THE SPACE-FRACTIONAL TELEGRAPH EQUATION AND THE RELATED FRACTIONAL TELEGRAPH PROCESS

Chin. Ann. Math.24B:1(2003),45-56.THE SPACE-FRACTIONAL TELEGRAPHEQUATION AND THE RELATEDFRACTIONAL TELEGRAPH PROCESS***E. ORSINGHER*ZHAO XUELEI**AbstractThe apace fractional telegraph equation is analyzed and the Fourier transform of ite funda-mental solution is obtained and discussed.A symmetric process with discontinuous trajectories, whose transition function satisfies thespace fractional telegraph equation, is presented. Its limiting behaviour and the connectionwith Bymmetric stable processes is also examined.Keywords Fractional calculus, Marchaud's derivative, Weyl's derivative, Riesz potential,Telegraph equation, Stable processes2000 MR Subject Classfcation 60H30, 35G05Chinese Library Classifcation 0211.6, O175.2, 0211.9,Document Code AArticle ID 0252 -9599(2003)01-0045-12g1. IntroductionIt is well known that Brownian motion B is the limit, in some sense, of the telegrapher'sprocess T (see [3]).The transition function pB(x,t) = pB of B is the fundamental solution of{器=號x∈R, t>0,(1.1)\p(x,0)= 8(x),while the transition function pr(x,t) = Pr of T is the fundamental solution of(器+2λ器=c2品,o∈R t>0,pP(x,0) =(x),(1.2)( P+(x,0)=0,”δ being the Dirac's delta function.It was discovered by Riesz5l that the transition functions of symmetric stable processeswith characteristic functionsU(r,t) =e-hI*e/2(1.3)of degree0 0.中國煤化工MYHCNMHG辨嗓畢出9HWN)工北粼國中(9°8)"[(n+x)*s+(n-)°f]-o In oJ 6-D)I(-)()It-oZ=mpτ--rMτ- 8+$-(m-I) | rP1n+2)4+(n-myad t-oC=I-ont/o(zn- zn)apnp(n+)*f+(n-x)g] I 0=z/o(zB-z)np(n+2)1+(n-) aJP"(A-m+2)/+(n+m-x)可.J m.J=fARq毗eAoqe se uoequoysteId aures aq4 jo sueou ia('8)"[([+ x)f +(a-)4] I“J(語(yǔ)-)I(( - T)I()近'P:-{t-D)r-v*]1(a+)*+(a-x)°fI5zp I 1-oz=1o6吧" J(+)+(-x)"1ap g三npsuogn4gsqns aAssaoons ot$ ID2Je‘a(chǎn)ABq M aIoJaLn=h- m 'a =h+ m uoeuoustreoe otq 8upu1orod Aq pearenpea oq ue中m(")(h+m+2)4+(h-m-啊。) pj=Ito u1ure mo Mou的eTUeO aM(εε)of導nofop”"(n+m+2)*f+(h-m+)"f-(h+m-x)f-(h-m-x)Tpzo==rh+my-(t-可.J ]}號Jpc0=z/ol1ez/olme]Ppo=' z/01ple l z/olpleOJ!IM ues @M_ghi°f衛，(zz)(hi+1)6-(h-)) 0hp(h+1)f-(h-x)5 a五幾塑緩-)黃s0_。P[華/(結-I).物工)]告(二山]血80__PI(2)(10)z/oe:SMoIToJ se i @1!IM 04 Aue?UAUOD s! 4I "Joo1dI9 NOIIVNOH HDVHDETaL TVNOILOVHJ~EOVdS SHL“7 X‘OVHZ智HHDNISHO 9 T'ON5CHIN. ANN. MATH.Vol.24 Ser.BNow, taking into account the refection formular(z)F(1-z)= gsinπz，z≠0,土1,土2,..and the Euler's duplication formula[(z+ )=2)-2T()HE)2z≠0,-1,-2,-we readily have20-1()1(1一號) _ 2-1T(號- )5(1一號)(一號)r(I)r()-四空亞1=2-+T(- [ r()r一)_π\_1F(1-號)=2-T(1 -凱(++ 0)12-21-1(2-可2)r2(1-號)1+cos弩(3.7)T(2-a) cos譬Formula (3.3) can be written asc2dr2(1-號)1+Co8登d fo f1e(x-4)+ f,(+w)au22 cos32于T2(1一號) r(2-a)- COs號 dx Joua-1dp° f(x-y)+ f(x+四d2cos號T(2-a)dx2 ]=(- 0()x)(3.8)可x|a,In the last step we have considered the defnition (2.3) for 1 0.We frst assume that during every time interval [t,t + Ot) a particle can either make ajump in the positive direction (with probability 1/2) or a jump in the negative direction(with the same probability).We also assume that the distribution of Y is the following one:for (Ot)吞 0and by X = X(t),t> 0 the curent position of the particle, our task here is to derivethe equations governing the fllowing probabilitiesf(x,t)dx = Pr {X(t)∈dx,N(1) is even},b(x,t)dx = Pr{X(t)∈dx,N(t) is odd}.(4.3)中國煤化工MYHCNMHG54CHIN. ANN. MATH.Vol.24 Ser.BTheorem 4.1. The integro diferential systemn governing (4.) is{8f_1_d=號]。(U(x-y,t)- f(t)y+a/2d39 I [f(x+y,t)- f(x,，+a/2 +2(b-f),(4.4)dyo=8。[b(x-y,t)- b(,1)y+0/2告1。(x+,1)-6(x,1 +\(f-b).Jy1+a/2Proof. We derive only the first equation since the other one follows in tbe same way. Wesuppose, for the sake of defniteness, that we evaluatef(x,t+ Ot)dx = Pr{X(t + Ot)∈dx, N(t + Ot) is even}.In the case where no Poisson event happens in the interval of time [t,t + Ot) and N(t) isan even number, a point r can be reached, at time t+ St, if either a jump upward (withprobability 1/2) or a jump downward (with the same probability) occur8. Another case iswhere the particle is located around x and a Poisson event happens during [t,t + Ot) wbenthe cumulative number N(t) was odd at time t. For the random movement occurring everyOt instants we can therefore writef(x,t+Ot)=(1-\O){言f(x-y,19 ji+o/ndyua_ Ot+ ,。 f(=+u,0號郵} + \Qtb+ o(Q2)St=(1- \O){三[0(z+.t)- (.,1能adOtgf(a,0io/adu} + X0tb(,t)+ o(Qt)ly2[f(+y,t)- f(.,t),后}J(d)2/a"+ (1 -入Ot)f(x,t) +入Otb(x,t) + o(Ot).(4.5)Now expanding f(x,t + Ot), simplifying in both members and letting Ot→0 we obtain thefirst equation in (4.4).Remark 4.1. We recall the relationshipe between the righthanded and left- handed中國煤化工MYHCNMHGNo.1 E. ORSINGHER & ZHAO, x L. THE SPACE FRAOTIONAL TELEGRAPH EQUATION 55Marchaud's and Weyl's derivatives:1poo f(x,t)-f(x-y,t)*D/*9f(0,)= 0-鄂Jyl+a/21__ d p f(u,t)= (一)2 J_~ (-y72y = W/Pf(a,),(4.6a)x_ 1ro f(x,t)- f(x+ y,t)yi+a/2= ~或-哥)(4.6b)for0 0. Clearly X'"(t) takes values on the imaginary axis, but develops according tothe rules governing the evolution of X(t),t > 0. Furthermore, if N(t) is even,x'=r(1-號)8平and, if N(t) is odd,icx= (0-)0籮so that, in terms of the coordinate x', the system (4.7) can be rewritten as(號=-ic的+\(b- f),ga72b(4.9)l器=icjq7s +\(f-b).In deriving (4.9) we have taken into account that Weyl's derivative can be witten down asfollows:w/2f(a,)=z是[° (- w,t)dwn,)山Jwa/2we/jf(x,)=n1.. [^ +er+w;2dw.dxJoRemark 4.3. We can show that the process whose transition function is the fundamentalsolution of the fractional telegraph equation converges in the limit to the symmetric stable中國煤化工MYHCNMHGCHIN. ANN, MATH.Vol.24 Ser.Bprocess with caraceristic function (1.3). Our idea is to consider the Laplace transform of(2.8) and take the limit asc→∞,λ→∞, in sucha way thate2/λ→1:[. e("(,)at = (u+入+√x-2M1)(\+原-2月間)+(μ+λ-不- 2R)(vX-c2n1a-\)]VA- c2hra(\+ u)2- (\2一e24nla)](4.10)μ2 + 2Xμ + c2nylaIt is now a simple matter to observe thatlin e-l*U(r,t)dt=uty=j。 e(1-t1h*/2dt.This result corresponds to the fact that the fractional telegraph equation (1.2) converges,asλ→∞, c→∞, to the fractional heat- wave equation appearing in (1.4).Acknowledgement. The authors gratefully acknowledge the help of Dr. Luisa Beghin,who checked the calculations and purged the manuscript of several misprints.REFERENCES[1] Feller, W., On a generalization of Marcel Riesz' potential and semigroups generated by them, Medde-landen Universitets Matematiska Simninarium, Lund, 21(1952), 73-81.[2] Fujita, Y, Integrodiferential equation which interpolates tbe heat and wave equationa, Osaka Journalof Mathematica, 27(1990), 309 321; 797-804.3] Orsingher, E., Probability law, flow functions, maximum distributions of wave-governed random motionsand their connections with Kirchof's law, Stochastic Processes and Their Applications, 34(1990), 49-66，4] Orsingher, E, Motions with rflecting and absorbing barriers driven by the telegraph equation, RandomOperators and Stochastic Equations, 8:1995), 9-21.] Riesz, M, Lintegrale de Rieman-Liovuille et le problemee Caucby, Acta Mathematica,und,，81(1948), 1-223.[6] Saichev, A. I. & Zalavsky, G. M., Fractional kinetic equations: solutions and applications, Chaos,7(1997), 753-784.[7] Samko, s. G., Kilbas, A. A. &c Marichev, O. I, Fractional integrals and derivatives, Gordon and BreachScience Publishers, Amsterdan, 1993.I 8] Samorodnitaky, G. & Taqqu, M.S., Stable non-Gaussian random processes, Chapman and Hall, NewYork 1994.[9] Schneider, W. R. & Wys, W., Fractional difusion and wave equatjons, Journal of MathematicalPhysics, 30(1989), 134-144.中國煤化工MYHCNMHG