Particular solutions of a certain class of associated Cauchy-Euler fractional partial differential equations via fractional calculus

Transcription

1 Lin t al. Boundary Valu Problms 2013, 2013:126 R E S E A R C H Opn Accss Particular solutions of a crtain class of associatd Cauchy-Eulr fractional partial diffrntial quations via fractional calculus Shy-Dr Lin *, Chia-Hung Lu and Shou-Mi Su * Corrspondnc: shydr@cycu.du.tw Dpartmnt of Applid Mathmatics, Chung Yuan Christian Univrsity, Chung-Li, 32023, Taiwan, R.O.C. Abstract In rcnt yars, various oprators of fractional calculus that is, calculus of intgrals and drivativs of arbitrary ral or complx ordrs) hav bn invstigatd and applid in many rmarkably divrs filds of scinc and nginring. Many authors hav dmonstratd th usfulnss of fractional calculus in th drivation of particular solutions of a numbr of linar ordinary and partial diffrntial quations of th scond and highr ordrs. Th purpos of this papr is to prsnt a crtain class of th xplicit particular solutions of th associatd Cauchy-Eulr fractional partial diffrntial quation of arbitrary ral or complx ordrs and thir applications as follows: Ax 2 2 u u + Bx x2 x + Cu = u M α t α + u N β t β, P 1) A 2 u x + B u 2 x + Cu = u M α t α + u N β t β, P 2) whr u = ux, t); A, B, C, M, N, α and β ar arbitrary constants. MSC: 26A33; 33C10; 34A05 Kywords: fractional calculus; diffrntial quation; partial diffrntial quation; gnralizd Libniz rul; analytic function; indx law; linarity proprty; principl valu; initial and boundary valu 1 Introduction, dfinitions and prliminaris Th subjct of fractional calculus that is, drivativs and intgrals of any ral or complx ordr) has gaind importanc and popularity during th past two dcads or so, du mainly to its dmonstratd applications in numrous smingly divrs filds of scinc and nginring cf. [1 15]). By applying th following dfinition of a fractional diffrntial that is, fractional drivativ and fractional intgral) ofordrν R, manyauthors hav obtaind particular solutions of a numbr of familis of homognous as wll as nonhomognous) linar fractional diffr-intgral quations. In this papr, w prsnt a dirct way to obtain xplicit solutions of such typs of th associatd Cauchy-Eulr fractional partial diffrntial quation with initial and boundary valus. Th rsults ar a coincidnc that th solutions ar obtaind by th mthods apply Lin t al.; licns Springr. This is an Opn Accss articl distributd undr th trms of th Crativ Commons Attribution Licns which prmits unrstrictd us, distribution, and rproduction in any mdium, providd th original work is proprly citd.

2 Lintal. Boundary Valu Problms 2013, 2013:126 Pag 2 of 11 ing th Laplac transform with th rsidu thorm. In this papr, w prsnt som usful dfinitions and prliminaris for th papr as follows. Dfinitions 1.1 cf. [6 10]) If th function f z) is analytic and has no branch point insid and on C,whr C := {C, C + } 1.1) C is an intgral curv along th cut joining th points z and + it z), C + is an intgral curv along th cut joining th points z and + it z), and f ν z)= C f ν z):= Ɣν +1) C f ζ ) dζ ζ z) ν+1 ν R/Z ; Z := { 1, 2, 3,...} ) 1.2) { f n z):= lim fν z) } n N := {1,2,3,...} ), 1.3) ν n whr ζ z, π argζ z) π for C, 1.4) and 0 argζ z) 2π for C +, 1.5) thn f ν z)ν >0)issaidtobthfractional drivativ of f z)ofordrν and f ν z)ν <0)is said to b th fractional intgral of f z) ofordr ν, providd that f ν z) < ν R). 1.6) First of all, w find it is worthwhil to rcall hr th following usful lmmas and proprtis associatd with th fractional diffr-intgration which is dfind abov. Lmma 1.1 Linarity proprty) If th functions f z) and gz) ar singl-valud and analyticinsomdomain C, thn k 1 f + k 2 g) ν = k 1 f ν + k 2 g ν ν R, z ) 1.7) for any constants k 1 and k 2. Lmma 1.2 Indx law) If th function f z) is singl-valud and analytic in som domain C, thn f μ ) ν = f μ+ν =f ν ) μ f μ 0,f ν 0,μ, ν R, z ). 1.8)

3 Lintal. Boundary Valu Problms 2013, 2013:126 Pag 3 of 11 Lmma 1.3 Gnralizd Libniz rul) If th functions f z) and gz) ar singl-valud and analytic in som domain C, thn f g) ν = n=0 ) ν f ν n g n ν R, z ), 1.9) n whr g n isthordinarydrivativofgz) of ordr n n N 0 := N {0}), it bing tacitly assumd for simplicity) that gz) is th polynomial part if any) of th product f z)gz). Lmma 1.4 Cauchy s rsidu thorm) Lt b a simpl connctd domain, and lt C b a simpl closd positivly orintd contour that lis in. If f is analytic insid C and on C, xpct at th point z 1, z 2,...,z n that li insid C, thn C f z) dz = n Rs[f, z k ]. k=1 I) If f has a simpl pol at z 0, thn Rs[f, z 0 ]= lim z z0 z z 0 )f z); II) If f has a pol of ordr k at z 0, thn Rs[f, z 0 ]= 1 d k 1 lim k 1)! z z 0 dz z z 0) k f z). k 1 Proprty 1.1 cf. [6 10]) For a constant a, az ) ν = aν az a 0,ν R, z C). 1.10) Proof Th proofs btwn ν is not an intgr and ν is an intgr ar not coincidnt, so w mntion th proof as follows. In cas of arg a < π/2, w hav C aζ az ) v = C az ) Ɣv +1) = v = a v az Ɣv +1) 0+) ξ v+1) ξ dξ φ = arg a) iφ = a v az Ɣv +1) 0+) dζ put ζ z = η and aη = ξ, arg η π) ζ z) v+1 ξ v+1) ξ dξ for arg a < π/2 sinc 0+) ξ v+1) ξ dξ = Ɣv+1). In cas of π/2 < arg a π,whav az ) v = C + az ) v = Ɣv +1) C+ aζ dζ ζ z) v+1 put ζ z = η and aη = ξ,0 arg η 2π) for φ < π ) = a v az 2

4 Lintal. Boundary Valu Problms 2013, 2013:126 Pag 4 of 11 = a v az Ɣv +1) 0+) ξ v+1) ξ dξ φ = arg a) iφ = a v az Ɣv +1) = a v az. 0+) ξ v+1) ξ dξ for π2 < φ π ) Thrfor w hav Proprty 1.1 for arbitrary a 0. Proprty 1.2 For a constant a, az ) ν = iπν a ν az a 0,ν R, z C). 1.11) Proprty 1.3 For a constant a, z a ) ν = iπν Ɣν a) Ɣ a) za ν ν R, z C, Ɣν a) Ɣ a) ). < 1.12) Proprty 1.4 cf. [2, 16, 17]) Th fractional drivativ of a causal function f t) is dfind by d α dt f t):= f n) t) if α = n N, α 1 t f n) τ) Ɣn α) 0 dτ if n 1<α < n, t τ) α+1 n whr f n) t) dnots th ordinary drivativ of ordr n and Ɣ is th gamma function. Th Laplac transform of a function f t) is dnotd as L { f t) } s)= ) st f t) dt, 1.14) whr s is th Laplac complx paramtr. W rcall from th fundamntal formula cf. [16]) { d α } L dt f t) s)=s α L { f t) } s) α 2 Main rsults Thorm 2.1 Th fractional partial diffrntial quation n s α k f k 1) 0), n 1<α n, n N. 1.15) k=1 Ax 2 2 u x 2 + Bx u x + Cu = M α u t α + N β u t β, 2.1) with u = ux, t), n 1<α, β n, n N and A 0),B, C, M, Narconstants, has its solutions of th form givn by a) ux, t)= λt[ k 1 x A B+ A B) 2 4AC Mλ α Nλ β ) + k 2 x A B whn th discriminant A B) 2 4AC Mλ α Nλ β )>0; A B) 2 4AC Mλ α Nλ β ) ], 2.2)

5 Lintal. Boundary Valu Problms 2013, 2013:126 Pag 5 of 11 b) ux, t)= λt[ k 1 x m + k 2 x m ln x ], c) whn th discriminant A B) 2 4AC Mλ α Nλ β )=0, and th roots m 1, m 2 of Equation 2.5) ar rpatd; that is, m 1 = m 2 = m; ux, t)= λt[ k 1 x a+bi + k 2 x a bi], whn th discriminant A B) 2 4AC Mλ α Nλ β )<0, and a + bi, a bi ar th conjugat pair roots of Equation 2.5). Proof Suppos that ux, t)=x m λt.whav u x = mxm 1 λt, 2 u x = mm 2 1)xm 1 λt, α u t = α λα x m λt, and β u t = β λβ x m λt. 2.3) So that th givn quation 2.1)bcoms Amm 1)x m λt + Bmx m λt + Cx m λt Mλ α x m λt Nλ β x m λt =0. 2.4) Equation 2.4) lads to th auxiliary quation Amm 1)+Bm + C Mλ α Nλ β =0 or Am 2 A B)m + C Mλ α Nλ β) =0 ). 2.5) That is, and m 1 = A B + A B) 2 4AC Mλ α Nλ β ) m 2 = A B A B) 2 4AC Mλ α Nλ β ) ar th two roots of th auxiliary quation 2.5). Thus, ux, t)= 2 i=1 k ix m i λt is a solution of th fractional partial diffrntial quation 2.1)whnvrm i i = 1, 2) is a solution of th auxiliary quation 2.5). Thr ar thr diffrnt cass to b considrd, dpnding on whthr th roots of this quadratic quation 2.5) ar distinct ral roots, qual ral roots rpatd ral roots), or

6 Lintal. Boundary Valu Problms 2013, 2013:126 Pag 6 of 11 complx roots roots appar as a conjugat pair). Th thr cass ar du to th discriminant of th cofficints A B) 2 4AC Mλ α Nλ β ). CasI:DistinctralrootswhnA B) 2 4AC Mλ α Nλ β )>0). Lt m 1 and m 2 dnot th ral roots of Equation 2.5)suchthatm 1 m 2. Thn th gnral solution of Equation 2.1)is ux, t)= λt k 1 x m 1 + k 2 x m 2 ), whr k i i =1,2)arconstants. Cas II: Rpatd ral roots whn A B) 2 4AC Mλ α Nλ β )=0). If th roots of Equation 2.5) ar rpatd, that is, m 1 = m 2 = m, thn th gnral solution of Equation 2.1)is ux, t)= λt k 1 x m + k 2 x m ln x ), whr k i i =1,2)arconstants. Cas III: Conjugat complx roots whn A B) 2 4AC Mλ α Nλ β )<0). If th roots of Equation 2.5) ar th conjugat pair m 1 = a + bi and m 2 = a bi,thn a solution of Equation 2.1)is ux, t)= λt k 1 x a+bi + k 2 x a bi), whr k i i =1,2)arconstants. In gnral, ux, t)= λt[ k 1 x A B+ A B) 2 4AC Mλ α Nλ β ) + k 2 x A B A B) 2 4AC Mλ α Nλ β ) ] forms a fundamntal solution, whr k 1, k 2 and λ ar constants. Thorm 2.2 Th fractional partial diffrntial quation A 2 u x 2 + B u x + Cu = M α u t α + N β u t β, 2.6) with u = ux, t), n 1<α, β n, n N and A 0),B, C, M, Narconstants, has its solutions of th form givn by a ) b ) ux, t)= λt[ k 1 B+ B 2 4AC Mλ α Nλ β ) )x + k 2 B whn th discriminant B 2 4AC Mλ α Nλ β )>0; B 2 4AC Mλ α Nλ β ) )x ], 2.7) ux, t)= λt[ k 1 mx + k 2 x mx], whn th discriminant B 2 4AC Mλ α Nλ β )=0, and th roots m 1, m 2 of Equation 2.10) ar rpatd; that is, m 1 = m 2 = m;

7 Lintal. Boundary Valu Problms 2013, 2013:126 Pag 7 of 11 c ) ux, t)= λt[ k 1 x a+bi)x + k 2 x a bi)x], whn th discriminant B 2 4AC Mλ α Nλ β )<0, and a + bi, a bi ar th conjugat pair roots of Equation 2.10). Proof Th similarity btwn th forms of solutions of Equation 2.1) and solutions of a linar quation with constant cofficints of Equation 2.6)is not just a coincidnc. Suppos that ux, t)= mx λt.whav u x = mmx λt, 2 u x = 2 m2 mx λt, α u t = α λα mx λt, β u t = β λβ mx λt. and 2.8) So that th givn quation 2.6)bcoms Am 2 mx λt + Bm mx λt + C mx λt Mλ α mx λt Nλ β mx λt =0. 2.9) If m 1 and m 2 ar th two roots of th auxiliary quation Am 2 + Bm + C Mλ α Nλ β =0, 2.10) thn and m = m 1 = B + B 2 4AC Mλ α Nλ β ) m = m 2 = B B 2 4AC Mλ α Nλ β ). ThanalysisofthrcassissimilartoThorm2.1, w can obtain ach solution of th formsasfollows: ux, t)= λt k 1 m 1x + k 2 m 2x ) with m 1 m 2, two distinct ral roots, ux, t)= λt k 1 mx + k 2 x mx ) with m 1 = m 2 = m, rpatd ral roots, and ux, t)= λt k 1 a+ib)x + k 2 a ib)x ) with th conjugat complx roots. Rmark Th constant λ in Equations 2.2) and2.7) can b solvd dirctly by constant initial valu and constant boundary valus or by th numrical mthods). Corollary 2.1 Th fractional partial diffrntial quation Ax 2 2 u x 2 + Bx u x + C = M α u t α, 2.11)

8 Lintal. Boundary Valu Problms 2013, 2013:126 Pag 8 of 11 with u = ux, t), n 1<α n, n N and A 0),B, C, Marconstants, hasitssolutionsof th form givn by a ) ux, t)= λt[ k 1 x A B+ A B) 2 4AC Mλ α ) + k 2 x A B A B) 2 4AC Mλ α ) ], 2.12) b ) whn th discriminant A B) 2 4AC Mλ α )>0; ux, t)= λt[ k 1 x m + k 2 x m ln x ], c ) whn th discriminant A B) 2 4AC Mλ α )=0, and th roots m 1, m 2 of Equation 2.5) with N =0ar rpatd; that is, m 1 = m 2 = m; ux, t)= λt[ k 1 a+bi + k 2 a bi], whn th discriminant A B) 2 4AC Mλ α )<0, and a + bi, a bi ar th conjugat pair roots of Equation 2.5) with N =0. Corollary 2.2 Th fractional partial diffrntial quation A 2 u x 2 + B u x + Cu = M α u t α, 2.13) with u = ux, t), n 1<α n, n N and A 0),B, C, Marconstants, hasitssolutionsof th form givn by a ) ux, t)= λt[ k 1 B+ B 2 4AC Mλ α ) )x + k 2 B B 2 4AC Mλ α ) )x ], 2.14) b ) whn th discriminant B 2 4AC Mλ α )>0; ux, t)= λt[ k 1 mx + k 2 x mx], c ) whn th discriminant B 2 4AC Mλ α )=0, and th roots m 1, m 2 of Equation 2.10) with N =0ar rpatd; that is, m 1 = m 2 = m; ux, t)= λt[ k 1 a+bi)x + k 2 a bi)x], whn th discriminant B 2 4AC Mλ α )<0, and a + bi, a bi arthconjugatpair roots of Equation 2.10) with N =0.

9 Lintal. Boundary Valu Problms 2013, 2013:126 Pag 9 of 11 3 Exampls Exampl 3.1 If th two-dimnsional harmonic quation 1 2 u = 0 is transformd to plan polar coordinats r and θ,dfindbyx = r cos θ, y = r sin θ, it taks th form 2 u r + 1 u 2 r r u =0, 3.1) r 2 θ2 thn it has solutions of th form ur, θ)= k 1 r iλ + k 2 r iλ) λθ, whr k 1, k 2 and λ ar constants. Solution Equation 3.1) is coincidnt to r 2 2 u r + r u 2 r + 2 u θ =0. 2 W hav th solution ur, θ)= k 1 r iλ + k 2 r iλ) λθ by taking A =1,B =1,C =0,M = 1,N =0andα =2inThorm2.1. Exampl 3.2 Th fractional partial diffrntial quation 2 u x + u 2 x = α u with n 1<α n, n N. t α Solution Putting A =1,B =1,C =0andM =1inCorollary2.2, w obtain th solution ux, t)= λt[ c λ α 2 )x + c λ α 2 )x ], whr th discriminant 1 + 4λ α >0. If 1 + 4λ α =0, ux, t)= 1 2 x t) c 1 + c 2 x). Th analysis of th cas 1 + 4λ α <0issimilartoThorm2.2. Exampl 3.3 Th fractional partial diffrntial quation x u x x u 2 x = 2 2 u t 1 2 with u1,t) =u, t) =0. Solution Putting A =1,B = 1,C =0andM = 2andα = 1 in Corollary 2.1, wobtain 2 th solution ux, t)=x λt [k 1 cos = x λt[ k 1 cos 4 2λ ln x + k 2 sin 2λ ln x + k 2 sin 4 2λ λ ln x ], ] ln x

10 Lin t al. Boundary Valu Problms 2013, 2013:126 Pag 10 of 11 u1, t)=k 1 λt = 0 implis k 1 =0. Thn ux, t)=k 2 x λt sin 2λ ln x, u, t)=k 2 λt+1 sin 2λ = 0 implis that 2λ 1 2 1=nπ, n Z. That is, λ = n2 π 2 +1) 2. 4 Thus, ux, t)=k 2 x n2 π 2 +1) 2 4 t sin nπ ln x, n Z. If th discriminant 2λ = 0, th solution is trivial. If th discriminant 2λ 2 1 1<0, thn th solution is ux, t)=k 1 n2 +π 2 ) 2 t 4 x 1+inπ x 1 inπ), n N. Exampl 3.4 Th fractional partial diffrntial quation 2 3 u t 2 3 = 1 2 u with u0, t)= 2t. 4 x 2 Solution Putting α = 2 3, A = 1, B =0,C =0andM =1inCorollary2.2, thdiscriminant 4 is λ 2 3 =λ3 1 ) 2 0, but λ = 0 lads to a contradiction, hnc thr ar diffrnt ral roots m 1 =2λ3 1 and m 2 = 2λ3 1,sothatwhav ux, t)= λt k 1 2λ 1 3 x + k 2 2λ 1 3 x ). By th boundary condition u0, t)= 2t,wobtainλ = 2andk 1 + k 2 =1.So, ux, t)= 2t 2 3 2x 2k 1 2t sinh 2 3 2x ) = 2t 2 3 2x +2k 2 2t sinh 2 3 2x ), and th particular solution is ux, t)= 2t 2 3 2x or ux, t)= 2t 2 3 2x ). If w apply th Laplac transform to φt) =u0, t) = 2t,thnL{φt)}s) = 1 s+2 and ũx, s)= 1 s+2 2xs 13.Usingthrsiduthorm, ux, t)= 1 st 1 1 C s +2 2xs 3 ds = 1 = 2t 2 3 2x. lim s 2 1 s +2) 2xs 3 st s +2 Th solution obtaind by th mthod of Laplac transform and th rsidu thorm is a coincidnc, which is our rsult abov.

11 Lin t al. Boundary Valu Problms 2013, 2013:126 Pag 11 of 11 Compting intrsts Th authors dclar that thy hav no compting intrsts. Authors contributions SDL carrid out th molcular gntic studis, participatd in th squnc alignmnt and draftd th manuscript. CHL carrid out th immunoassays. SMS participatd in th squnc alignmnt. Acknowldgmnts Ddicatd to Profssor Hari M Srivastava. Th authors ar dply apprciativ of th commnts and suggstions offrd by th rfrs for improving th quality and rigor of this papr. Th prsnt invstigation was supportd, in part, by th National Scinc Council of th Rpublic of China undr Grant NSC M Rcivd: 30 January 2013 Accptd: 1 May 2013 Publishd: 16 May 2013 Rfrncs 1. Hilfr, R d.): Applications of Fractional Calculus in Physics. World Scintific, Rivr Edg 2000) 2. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Thory and Applications of Fractional Diffrntial Equations. Elsvir, Amstrdam 2006) 3. Lin, S-D, Chang, L-F, Srivastava, HM: Som familis of sris idntitis and associatd fractional diffrintgral formulas. Intgral Transforms Spc. Funct. 2010), ) 4. Lin, S-D, Tu, S-T, Srivastava, HM: Explicit solutions of crtain ordinary diffrntial quations by mans of fractional calculus. J. Fract. Calc. 20, ) 5. Mainardi, F: Th tim fractional diffusion-wav quation. Izv. Vysš. Učbn. Zavd., Radiofiz ), ) 6. Nishimoto, K: Fractional Calculus, vol. 1. Dscarts Prss, Koriyama 1984) 7. Nishimoto, K: Fractional Calculus, vol. 2. Dscarts Prss, Koriyama 1987) 8. Nishimoto, K: Fractional Calculus, vol. 3. Dscarts Prss, Koriyama 1989) 9. Nishimoto, K: Fractional Calculus, vol. 4. Dscarts Prss, Koriyama 1991) 10. Nishimoto, K: Fractional Calculus, vol. 5. Dscarts Prss, Koriyama 1996) 11. Podlubny, I: Fractional Diffrntial Equations, Mathmatics in Scinc and Enginring. Acadmic Prss, Nw York 1999) 12. Srivastava, HM, Owa, S, Nishimoto, K: Som fractional diffrintgral quations. J. Math. Anal. Appl. 106, ) 13. Tu, S-T, Chyan, D-K, Srivastava, HM: Som familis of ordinary and partial fractional diffrintgral quations. Intgral Transforms Spc. Funct. 11, ) 14. Eidlman, SD, Kochubi, AN: Cauchy problm for fractional diffusion quation. J. Diffr. Equ. 1992), ) 15. Schnidr, WR, Wyss, W: Fractional diffusion and wav quation. J. Math. Phys. 301), ) 16. Caputo, M: Elasticita Dissipazion. Zanichlli, Bologna 1969) 17. Mainardi, F: Th fundamntal solutions for th fractional diffusion-wav quation. Appl. Math. Ltt. 96), ) doi: / Cit this articl as: Lin t al.: Particular solutions of a crtain class of associatd Cauchy-Eulr fractional partial diffrntial quations via fractional calculus. Boundary Valu Problms :126.