Analysis and Numerical Approximation of a Singular Boundary Value Problem for the One-dimensional p-laplacian

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1 Proceedigs of the 4th WSEAS Iteratioal Coferece o Fluid Mechaics Gold Coast Queeslad Australia Jauary Aalysis ad Numerical Approximatio of a Sigular Boudary Value Problem for the Oe-dimesioal p-laplacia PEDRO LIMA Cetro de Matemática e Aplicações Uiversidade Técica de Lisboa Av. Rovisco Pais Lisboa PORTUGAL LUISA MORGADO Departameto de Matemática Uiversidade de Trás-os-Motes e Alto Douro Vila Real PORTUGAL Abstract: I this work we are cocered about a sigular boudary value problem ivolvig the p-laplacia which arises i mathematical models of fluid mechaics. We aalyze the asymptotic behavior of the solutios of the cosidered ordiary differetial equatio ear the sigularities ad itroduce a computatioal method which takes this behavior ito accout. Key Words: Sigular boudary value problem p-laplacia upper ad lower solutios oe-parameter families of solutios asymptotic expasios. Itroductio The mathematical modellig of various pheomea of mechaics ad physics leads to the followig Dirichlet problem: div g p g ) = f x g) x B; gx) > 0 x B; gx) =0 x B ) where B is the uit ball cetered at the origi i IR N N> p>. The oliear differetial operator i the lefthad side of ) is kow as the p-laplacia ad i the case of p = reduces to the usual laplacia. It is worth to remark that the laplacia operator arises i the modelig of may classical problems of fluid dyamics as the result of the applicatio of a liear flow law see ). For example i the case of heat coductio problems if we deote by g the steady state temperature accordig to the Fourier s law the heat flow vector is proportioal i module to g. Whe this law is applicable the heat diffusio is give by div g) which yields the laplacia operator. However i the modelig of certai processes for example the meltig of ice at a costat temperature the heat flow is better described by a geeralized Fourier s law which states that the flow vector is a oliear fuctio of the temperature gradiet with the form g p g see for example ). The the heat diffusio will be represeted by the p-laplacia. Similar situatios whe the geeralizatio of classical liear problems leads to oliear oes ivolvig the p-laplacia also occur whe modelig flows of o-ewtoia fluids ad i elasticity theory whe modellig the deformatio of oliear membraes 3. Oe is ofte iterested i radial solutios of problem ) that is solutios that deped oly o x. I that case problem ) reduces to a boudary value problem for a ordiary differetial equatio: r N r N g p g ) = fr g) r 0 ); gr) > 0 r 0 ); g 0) = 0g) = 0 ) where N is the space dimesio. We shall assume that f :0 ) 0 ) IR is cotiuous i the cosidered domai ad has sigularities at y =0 or r =0. The existece ad the properties of solutios to problem ) uder differet assumptios o the fuctio f were ivestigated by may authors. The existece ad uiqueess of solutio i the case p = N = with fr g) = rg < 0 was proved by Nachma ad Callegari 4 i coectio with a boudary layer problem i fluid mechaics. I 56 ad 7 the authors also cosidered the case p = but with fr g) =r σ g σ >. I these works the asymptotic behavior of the solutios ear the sigularity at r = has bee aalyzed ad umerical methods were itroduced which take ito accout this behavior. Moreover upper ad lower solutios were determied ad i some particular cases a explicit formula for the exact solutio was obtaied.

2 Proceedigs of the 4th WSEAS Iteratioal Coferece o Fluid Mechaics Gold Coast Queeslad Australia Jauary Cocerig the case p i 8 ad 9 Wog has obtaied sufficiet coditios for existece ad uiqueess of solutios of problem ) for N =. I a recet work Ji Yi ad Wag 0 usig the method of lower ad upper solutios have ivestigated the existece of solutios to problem ) i the geeral case N. I the preset paper we will cotiue the ivestigatio of problem ) i the case N = for p> ad fr g) =ar σ g where a is a egative costat. The values of σ ad for which the existece of solutio is guarateed deped o p. Accordig to the Existece Theorem of 8 a sufficiet coditio for the existece of solutio is σ 0 p + < 0 p. O the other had uiqueess of solutio i this case is guarateed by Theorem.3 of 9. As we shall see i the ext sectio existece of solutio ca be guarateed uder weaker coditios. Whe <0 this problem has a sigularity at r = sice the secod derivative of g is ot cotiuous at this poit. The first derivative has also a discotiuity at this poit if <. Moreover the problem ca have also a sigularity at r =0 for some values of σ ad p. This will be discussed i the ext sectios. Behavior of the solutio i the eighborhood of r =0 gr) =C C r k + o) g r) = kc r k + o) g r) = kk )C r k + o) r ) Substitutig i 3) we obtai r 0 k k ) p )C + r k) p σ C C C r k = a 7) which implies k = p+σ σ+ k = ad with C = ) b>0we have C =. a) b p+σ) σ+) Defiig yr) by g r) =C C r k + y r) we obtai the Cauchy problem i y ) σ+ +y + r k y p k )k + y)+kry + r y = 8) b C r k + y) y0) = r 0 ry r) =0 9) + where r =0is a regular sigular poit of 8). For each b 0 this problem has a particular solutio that ca be represeted by Cosider the sigular Cauchy problem: g r) p g r)) = ar σ g r) 0 <r<r 0 3) r 0+ g 0) = 0. 4) I this sectio we will aalyze the asymptotic behavior of the solutios of this Cauchy Problem whe r 0+. Our mai result is resumed i the followig propositio. Propositio Assume that i 3) p> σ 0 < 0 ad r 0 > 0. The Cauchy problem 3)-4) has i the eighborhood of the sigular poit r =0 a oe parameter family of solutios that ca be represeted by: g r b) =C br k where k = p+σ C = b>0is the parameter. b + σ) C p +σ ) rk + o r k) 5) a) b p+σ) σ+) ) ad Proof. I the eighborhood of r =0 let us look for a solutio of 3) 4) i the form y par r b) = + l=0j=0l+j p+σ l+j y lj b) r where 0 <r δ b) δb) 0 ad the coefficiets y lj depedig o b may be determied substitutig y par i 8) which gives i the case l =0 j = b + σ) y 0 = C p +σ ). Writig out the leadig liear homogeeous terms of 8) for solutios that satisfy 9) we obtai the equatio r y + 3p + pσ ry p + σ)σ +) + y =0 r 0 + p p whose characteristic expoets are λ = p+σ < 0 ad λ = σ +)< 0. Therefore for each b 0 this problem has o other solutio tha y par r b). From Propositio we ca coclude that if g is a solutio of problem 3)-4) the g will have a discotiuity at r =0wheever k< that is whe σ<p.

3 Proceedigs of the 4th WSEAS Iteratioal Coferece o Fluid Mechaics Gold Coast Queeslad Australia Jauary The sigularity at r = r 0 I order to obtai the behavior of the solutio i the eighborhood of the sigular poit r = r 0 let us ow cosider the sigular Cauchy problem: g r) p g r)) = ar σ g r) 0 <r<r 0 0) gr 0 ) = r 0 r) g r) = 0. ) As we shall see the asymptotic behavior of the solutios whe r r 0 + may chage sigificatly depedig o the value of as it happes i the case p =). Here we will focus o the two mai cases: <<0ad <. The cases of = ad will ot be treated here. Details about these cases ca be foud i. Propositio Assume that i 0) p> σ 0 < 0 ad r 0 > 0. The Cauchy problem 0)-) has i the eighborhood of the sigular poit r = r 0 a oe parameter family of solutios that ca be represeted by: { g r c) if < gr c) = ; g 3 r c) if > ; ) as r r0 where c is the parameter ar σ 0 ) p ) g r c) = r p )+) 0 r) +)pσ )p+p )r 0 r 0 r)+c r 0 r) p+) +Or 0 r) +μ ) { } 3) μ =mi p+) ; ) p ar0 g 3 r c) = σ )+)+)c r 0 r)+ c r 0 r) + +y 0 c)r 0 r) + + o Proof. First cosider the case <. It ca be easily proved that i the eighborhood of r = r 0 a solutio of problem 0)-) may be give by g r) =C r 0 r) k + o) r r 0 5) p where k = > 0 k k ) = p+) < 0 ) ad C = arσ 0 )p p )+) ). If i 0)-) we perform the variable substitutio g r) =C r 0 r) k + y r) 6) p r 0 r) +) 4) we obtai the Cauchy problem i y ) p p+) +y p p r 0 r)y r 0 r) y p r 0 r) y 7) + p+) + y) ) r r 0 ) σ + y) =0 0 <r<r 0 y r 0 ) = r0 r) y r) =0 8) where r = r 0 is a regular sigular poit of equatio 7). Writig out the leadig liear homogeeous terms of 7) for solutios that satisfy 8) we obtai the equatio r 0 r) y p+p )+) r 0 r) y + p+) y =0 u r 0 9) whose characteristic expoets are λ = < 0 ad λ = p+) > 0. Whe = we obtai λ = ad therefore the differece betwee the characteristic expoets is a iteger. I this case the expasio of y has a differet form ad it is out of scope of the preset work. For ay other value of < the problem 7)-8) has a particular solutio y par r) holomorphic i r = r 0 ad if r r 0 y par r) = + l= y l r 0 r) l r 0 r δ δ > 0 where the coefficiets y l may be determied substitutig y par r) i 7). For example with l =we get + ) pσ y =. p )p + p ) r 0 The oe-parameter family of solutios of 7) 8) which reduces to y par whe the parameter c is zero ca be represeted by: y r c) =y r 0 r)+o r 0 r) ) +c r 0 r) λ + o) 0) whe r r0. Moreover takig ito accout that 7) is a asymptotically autoomous equatio this family may be represeted i the form of a coverget series: y r c) =y par r)+cr 0 r) λ + b r 0 r)+ b r 0 r) + +c r 0 r) λ c 0 + c r 0 r)+c r 0 r) + + r 0 r Δc) Δc) > 0 )

4 Proceedigs of the 4th WSEAS Iteratioal Coferece o Fluid Mechaics Gold Coast Queeslad Australia Jauary ad the oe-parameter family of solutios of the Cauchy problem 0) ) that we were lookig for i the case < ad is give by 3). The ) 3) has o other solutio tha y par r c). Summarizig we coclude that for ay <<0 the Cauchy problem 0) ) has a oe-parameter family of solutios that ca be represeted by 4). Let us ow cosider the case >. I the eighborhood of r = r 0 we shall look for a solutio of this problem i the form: g r) = C r 0 r)+c r 0 r) k + o ) g r) = C kc r 0 r) k + o ) g r) = k k ) C r 0 r) k + o ) r r0. ca be foud i ). From Propositio some coclusios ca be take cocerig the sigularity of the problem 0)) at r = r 0. Whe <<0 g is cotiuous at r = r 0 butg is t. Whe < the first derivative is also discotiuous at this poit. The same hapes whe = details about this case From 0)-) p+)k k ) C r 0 r) k C p r r = ar0 σ 0 which implies k = + > 0 k = + > 0 ad with C ) = c < 0 we get C = p. ar σ 0 p+)+)+)c Next we defie the fuctio y r) by g r) =C r 0 r)+c r 0 r) k + y r). Substitutig i 0)-) we obtai the Cauchy problem i y + c C kr 0 r) k c C r 0 r) k y p r 0 r) y k r 0 r) y + k k ) + y) ) σ = kk ) r r 0 + c C r 0 r) k + y) 0 <r<r 0 ) y r 0 ) = r 0 r) y r) = 0 3) where r = r 0 is a regular sigular poit of ) -3). For ay c 0 this problem has a particular solutio that ca be represeted by the coverget series y par r c) = + l=0j=0l+j y lj c)r 0 r) l+j+) where 0 r 0 r δ c) δ c) 0 ad the coefficiets y lj may be determied substitutig y par i ). If we write the leadig liear homogeeous terms of ) we obtai r 0 r) y +)r 0 r) y + + ) +)y =0 4) as r r0. This equatio has the characteristic expoets λ = <0ad λ = <0. 4 Lower ad Upper Solutios Let us cosider agai the boudary value problem g r) p g r)) = ar σ g r) 0 <r<r 0 5) r 0 + g 0) = 0 6) r r 0 with p> σ> ad <. gr) =0 7) Defiitio 3 We shall say that h r) is a lower resp. upper) solutio of the problem 5)-7) if h r) C 0r 0 C 0r 0 ) ad satisfies h r) p h r)) + ar σ h r) 0 resp. 0) 0 <r<r 0 h 0) 0 resp. 0) h r 0 )=0. We wat upper ad lower solutios to verify r 0 h r) =0 + h r) =0 hr) gr) = α 0 where the last coditio meas that the upper ad lower solutios are asymptotically equivalet to the exact solutio i the eighborhood of the sigular poit r = r 0. Takig ito accout the asymptotic behavior of the solutios at r =0ad r = r 0 studied i the previous sectios we will look for lower ad upper solutios i the form hr) =B r0 k r k) p 8) where k = σ+p ad B is a positive costat. Note that i the case p =we obtai the upper ad lower solutios itroduced i 7. We have k> which follows from σ> ad p>. We wat to defie

5 Proceedigs of the 4th WSEAS Iteratioal Coferece o Fluid Mechaics Gold Coast Queeslad Australia Jauary B i such a way that h satisfies equatio 5) that is h must verify h r) p h r)) = ahr) r σ 9) With this purpose we ote that = h r) p h r)) = Bkp +p)rσ r k 0 rk ) p Bkp ) p qr) 30) where qr) = p )k )r k k )p) r k r 0r 0. O the other had from 8) we have hr) = B r k 0 r k) p By substitutig 30) ad 3) ito 9) we obtai 3) ) Bkp p Bkp +p) qr)+ab =0. p 3) Note that if + +k )p =0 33) the qr) reduces to a costat: qr) r k 0k ) + p). 34) I this case that is if = k)p equatio 3) takes the form Bkp +p) Bkp ) p r k 0 k ) + p) +ab =0 35) Solvig 35) with respect to B we obtai ) ap ) B =. 36) r0 kkp) k ) Hece we coclude that whe the coditio 33) is satisfied the exact solutio of the problem 5)-7) has the form 8) with B give by 36). For values of p σ which do ot satisfy that coditio we caot fid a explicit formula for the exact solutio. However by aalyzig the fuctio qr) we ca obtai coditios o B uder which the fuctio h ca be a upper or a lower solutio accordig to Defiitio 3. This is summarized i the ext propositio. Propositio 4 Let σ> < p> ad let B > 0 be a costat. The the fuctio defied by 8) will be a lower solutio of the problem 5)-7) if B satisfies ) B B = a) p ) p +)σ+p) p r0 k whe k)p ad B B = a) ) σ+p p σ+p) σ+)r0 whe > k)p. ) a upper solutio of the problem 5)-7) if B satisfies B B whe > k)p ad B B whe k)p. We see that whe = k)p B = B = B where B is give by 36). As we have remarked above i this case we have a exact solutio i the form 8). The upper ad lower solutios ca be used to obtai suitable iitial approximatios for the applicatio of computatioal methods. O the other had we ca use them to apply the existece results obtaied i 0. Actually choosig a upper solutio g ad a lower solutio g of the form 8) the coditios of propositio. of that work are satisfied ad therefore the problem has at least oe solutio g such that g g g. Hece we coclude that the problem 5)-7) is solvable for p> σ> <. Note that the problem is also solvable i the case <0 but i this case the upper ad lower solutios must be sought with a differet form. 5 Numerical results Let us briefly describe the umerical algorithm we used i order to obtai the approximate solutios of problem 5)-7). Whe p>+σ for give values of a r 0 σ we have cosidered two regular Cauchy problems { g r) p g r)) = ar σ g r) 0 <r< r 0 gδ) =g δ b) g δ) =g δ b) { 37) g r) p g r)) = ar σ g r r) 0 <r<r 0 gr 0 δ) =g i r 0 δ c) g r 0 δ) =g i r 0 δ c) 38) which we deote Problem A ad Problem B respectively. Here i =or 3 depedig o δ is a small costat. The fuctios g g ad g 3 are computed

6 Proceedigs of the 4th WSEAS Iteratioal Coferece o Fluid Mechaics Gold Coast Queeslad Australia Jauary accordig to the formulae 5) 3) ad 4) respectively igorig the remaiders of the series. We solve problems A ad B for certai values of the parameters b ad c. With this purpose we use the program ND- Solve of Mathematica. The followig the idea of the shootig method we determie the values of b ad c so that g ad g are cotiuous at r = r 0 /. This gives a system of two oliear equatios which is solved by the Newto s method. Whe p +σ sice we have o sigularity at r =0 we solve the auxiliary problem B) startig from r 0 δ for a certai value of the parameter c. The we compute c by the shootig method requirig that the boudary coditio at r =0is satisfied. I our calculatios we have used δ =0.00. I figures we plot some approximate solutios of problem 5)-7) for differet values of σ r 0 a ad p Figure : Solutio of the boudary value problem with r 0 =5σ = 0.5= 5 3 a= p= Figure : Solutio of the boudary value problem with r 0 =5σ =0.5= 0.6a= p=.8. I table we compare the computed values of the solutios at r = r 0 / with the exact oes i some cases where the exact solutio is kow. The obtaied umerical results show that the used method has a accuracy of about 8 digits. Refereces: A. Acker ad R. Meyer A free boudary problem for the p-laplacia: uiqueess covexity ad sucessive approximatio of solutios Electroic Joural of Differetial Equatios pp. 0. p σ r 0 a) exact value appr. value ) ) ) ) ) ) Table : Compariso betwee the exact ad the approximate solutio at r = r 0 J.M. Urbao A free boudary problem with covectio for the p-laplacia Redicoti di Matematica Serie VII7 997 pp F. Cuccu B. Emamizadeh ad G. Porru Noliear elastic membraes ivolvig the p-laplacia operator Electroic Joural of Differetial Equatios pp A. J. Callegari ad A. Nachma Some sigular oliear differetial equatios arisig i boudary layer theory J. Math. Aalysis Applic pp N.B. Koyukhova P.M. Lima ad ad M.P. Carpetier Asymptotic ad umerical approximatio of a oliear sigular boudary value problem Tedêcias em Matemática Aplicada e Computacioal 3 00 pp P.M.Lima ad A.M.Oliveira Numerical methods ad error estimates for a sigular boudary-value problem Mathematical Modelig ad Aalysis 7 00 pp P. Lima ad L. Morgado Aalysis of sigular boudary value problems for a Emde-Fowler equatio Commuicatios o Pure ad Applied Aalysis pp F. Wog Existece of positive solutios for m- laplacia boudary value problems Appl. Math. Let. 999 pp F. Wog Uiqueess of positive solutios for Sturm-Liouville boudary value problems Proc. Amer. Math. Soc pp C. Ji J. Yi ad Z. Wag Positive radial solutios of p-laplacia equatio with sig chagig oliear sources Math. Meth. Appl Sci. to be published) P. Lima ad L. Morgado Aalytical-umerical ivestigatio of a sigular boudary value problem for a geeralized Emde-Fowler equatio submitted.

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

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