ridual..2 Soluion 5 Error Eima Ridual u().5. r Figur. Problm(4.6), A=, γ=.2: Thnumricalapproximaionforholuion componn u(), h rror ima and h ridual. ThlaingdicudinThorm3.6i A = and a >. Wu a = 2, all h ohr paramr rmain unchangd, Fig. o 5 for h numrical imulaion corrponding o h abov valu of γ. 7

25 Soluion Error Eima Ridual u() r Figur. Problm(4.6), A=, a >, γ=2.5: Thnumricalapproximaionforh oluion componn u(), h rror ima and h ridual. u() Soluion Error Eima r Ridual Figur2. Problm(4.6), A=, a >, γ=2: Thnumricalapproximaionforh oluion componn u(), h rror ima and h ridual Soluion 5 Error Eima 7 Ridual.28 8 u() r Figur3. Problm(4.6), A=, a >, γ=.5: Thnumricalapproximaionforh oluion componn u(), h rror ima and h ridual. Soluion Error Eima Ridual u().22 r Figur4. Problm(4.6), A=, a >, γ=.3: Thnumricalapproximaionforh oluion componn u(), h rror ima and h ridual. 7

26 Soluion Error Eima Ridual u() r Figur5. Problm(4.6), A=, a >, γ=.2: Thnumricalapproximaionforh oluion componn u(), h rror ima and h ridual. All numrical rul how a good agrmn wih Thorm 3.6. Boh h rror ima 2 andhridualhowhaholuionaccuracyixclln.toviualiz oluion of problm(.4) pod on h mi-infini inrval, w hav o ranform hnumricaloluionobaindon [, ]backohoriginalinrval [, ).Tohi ndwu ( ) := z 2 = u(), [, ), (, ], oobainhvalufor. Wagaindicuhrdiffrning,whrforallxprimn b = a =. For A = and b = 2 a =,Fig.6howhnumricaloluionof(.4)diplayd onahorandalonginrval. Soluion Soluion Soluion Soluion Soluion Soluion Figur6. Problm(.4), A=, b =: Soluion onhinrval[,](abov)and inrval[,](blow)forvaluof γ=2.5, γ=.5and γ=.3(fromlfo righ). 2 OfnwihinhlvlofhmachinaccuracyofMalab. 72

27 Forabrilluraionofholuionbhaviorfor γ = 2.5diplaydonhlong inrvalinfig.6,wdpichioluioninfig.7onhrfurhrinrvalof mallr lngh, alo Fig.. Soluion Soluion Soluion Figur7. Problm(.4), A=, b =, γ=2.5: Soluion onhinrval[,2], [,5],and[,](fromlforigh). For γ 3 2, z (+) Rholdandwknowhaholuionof(4.6)ipoiiv wih >. Alo,for λ =, γ z () = 8 γ. Inprincipl,whould bablovrifyhilariuinghvaluofhnumricaloluionah mhpoin approaching zro and h rlaion (4.7) γ z () = v()/(2 2γ ) =: w(), cf.(4.6b). For γ =.5(and γ =.6)whavplod w()uingivaluah mhpoinandfoundouha w() ( 8 γ ) 5. InFig.8hnumricaloluionof(.4)for A = and b = ihown. Soluion.25 Soluion Soluion Soluion.25 Soluion Soluion Figur8. Problm(.4), A=, b =: Soluion onhinrval[,](abov)and inrval[,](blow)forvaluof γ=2.5, γ=.5and γ=.3(fromlfo righ). 73

28 Hr,axpcd, z (+) = holdforallvaluof γ.alo,. Finally,wconidr A = and b =.Thnumricalrulforhiingand habovfivvaluof γargivninfig.9. Wih z (+) R, γ z () 8 γ for γ =.5,and hnumricaloluionagainvrywllrflc h propri of h analyical oluion. Soluion.3 Soluion.285 Soluion Soluion.3 Soluion.28 Soluion Figur9. Problm(.4), A=, b =: Soluion onhinrval[,](abov)and inrval[,](blow)forvaluof γ=2.5, γ=.5and γ=.3(fromlfo righ). Rfrnc [] R. P. Agarwal, D. O Rgan: An infini inrval problm ariing in circularly ymmric dformaion of halow mmbran cap. In. J. Non-Linar Mch. 39(24), zbl [2] R. P. Agarwal, D. O Rgan: Singular problm ariing in circular mmbran hory. Dyn. Conin. Dicr Impul. Sy., Sr. A, Mah. Anal. (23), zbl [3] R. P. Agarwal, S. Saněk: Nonngaiv oluion of ingular boundary valu problm wih ign changing nonlinarii. Compu. Mah. Appl. 46(23), zbl [4] W. Auzingr, O. Koch, E. Winmüllr: Efficin collocaion chm for ingular boundary valu problm. Numr. Algorihm 3(22), zbl [5] W. Auzingr, G. Knil, O. Koch, E. Winmüllr: A collocaion cod for boundary valu problm in ordinary diffrnial quaion. Numr. Algorihm 33(23), zbl [6] W. Auzingr, O. Koch, E. Winmüllr: Analyi of a nw rror ima for collocaion mhod applid o ingular boundary valu problm. SIAM J. Numr. Anal. 42(25), zbl [7] W. Auzingr, O. Koch, E. Winmüllr: Efficin mh lcion for collocaion mhod applid o ingular BVP. J. Compu. Appl. Mah. 8(25), zbl [8] J. V. Baxly, S. B. Robinon: Nonlinar boundary valu problm for hallow mmbran cap. II. J. Compu. Appl. Mah. 88(998), zbl [9] C. J. Budd, O. Koch, E. Winmüllr: Slf-Similar Blow-Up in Nonlinar PDE. AU- RORA TR Iniu for Analyi and Scinific Compuing, Vinna Univ. 74

29 of Tchnology, Auria, 24, availabl a hp:// publicaion/. [] C. J. Budd, O. Koch, E. Winmüllr: Compuaion of lf-imilar oluion profil for h nonlinar Schrödingr quaion. Compuing 77(26), zbl [] C.J.Budd,O.Koch,E.Winmüllr:FromnonlinarPDEoingularODE.Appl. Numr. Mah. 56(26), zbl [2] C. D Cor, P. Hab: Th lowr and uppr oluion mhod for boundary valu problm. Handbook of Diffrnial Equaion, Ordinary Diffrnial Equaion, Vol. I (A. Caňada, P. Drábk, A. Fonda, d.). Elvir/Norh Holland, Amrdam, 24, pp zbl [3] F. d Hoog, R. Wi: Collocaion mhod for ingular boundary valu problm. SIAM J. Numr. Anal. 5(978), zbl [4] R. W. Dicky: Roaionally ymmric oluion for hallow mmbran cap. Q. Appl. Mah. 47(989), zbl [5] K. N. Johnon: Circularly ymmric dformaion of hallow laic mmbran cap. Q. Appl. Mah. 55(997), zbl [6] R. Kannan, D. O Rgan: Singular and noningular boundary valu problm wih ign changing nonlinarii. J. Inqual. Appl. 5(2), zbl [7] I. T. Kiguradz, B. L. Shkhr: Singular boundary valu problm for cond ordr ordinary diffrnial quaion. Iogi Nauki Tkh., Sr. Sovrm. Probl. Ma. 3(987), 5 2.(In Ruian.) zbl [8] G. Kizhofr: Numrical ramn of implici ingular BVP. PhD. Thi. Iniu for Analyi and Scinific Compuing, Vinna Univ. of Tchnology, Auria. In prparaion. [9] G. Kizhofr, O. Koch, E. Winmüllr: Collocaion mhod for h compuaion of bubbl-yp oluion of a ingular boundary valu problm in hydrodynamic. J. Sci. Compu. To appar. Availabl a hp:// [2] O. Koch: Aympoically corrc rror imaion for collocaion mhod applid o ingular boundary valu problm. Numr. Mah. (25), zbl [2] I. Rachůnková, O. Koch, G. Pulvrr, E. Winmüllr: On a ingular boundary valu problm ariing in h hory of hallow mmbran cap. J. Mah. Anal. Appl. 332 (27), zbl [22] I. Rachůnková, S. Saněk, M. Tvrdý: Singularii and Laplacian in Boundary Valu Problm for Nonlinar Ordinary Diffrnial Equaion. Handbook of Diffrnial Equaion. Ordinary Diffrnial Equaion, Vol. 3(A. Caňada, P. Drábk, A. Fonda, d.). Elvir, Amrdam, 26. [23] U.M.Achr,R.M.M.Mahij,R.D.Rull:NumricalSoluionofBoundaryValu Problm for Ordinary Diffrnial Equaion. Prnic-Hall, Englwood Cliff, 988. zbl [24] E. Winmüllr: Collocaion for ingular boundary valu problm of cond ordr. SIAM J. Numr. Anal. 23(986), zbl Auhor addr: I. Rachůnková, Dparmn of Mahmaical Analyi, Faculy of Scinc, Palacký Univriy, ř. 7. liopadu 2, Olomouc, Czch Rpublic, -mail: rachunko@inf.upol.cz; G. Pulvrr, Iniu for Analyi and Scinific Compuing, Vinna Univriy of Tchnology, Widnr Haupra 6-, A-4 Vinna, Auria, -mail: grnopulvrr@gmail.com; E. B. Winmüllr, Iniu for Analyi and Scinific Compuing, Vinna Univriy of Tchnology, Widnr Haupra 6-, A-4 Vinna, Auria, -mail:.winmullr@uwin.ac.a. 75