Invariance of a Partial Differential Equation of Fractional Order under the Lie Group of Scaling Transformations

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 7, ARTICLE NO AY Invariance of a Partial Differential Equation of Fractional Order under the Lie Group of Scaling Tranformation Evelyn Bucwar and Yuri Lucho Department of Mathematic and Computer Science, Free Unierity of Berlin, D Berlin, Germany Submitted by William F Ame Received February 16, 1998 In thi article a ymmetry group of caling tranformation i determined for a partial differential equation of fractional order, containing among particular cae the diffuion equation, the wave equation, and the fractional diffuion-wave equation For it group-invariant olution, an ordinary differential equation of fractional order with the new independent variable z xt i derived The derivative then i an ErdelyiKober derivative depending on a parameter It complete olution i given in term of the Wright and the generalized Wright function 1998 Academic Pre 1 INTRODUCTION In thi paper we preent the group-invariant olution of a partial differential equation of fractional order containing among particular cae the diffuion equation Ž 1, the wave equation Ž, and the ocalled fractional diffuion-wave equation Ž 1 Thi equation i obtained by replacing the firt- or econd-order time derivative in the diffuion or wave equation, repectively, by a generalized derivative of order 1, defined in the ene of the RiemannLiouville fractional calculu: už x, t už x, t D, x 0, t 0, D 0, 1, Ž 1 t x * bucwar@mathfu-berlinde; lucho@mathfu-berlinde X98 $500 Copyright 1998 by Academic Pre All right of reproduction in any form reerved

2 8 BUCKWAR AND LUCHKO with u x, t t n už x, t,, t n n n 1 t n1 H Ž t už x, d, Ž n t n 0 n 1 n Such equation have already appeared in text on phyic and mathematic Mathematical apect of the boundary value problem for thi equation and for more general one and their application in phyic have been treated in paper by Engler 1, Fujita, Mainardi 101, Pru 16, Saichev and Zalavi 17, Schneider and Wy 18, and Wy 0 In a erie of paper Žee 1 and the reference there the two baic boundary value problem for the fractional diffuion-wave equation Ž 1 were conidered Ž a Cauchy problem: už x,0 gž x, x ; už, t 0, t 0, b Signaling problem: už x,0 0, x 0; už 0, t hž t, už, t 0, t 0 By uing integral tranform Ž Laplace, Fourier, or Mellin type, the Green function G Ž x, t, and G Ž x, t, c for thee problem were expreed in term of ome pecial function Žof Wright or MittagLeffler type, Fox H-function with the imilarity argument z xt ' DŽ x 0, t 0 We will explain thi fact in our article and determine by uing the method of group analyi that Eq Ž 1 i in fact invariant under a ymmetry group of caling tranformation The method of group analyi of differential equation began with the wor of Sophu Lie more than 100 year ago Roughly peaing, a ymmetry group of a ytem of differential equation i a group that tranform olution of the ytem to other olution For partial differential equation one can determine pecial type of olution, which are invariant under ome ubgroup of the full ymmetry group of the ytem Thee group-invariant olution are found by olving a reduced ytem of equation having fewer independent variable than the original ytem In recent year the idea of the Lie group approach have been extended to difference equation and to integro-differential equation We refer to the handboo 46 for reult and literature and epecially to 3 for the cae of integro-differential equation

3 INVARIANCE OF A DIFFERENTIAL EQUATION 83 In 14 one can find the full ymmetry group of the diffuion and the wave equation Here we will focu on the o-called imilarity method, developed by G D Birhoff in the 1930, and conider the pecial cae of caling tranformation In the general cae Ž one cannot ue the chain rule for the operation of differentiation to get a reduced equation for the cale-invariant olution of Ž 1 a in the cae of partial differential equation Ž 1, In pite of thi we will tranform Eq Ž 1 into an ordinary differential equation of fractional order with the new independent variable z xt The derivative then i an ErdelyiKober derivative depending on a parameter For 1 and Žthe diffuion and the wave equation thi reduced equation correpond to the ordinary differential equation well nown in the literature Žee 14 We preent alo the general olution of the obtained differential equation of fractional order with the ErdelyiKober derivative in term of the Wright and the generalized Wright function THE EQUATION FOR THE SCALE-INVARIANT SOLUTIONS In thi article we will apply the imilarity method to the partial differential equation of fractional order Ž 1 At firt we determine a ymmetry group T of caling tranformation for thi equation We have Ž x x, b t t, u u : už x, t už x, t, x x t n nb n už x, t už x, t, for n t In the cae n 1 n, n, uing variable ubtitution we get 1 n t n1 už x, t H Ž t už x, x d n t Ž n t 0 nb n b t b n1 b H Ž t už x, d Ž n t n 0 b n t n1 H Ž t už x, d Ž n t n 0 b už x, t t

4 84 BUCKWAR AND LUCHKO It follow that u u u u b D D 0, if b t x t x Thi mean that the invariant of the caling tranformation T for Eq Ž 1 ha the form Ž x, t, u xt 1 b xt A in the cae of partial differential equation, we ue the tranformation už x, t Ž z, z xt to determine the cale-invariant olution of the partial differential equation of fractional order Ž 1 Let u olve for the partial derivative u xx and partial derivative of fractional order u t, 0, in term of partial derivative of We find u z t, u x xx Ž z t Ž In the cae 1 and, the derivative u t can be expreed in term of a linear ordinary differential operator, applied to, of order 1 or, repectively Žee 14 In the general cae we have THEOREM 1 The partial deriatie of fractional order u t, 0 of the function u x, t z, z xt i gien by the relation u 1, t Ž P Ž z, z 0, Ž 3 t with the ErdelyiKober fractional differential operator P, of order 0: 1 1 d,, Ž P g Ž z Ł j z Ž K g Ž z, Ž 4 j0 dz 1,, ½,, where H Ž u 1 u g Ž zu du, 0,, Ž K g Ž z Ž 1 gž z, 0 i the ErdelyiKober fractional integral operator

5 INVARIANCE OF A DIFFERENTIAL EQUATION 85 Ž Proof Let n 1 n, n 1,, 3, We then have z xt where / u n 1 t n1 H Ž t Ž x d Ž 5 n ž t t Ž n 0 n n 1, n t Ž K Ž z, t n 1 1, n n1 n1 H 1 K z u 1 u zu du n i the ErdelyiKober fractional integral operator Ž In view of the relation z xt we arrive at d 1 t Ž z tx t Ž z z Ž z, Ž 6 t dz n n 1, n ž t Ž K Ž z / t n / n1 d 1, n Ł j0 dz n1 d n1 1, n t n z Ž K Ž z n1 ž t dz t 1 j z Ž K Ž z t Ž P 1, Ž z In the cae n 1,, 3, we ue the relation Ž 6 once again and get Ž n z xt / 1 dz / n1 n d Ł ž dz / j0 u n n-1 1 Ž z t t Ž z n n1 ž t t t t n1 n d t z z n1 t ž n n 1n, n t 1 n j z z t P z

6 86 BUCKWAR AND LUCHKO Now we ubtitute the expreion Ž and Ž 3 into the partial differential equation of fractional order Ž 1, and we find t Ž P 1, z Dt Ž z, z 0, D 0 A could be expected, thi equation i equivalent to one in which the variable t doe not occur: Ž P 1, z D Ž z, z 0, 1, Ž 7 with the ErdelyiKober fractional differential operator P 1, Remar 1 Following from the definition of the ErdelyiKober fractional differential operator Ž 4, in the cae n the reduced equation Ž 7 for the cale-invariant olution i a linear ordinary differential equation of order max, n 4 In the cae 1 Ž the diffuion equation thi equation tae the form 0,1 1 Ž P Ž z z Ž z D Ž z, and in the cae Ž the wave equation the form Ž P 1, z z z z z D 1 Ž z One can find a complete dicuion of thee cae in, for example, 14 3 THE SCALE-INVARIANT SOLUTIONS In thi part of our article we preent the exact form for the cale-invariant olution of the partial differential equation of fractional order Ž 1 Note here that uch olution are well nown for the diffuion equation Ž 1 and for the wave equation Ž Žee 14 In the general cae, to the author nowledge, equation of type Ž 7 have not yet been conidered in the literature on fractional calculu and fractional differential equation It turn out that the olution of Eq Ž 7 have different tructure in the cae 1,, and We begin with the firt cae: THEOREM The cale-inariant olution of the partial differential equa- Ž Ž tion of fractional order 1 in the cae 1 hae the form z xt Ž z C1 1Ž z C Ž z, with arbitrary real contant C1 and C, Ž 8 z W z' D ;,1, z W z' 1 D ;,1 Ž 9

7 INVARIANCE OF A DIFFERENTIAL EQUATION 87 with the Wright function z WŽ z;, Ý!Ž 0 Proof Let u conider the cae 1 Then Eq Ž 7 for the cale-invariant olution i the linear ordinary differential equation of order Žee Remar 1, which ha two independent olution Since the function Ž z 1 and Ž z are entire function in the complex plane for Žee 19, we immediately have 1 Ž ' z D 1 ' Ý D 1 z z z 1! 1 Ž ' 1 z D Ý 1! 1 Ž z' D Ý 1!Ž D z, 1 and, analogouly, z z D z, 1 which complete the proof in the cae 1 Ž In the cae 1 after ubtituting y, Eq 7 tae the form 1 yž z Ž L yž z cz 1 c, z 0, D 0, Ž 10 D where c, c 1 are arbitrary real contant, and Ž L yž z P 1, I y Ž z, 0 with the RiemannLiouville fractional integral operator 1 t 1 I0y Ž t Ž t yž d H Ž 0

8 88 BUCKWAR AND LUCHKO Equation Ž 10 i a fraction integro-differential equation of the econd ind of order 0, and it ha a unique olution if the right-hand ide ha the form fž z czc Žee 9, 1 1 Thi mean that the initial equation Ž 7 ha no more than two linearly independent olution Let u prove that the function Ž 9 atify Eq Ž 7 Firt we have 1 ' Ž z D Ž 1 1 Ž z' D 1 Ý Ý D!Ž 1 Ž D 0!Ž 1 Ž Ž 11 z 1 W z' D ;,1, D 1 ' Ž z D Ž 1 1 Ž z' D Ý Ý D!Ž 1 Ž D 0!Ž 1 Ž Ž 1 z 1 W z' D ;,1 D 1, Unfortunately, it i not poible to apply the operator P term by term to the power erie Ž 9 becaue of divergence of the correponding integral Intead we ue here method baed on the Pareval equality for the Mellin integral tranform and the well-nown integral repreentation of the Wright function LEMMA 1 Let z and z be defined by 9Then the relation 1 hold for 1 Proof 1, Ž P Ž z W z' D ;,1, 1, Ž P z W z' ž D ;,1 / 1 In the cae 1, the left-hand ide of Eq Ž 7 ha the form d d 1, 1, P Ž z 1 z z K Ž z, dz dz Ž 13

9 INVARIANCE OF A DIFFERENTIAL EQUATION 89 where 1 1, 1 3 H 1 K z t 1 t zt dt i the ErdelyiKober fraction integral operator According to Wright Ž19, e Z 1 1 z A0 O Z, for z, 14 ' Z with 11 1 Ž 1 Z Ž 1 Ž z' D,, A0, ' ' Ž z OŽ z, for z Ž 15 1, It follow from thee two relation and 13 that the operator P i well defined on the function Ž z and Ž z 1 To get it effect on the function Ž z, we repreent the operator Ž 13 1 in the form Ž7, Ž 1, P Ž z Ž z d, Ž 16 i 1 Ž H L i where H 1 Ž Ž t t d 0 i the Mellin integral tranform of Ž t and L C: Ž 0 4 i By uing reult from 13 we arrive at Ž 1' 1 Ž D Ž 17 1 Ž

10 90 BUCKWAR AND LUCHKO We have then 1 Ž 1 Ž 1, Ž P 1 Ž z H 1 Ž z d i L i Ž 1 Ž 1 Ž 1 Ž Ž H i 1 Ž 1 Ž L i ' H Ž ' Ž 1 D z d 1 z D d i L i 1 The lat integral i a particular cae of the Fox H-function Žee 15 Repreenting it a a erie, we get the tatement of our Lemma 1 for the function Ž z 1 Thi method could not be applied to the function Ž z, ince the Mellin integral tranform of thi function doe not exit Intead we ue the modified integral repreentation of the Wright function Žee 19, which ha the form 1 e z 1 exp z ' H Ž D 4 1 d i L Here the contour L i the cut in the complex plane coinciding with the negative real emi-axi Applying now the operator Ž 13 and changing the order of integration, we have / 1 d d 1, Ž P Ž z ž 1 z z Ž 1 dz dz Ž 18 1 e ' i H Ž H L 1 Ž 4 t 1 t exp z t D 1 dt d Uing the Pareval equality for the Mellin integral tranform from 13, we have ' H 4 Ž t 1 t exp z t D dt Ž Ž 1 Ž Ž H Ž z ' D d, Ž 3 i L i Ž 3 Ž Ž 19

11 INVARIANCE OF A DIFFERENTIAL EQUATION 91 where 4 L, L C: 0 Ž 1 i The contour Li in the lat repreentation can be tranformed into the contour L, which goe from the point ai, a 0 to the point bi, b 0 in uch a manner that Ž C, Ž 0 for all L, C i a contant, and the point 1,, 3, and,4,6, are on the right ide of thi contour, wherea the point 0 i on the left ide We ubtitute the repreentation Ž 19 Žwith the contour L intead of L into Ž 18 and, uing the identity i d d 1 z z z 1 z, dz dz we arrive at Ž P 1, Ž z 1 1 e H Ž 1 i L 1 1 Ž Ž H Ž z ' D dd i 1 Ž L Changing the order of integration in the lat formula, uing the well-nown repreentation of the -function Žee H e d, if Ž 1, Ž i L and repreenting the obtained Fox H-function a a erie, we finally get 1 1 Ž 1 Ž P Ž 1, Ž Ž z H Ž 1 i 1 Ž L 1 ' Ž1Ž Ž z D H e d d i L 1 1 Ž H Ž 1 i 1 Ž ' z D L 1 Ž z' D Ý Ž 1! 1 1 W z' D ;,1

12 9 BUCKWAR AND LUCHKO Thi lat relation complete our proof of Lemma 1 The tatement of Theorem follow now by comparing relation Ž 11 and Ž 1 and the reult proven in Lemma 1 Remar In the cae 1 Ž ie, for the diffuion equation the cale-invariant olution Ž 9 have the form n1 ' Ž z D Ž z' D 1 Ý Ý 0!Ž 1 Ž 1 n0 Ž n 1!Ž 1 n n n1 Ž 1 zž ' D ' Ý Ž Ž ' Ž n! Ž n 1 z erf z D, and, analogouly, n0 Ž Ž ' z 1 erf z D, where erf i the error function, which i in agreement with the nown reult Žee 14 Remar 3 The cale-invariant olution Ž 9 can alo be ued in the cae Ž wave equation In thi cae we hould write them in the form Ž z D u1ž z Ž 1Ž z Ž z Ý, 0 Ž 1 Ž 1 už z D ž 1 / Ž Ž z 1Ž z Ž z D 1 z Ý Ž Ž 1 0 Uing the relation Ž Ž 1 1 Ž 1 Ž 1 Ž!, 0,1,, we arrive for at u1ž z 1, z D! Ž z D už z z Ý z Ý Ž 1! 1 0 ' ' ' 0 D D z ln, z ' D D z

13 INVARIANCE OF A DIFFERENTIAL EQUATION 93 Remar 4 In a erie of paper Žee 1 and reference there ome boundary-value problem for Eq Ž 1 were conidered Uing the aymptotic formulae Ž 14 and Ž 15, we can get the boundary value of ignaling type for our cale-invariant olution of Eq Ž 1 : Now we conider the cae : u1ž x,0 1 0, x 0, u1ž 0, t 1Ž 0 1, t 0, u1ž, t 1 0, t 0, už x,0, x 0, už 0, t Ž 0 1, t 0, už, t, t 0 THEOREM 3 The cale-inariant olution of the partial differential equation of fractional order Ž 1 in the cae,, hae the form Ž z xt Ý j j j j0 Ž z C Ž z with arbitrary real contant C, 0 j, Ž 0 Ž 1,1, Ž 1 j, Ž Ž1j jž z z 1 ; Dz, Ž 1 Ž j, where p q Ž a 1, A 1,, Ž a p, A p ; z b, B b, B q Ž i the generalized Wright function ee 15 : a, A,, a, A p Ž 1 1 Ž p p Ł i1ž ai A z Ý i1 i i ; z Ž b, B b, B Ł Ž b B! p q q 1 1 q q 0

14 94 BUCKWAR AND LUCHKO Ž 1, Proof After ubtituting P z y z, Eq 7 ha the form 4Ž Ž1j yž z DŽ L yž z Ý cz j, j0 where c,0j are arbitrary real contant and j d 1, L y z K y z dz Thi lat equation i an integro-differential equation of the econd ind and of order 0 that ha a unique olution if the right-hand ide 4Ž Ž1j ha the form f z Ý cz Žee 9, 1 j0 j Thi mean that Eq Ž 7 ha no more than 1 linearly independent olution Let u prove that the function Ž z,0j j are it olution According to 15 the generalized Wright function are entire function in the cae under conideration We can differentiate the correponding erie term by term, and we have / j Ž z ž Ž 1 j 3 Ž 1 j z ž / Ž 1 j Ý Ž Dz 0 Ž j ž / 4 Ž 1 j 4Ž Ž1j z Ý Ž Dz Ž j 0 4Ž Ž1j Ž 3 1, We can alo apply the operator P term by term to the erie Ž 1 Specifying the well-nown formula for the effect of the ErdelyiKober fractional derivative on a power function Žee 7 for our cae, we get 1 Ž p 1, p p Ž P t Ž z z,1 p 0 1 Ž p

15 INVARIANCE OF A DIFFERENTIAL EQUATION 95 Uing the lat formula, we arrive at 1, Ž Ž1j j 0 Ý P z z Dz z Dz Ž Ž Ž 1 j Ž j Ž j Ž j Ž Ž1j Ý Ž Dz 0 Ž Ž 1 j Ž j 4Ž Ž1j Ý Ž Dz 0 4 Ž Ž 1 j Ž j Comparing thi relation with 3, we get the tatement of our Theorem 3 Finally we conider the cae of natural number n Uing exactly the ame method a in the proof of Theorem 3, we have THEOREM 4 The cale-inariant olution of the partial differential equation Ž n Ž n hae the form z xt n u Du, t 0, x 0, D 0 xx t n n1 Ý j j j j0 Ž z C Ž z with arbitrary real contant C, 0 j n 1, Ž 1,1, Ž 1 j, Ž nž1j jž z z 1 n ; Dz, Ž n j, n n1ž z 1 Ž 4 0 j n,

16 96 BUCKWAR AND LUCHKO Remar 5 In the cae of n we can ue the reult of Theorem 4 for z ' D We have 1Ž z 1, 1 1 Ž Dz Ž! 1 Ž ' Dz 1 0 Ý ' Ý 0 Ž D 0 1 z z 1 z ' D ln ' D z ' D Ž Combining thee relation with Remar 3, we get the well-nown ee 14 cale-invariant olution of the wave equation: x ' 1 ' Dt u x, t xt C C ln x Dt ACKNOWLEDGMENTS The author are incerely grateful to Prof Rudolf Gorenflo for many ueful dicuion and to the referee for hi comment and uggetion Partial financial upport wa provided by the Reearch Commiion of the Free Univerity of Berlin Ž Project Convolution REFERENCES 1 H Engler, Similarity olution for a cla of hyperbolic integrodifferential equation, Differential Integral Equation 10, No 5 Ž 1997, Y Fujita, Integrodifferential equation which interpolate the heat and the wave equation, Oaa J Math 7 Ž 1990, 30931, Yu N Grigoriev and S V Meleho, Group analyi of inetic equation, Ruian J Numer Anal Math Modeling 10, No 5 Ž 1995, N H Ibragimov Ž Ed, CRC Handboo of Lie Group Analyi of Differential Equation, Vol 1, Symmetrie, Exact Solution and Conervation Law, CRC Pre, Boca Raton, FL, N H Ibragimov Ž Ed, CRC Handboo of Lie Group Analyi of Differential Equation, Vol, Application in Engineering and Phyical Science, CRC Pre, Boca Raton, FL, N H Ibragimov Ž Ed, CRC Handboo of Lie Group Analyi of Differential Equation, Vol 3, New Trend in Theoretical Development and Computational Method, CRC Pre, Boca Raton, FL, V Kiryaova, Generalized fractional calculu and application, Pitman Re Note in Math 301 Ž J D Logan, An Introduction to Nonlinear Partial Differential Equation, Wiley, New Yor, 1994

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