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Harmonic analysis of random fractional diffusion–wave equations
Applied Mathematics and Computation 141 (2003) 77–85 www.elsevier.com/locate/amc
Harmonic analysis of random fractional diffusion–wave equations q V.V. Anh a
a,*
, N.N. Leonenko
b,1
School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane Qld 4001, Australia b School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH, UK
Abstract This paper presents the Green functions and spectral representations of the meansquare solutions of the fractional diffusion–wave equations with random initial conditions. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Fractional diffusion equation; Diffusion–wave equation with random data; Spectral representation; Mittag–Leffler function; Volterra-type fractional integral equation
1. Introduction We consider the n-dimensional fractional diffusion–wave equation ob u c=2 a=2 ¼ lðI DÞ ðDÞ u; otb
l > 0;
ð1:1Þ
where u ¼ uðt; xÞ; t 2 R1 , x 2 Rn is assumed to be a causal function in time, i.e., vanishing for t < 0, and the fractional derivative in time is taken in the Caputo–Djrbashian sense:
q Partially supported by the Australian Research Council grants A89601825, C19600199, and NATO grant PST.CLG.976361. * Corresponding author. E-mail addresses: [email protected] (V.V. Anh), leonenkon@Cardiff.ac.uk (N.N. Leonenko). 1 Address: Department of Mathematics, Kyiv University (National), Kyiv 252601, Ukraine.
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. doi:10.1016/S0096-3003(02)00322-3
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8 m o uðt; xÞ > > ; ob u < otm Z t ¼ m 1 otb > mb1 o uðs; xÞ > : ðt sÞ ds; Cðm bÞ 0 osm
b ¼ m 2 N; ð1:2Þ m 1 < b < m;
(see Caputo [6], Djrbashian and Nersesian [9], Djrbashian [8]). Here, D is the n-dimensional Laplace operator, and the operators ðI DÞc=2 , c P 0, and ðDÞa=2 ; a > 0, are interpreted as inverses of the Bessel and Riesz potentials respectively (see Appendix B of Anh and Leonenko [4]). Both Bessel and Riesz potentials are considered to be defined in a weak sense in the frequency domain in terms of fractional Sobolev spaces. We refer to Eq. (1.1) as a fractional diffusion equation when 0 < b 6 1 and as a fractional wave equation when 1 < b 6 2. In the stochastic situation, linear and non-linear heat and wave equations have been studied by Kampe de Feriet [7], Orsingher [20], Anh and Leonenko [2], Millet and Morien [19], Peszat and Zabczyk [21] (see also the references therein). Beghin et al. [5] considered linear Korteweg–de Vries equation with random data. Mathematical aspects of the initial-value problems and boundary-value problems for Eq. (1.1) with the fractional derivative in the sense of Caputo– Djrbashian, Riemann–Liouville or inverse Riesz potential, etc. (see Podlubry [22]) and for other equations of this type (with c ¼ 0) have been treated by several authors including Schneider and Wyss [25], Schneider [24], Kochubei [13], Mainardi [16], Saichev and Zaslavsky [23], Gorenflo et al. [11,12], Gorenflo et al. [10], Angulo et al. [1], Anh and Leonenko [3,4]. On the other hand, some partial differential equations of fractional order of type (1.1) were successfully used for modelling relevant physical processes (see for example, Podlubny [22], Anh and Leonenko [3,4]). We shall concentrate on the case of random initial conditions, that is, for 0 < b 6 1, uðt; xÞjt¼0 ¼ u0 ðxÞ ¼ nðxÞ;
x 2 Rn ;
ð1:3Þ
while for 1 < b 6 2, uðt; xÞjt¼0 ¼ u0 ðxÞ ¼ nðxÞ;
o uðt; xÞ ¼ u1 ðxÞ ¼ gðxÞ; ot t¼0
x 2 Rn ;
ð1:4Þ
where nðxÞ and gðxÞ are real measurable random fields defined on a complete probability space ðX; F; P Þ. The Green function of Eq. (1.1) is given in Section 2 together with its special cases. The initial-value problems (1.1), (1.3) and (1.1), (1.4) and the spectral representations of their mean-square solutions are then addressed in Section 3.
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2. Green functions We shall extensively use the following entire function of order q ¼ 1=a and type 1: Ea;b ðzÞ ¼
1 X k¼0
zk ; Cðak þ bÞ
a > 0; b 2 C; z 2 C:
ð2:1Þ
The function (2.1) is known as the generalized Mittag–Leffler function (see Djrbashian [8], Mainardi and Gorenflo [17]). Special simple cases are E1;1 ðzÞ ¼ ez ;
ez 1 ; E2;1 ðþz2 Þ ¼ coshðzÞ; z E2;2 ðzÞ ¼ sinhðz1=2 Þ=z1=2 :
E1;2 ðzÞ ¼
E2;1 ðz2 Þ ¼ cos z;
Consider now Eq. (1.1) subject to the initial condition uðt; xÞjt¼0 ¼ dðxÞ;
x 2 Rn ;
ð2:2Þ
where dðxÞ is the Dirac delta function. Let S ¼ SðRn Þ be the Schwartz space of rapidly decreasing C 1 ðRn Þ functions with its dual S0 ¼ S0 ðRn Þ, which is the space of tempered distributions. We shall denote by b u ¼ Fx ½u the Fourier transform of a distribution u 2 S0 b ¼G b ðt; nÞ; with respect to the space variable x 2 Rn : In particular, let G n n t > 0; n 2 R being the dual variable of x 2 R , be the Fourier transform of the fundamental solution (i.e., the Green function) of the Cauchy problem (1.1) and (1.4). By the Fourier transform with respect to x, problem (1.1) and (1.4) is equivalent to the Cauchy problem b db G a 2 c=2 b ¼ ljnj ð1 þ jnj Þ G ; dtb
b ð0; nÞ ¼ 1: G
ð2:3Þ
(see Anh and Leonenko [4]). To solve (2.3), we use some important results of Djrbashian and Nersesian [9, Theorems 5 and 6], and Luchko and Gorenflo [15] (see also Kochubei [13], Gorenflo et al. [10] or Anh and Leonenko [4]). The following result is obtained: Theorem 1. The Cauchy problem (2.3) has a unique solution given by
b ðt; nÞ ¼ Eb;1 ltb jnja ð1 þ jnj2 Þc=2 ; G
ð2:4Þ
where 0 < b 6 2; a > 0; c > 0 and Eb;1 ðxÞ; x P 0, is the Mittag–Leffler function (2.1) of the negative real argument. Moreover, for u0 ðxÞ 2 S (or u0 ðxÞ 2 S0 with compact support), the initial value problem (1.1), (1.3) for the fractional diffusion equation ð0 < b 6 1Þ or the initial value problem (1.1), (1.4) with gðxÞ 0 for the fractional wave equation ð1 < b 6 2Þ has the unique solution.
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uðt; xÞ ¼
Z
Gðt; x yÞu0 ðyÞ dy;
Rn
where the Green function G has Fourier transform (2.4). Remark 1. This theorem was formulated by Anh and Leonenko [4] for the case 0 < b 6 1. Let Gðt; xÞ; t > 0; x 2 Rn , be the fundamental solution of the Eq. (1.1), whose Fourier transform is given by (2.4). The inverse Fourier transform can be written as Z n Gðt; xÞ ¼ ð2pÞ eihk;xi Eb;1 ðltb jkja ð1 þ jkj2 Þc=2 Þ dk: ð2:5Þ Rn
For a > 0; b 2 ð0; 2 ; c P 0 such that c=2
Eb;1 ðltb jkja ð1 þ jkj2 Þ Þ 2 L1 ðRn Þ;
ð2:6Þ n
the Green function (2.5) Gðt; xÞ belongs to L1 ðR Þ for a fixed t > 0. Using the Hankel method, this inverse Fourier transform can be represented as Z 1 n=2 c=2 Gðt; xÞ ¼ ð2pÞ j xjð2nÞ=2 qn=2 Jðn2Þ=2 ðj xjqÞEb;1 ðqa ð1 þ q2 Þ tb lÞ dq; 0
ð2:7Þ where Jm ðzÞ is the Bessel function of the first kind of order m. The Hankel transform (2.7) exists for c P 0; b 2 ½0; 2 ; a > 0 such that
c=2 ð2:8Þ qðn1Þ=2 Eb;1 qa ð1 þ q2 Þ tb l 2 L1 ð½0; 1ÞÞ: For a fixed t, the condition (2.6) holds, for example, for every b 2 ð0; 1 if a þ c > n, while condition (2.8) is satisfied for a þ c > ðn þ 1Þ=2. From these ranges we see the important role of the parameter c in Eq. (1.1) (see Anh and Leonenko [4] for details). To the best of our knowledge, the Green function (2.5) or (2.7) of the fractional diffusion–wave equation (1.1) is the most general in the existing literature. If b ¼ 1, the Mittag–Leffler function E1;1 ðxÞ ¼ exp fxg; x P 0, and from (2.5) we obtain an explicit expression for the Green function Z n o n Gðt; xÞ ¼ ð2pÞ exp ihk; xi ltjkja ð1 þ jkj2 Þc=2 dk; a > 0; c P 0: Rn
ð2:9Þ For c ¼ 0; a ¼ 2; the Green function (2.9) reduces to the n-dimensional isotropic Gaussian density. For c ¼ 0; a ¼ 1, the Green function (2.9) is the
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81
density function of the n-dimensional isotropic Cauchy distribution, while for c ¼ 0, a 2 ð0; 2 , the function (2.9) reduces to the density function of the n-dimensional symmetric stable distribution. Note that for a > 2, the function (2.9) may become negative for some value of x (see Beghin et al. [5]). For c ¼ 0, we are able to give a new explicit expression for the Green function (2.5) in terms of H-functions of Fox (see Anh and Leonenko [4]). For example, if c ¼ 0, a ¼ 2; b 2 ð0; 1 , the Green function (2.5) is reduced to ! 2 j xj ð1; 1Þ ð1; bÞ n 2;1 n=2 ; ð2:10Þ Gðt; xÞ ¼ p j xj H2;3 4tb l ðn=2; 1Þ ð1; 1Þ ð1; 1Þ 2;1 where H2;3 are the Fox H-functions (see Srivastava et al. [26]). The Green function (2.10) can be reduced to the Green functions of Schneider and Wyss [25] or Kochubei [13] (see Anh and Leonenko [25] for details). Note that for n ¼ 1; 0 < b 6 2, c ¼ 0, a ¼ 2, our Green function coincides with the Green function of Mainardi [16], which was obtained in terms of the entire function of order 1=ð1 þ kÞ
Wa;b ðzÞ ¼
1 X k¼0
zj ; j!Cðak þ bÞ
a > 1; b 2 C; z 2 C;
which is known as WrightÕs function (see Gorenflo et al. [11]) or generalized Bessel function due to the relation z m 1 2 Jm ðzÞ ¼ W1;mþ1 z : 2 4 The Wright function is also a special case of H-function (see Srivastava et al. [26] or Gorenflo et al. [11]). In this case ðn ¼ 1; 0 < b 6 2; c ¼ 0; a ¼ 2Þ 1 j xj b Gðt; xÞ ¼ b=2 pffiffiffi M b=2 pffiffiffi ; ; x 2 R1 ; t > 0; 2t t l l 2 where
b M u; 2
¼ Wb;1b ðuÞ; 2
u P 0:
2
In the simplest cases, 1 1 u2 ¼ pffiffiffi exp M u; ; 2 4 p
u
1 ¼ 32=3 Ai 1=3 M u; 3 3
with the Airy function AiðzÞ. For n ¼ 1, b ¼ a, c ¼ 0, we obtain the neutral– fractional diffusion with the Green function of a fractional Cauchy type (see Gorenflo et al. [10]):
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Gðt; xÞ ¼
1 j xja1 ta sin ap=2 : p t2a þ 2j xja ta cos ap=2 þ j xj2a
Some other explicit expressions of the Green function of the fractional diffusion–wave equation can be found in recent papers by Anh and Leonenko [4] or Mainardi et al. [18].
3. Spectral representations We use some elements of the spectral theory of random fields (see, for example, Leonenko [14]. Consider now the Eq. (1.1) with 0 < b 6 1, the initial condition Eq. (1.3), where nðxÞ ¼ nðx; xÞ : X Rn ! R1 , is a real, mean-square continuous homogeneous (in the wide sense) random field with mean EnðxÞ ¼ 0, and covariance function Z Bn ðxÞ ¼ covðnð0Þ; nðxÞÞ ¼ eihk;xi Fn ðdkÞ; ð3:1Þ Rn
where F is the spectral measure, that is, a bounded non-negative measure on ðRn ; BðRn ÞÞ, BðRn Þ being the r-field of Borel sets on Rn . In view of KarhunenÕs Theorem, there exists a complex-valued orthogonally scattered random measure Zn , such that, for every x 2 Rn , the random field itself has the spectral representation (P ––a:s) Z 2 nðxÞ ¼ eihk;xi Zn ðdkÞ; EjZn ðAÞj ¼ Fn ðAÞ; A 2 BðRn Þ: ð3:2Þ Rn
From (2.4), (2.5) and (3.2), we obtain the solution of the initial-value problem (1.1) and (1.3) for 0 < b 6 1 or the initial-value problem (1.1) and (1.4) with gðxÞ ¼ 0 for 1 < b 6 2. This can be written as the convolution Z uðt; xÞ ¼ Gðt; x yÞnðyÞ dy n ZR ¼ eihk;xi Eb;1 ðltb jkja ð1 þ jkj2 Þc=2 ÞZn ðdkÞ; ð3:3Þ Rn
where Eb;1 is the Mittag–Leffler function (2.1), and the stochastic integrals (3.2) and (3.3) are interpreted in the mean-square sense. In addition Z ob u ¼ l eihk;xi jkja ð1 þ jkj2 Þc=2 Eb;1 ðltb jkja ð1 þ jkj2 Þc=2 ÞZn ðdkÞ otb Rn c=2
¼ lðI DÞ ðDÞ
a=2
u;
where the fractional derivatives in space are interpreted in the mean-square sense in the frequency domain (see Anh and Leonenko [4]). Thus, we can in-
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83
terpret (3.3) as the mean-square solution or L2 ðXÞ-solution. We arrive at the following result: Theorem 2. The mean-square solution of the initial-value problem (1.1) and (1.3) for 0 < b 6 1 or (1.1), (1.4) with gðxÞ 0 for 1 < b 6 2 is given by (3.3). Moreover, the covariance function of the random field (3.3) is of the form Z c=2 covðuðt; xÞ; uðs; yÞÞ ¼ eihk;xy i Eb;1 ðltb jkja ð1 þ jkj2 Þ Þ Rn c=2
Eb;1 ðlsb jkja ð1 þ jkj2 Þ ÞFn ðdkÞ; where Fn is the spectral measure of the initial-condition field (3.2). In general, we are not able to solve the initial-value problem (1.1) and (1.4) for 1 < b 6 2 in which the random field gðxÞ; x 2 Rn is not zero. But for 1 < b 6 2; a ¼ 2; c ¼ 0, we propose to consider, instead of the Cauchy problem (1.1) and (1.4) the following fractional integral equation of Volterra type: Z t 1 uðt; xÞ ¼ u0 ðxÞ þ tu1 ðxÞ þ ðt sÞb1 Duðs; xÞ ds; 1 < b 6 2; CðbÞ 0 ð3:4Þ which reduces to the integrated wave equation for b ¼ 2. The solution of (3.4) can be expressed in term of the initial condition (1.4) as Z Z uðt; xÞ ¼ Gðt; x yÞu0 ðyÞ dy þ Gð1Þ ðt; x yÞu1 ðyÞ dy; ð3:5Þ Rn
Rn
where Gðt; xÞ is defined as Z eihk;xi Gðt; xÞ dx ¼ Eb;1 ðjkj2 tb Þ
ð3:6Þ
Rn
and ð1Þ
G ðt; xÞ ¼
Z
t
Gðs; xÞ ds: 0
In particular, for n ¼ 1 and the limiting case b ¼ 2 Gðt; xÞ ¼
dðt xÞ þ dðt þ xÞ ; 2
ð3:7Þ
and if u1 ðxÞ 0, we have dÕAlembert formula uðt; xÞ ¼ 12½u0 ðx tÞ þ u0 ðx þ tÞ : From (3.1)–(3.3) and (3.6)–(3.8) we arrive at the following result:
ð3:8Þ
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Theorem 3. Consider the solution uðt; xÞ of the fractional integral equation of Volterra type (3.4) for 1 < b 6 2 ða ¼ 2; c ¼ 0Þ, in which u0 ðxÞ ¼ nðxÞ; u1 ðxÞ ¼ gðxÞ, where nðxÞ and gðxÞ; x 2 Rn , are two uncorrelated measurable mean-square continuous homogeneous random fields with zero means and covariance functions Bn ðxÞ and Bg ðxÞ of the form (3.1) with spectral measures Fn and Fg having the spectral representations of the form (3.2) with complex-valued random spectral measures Zn and Zg respectively. Then the random field uðt; xÞ has the following spectral representation: Z Z uðt; xÞ ¼ eihk;xi Eb;1 ðtb jkj2 ÞZn ðdkÞ þ t eihk;xi Eb;2 ðtb jkj2 ÞZg ðdkÞ; Rn
Rn
1 < b 6 2; t > 0; x 2 Rn ; with the covariance structure Z eihk;xy i Eb;1 ðtb jkj2 ÞEb;1 ðsb jkj2 ÞFn ðdkÞ þ ts covðuðt; xÞ; uðs; yÞÞ ¼ Rn Z eihk;xy i Eb;2 ðtb jkj2 ÞEb;2 ðsb jkj2 ÞFg ðdkÞ; Rn
where Eb;1 ðxÞ; x P 0 and Eb;2 ðxÞ; x P 0 are the Mittag–Leffler functions (2.1). Remark 2. Note that for 1 < b 6 2; the Mittag–Leffler functions Eb;1 ðxÞ; x P 0 and Eb;2 ðxÞ; x P 0 have zeroes in contrast to the case 0 < b 6 1, in which the Mittag–Leffler function Eb;1 ðxÞ; x P 0 is completely monotonic (see, for example, Schneider [24]). Now, let n ¼ 1; b ¼ 2 and consider the dÕAlembert formula (3.8), that is, the case gðxÞ ¼ 0. Then, for the wave equation ðb ¼ 2; a ¼ 2; c ¼ 0; l ¼ 1Þ with the initial random field (3.2), we obtain the following spectral representation Z uðt; xÞ ¼ eikx cosðktÞZn ðdkÞ ð3:9Þ R1
with covariance function covðuðt; xÞ; uðs; yÞÞ ¼
Z
cosðkðx yÞÞ cos2 ðktÞFn ðdkÞ: R1
We call (3.9) the dÕAlembert random field.
References [1] J.M. Angulo, M.D. Ruiz-Medina, V.V. Anh, W. Grecksch, Fractional diffusion and fractional heat equation, Adv. Appl. Prob. 32 (4) (2000) 1077–1099. [2] V.V. Anh, N.N. Leonenko, Non-Gaussian scenarios for the heat equation with singular initial data, Stoch. Proc. Appl. 84 (1999) 91–114.
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[3] V.V. Anh, N.N. Leonenko, Scaling laws for fractional diffusion–wave equation with singular initial data, Stat. Prob. Lett. 48 (2000) 239–252. [4] V.V. Anh, N.N. Leonenko, Spectral analysis of fractional kinetic equations with random data, J. Stat. Phys., vol. 104(5/6) (2001) 1349–1387. [5] L. Beghin, V.P. Knopova, N.N. Leonenko, E. Orsingher, Gaussian limiting behaviour of the rescaled solution to the linear Korteweg-de Vries equation with random initial conditions, J. Stat. Phys. 99 (3/4) (2000) 769–781. [6] M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, 1969. [7] J.K. de Feriet, Random solutions of the partial differential equations, in: Proc. 3rd Berkeley Symp. Math. Stat. Prob., vol. III, University of California Press, Berkeley, CA, 1955, pp. 199– 208. [8] M.M. Djrbashian, Harmonic Analysis and Boundary Value Problems in Complex Domain, Birkh€ auser Verlag, Basel, 1993. [9] M.M. Djrbashian, A.B. Nersesian, Fractional derivatives and the Cauchy problem for differential equations of fractional order, Izv. Acad. Nauk Armjanskvy SSR 3 (1) (1968) 3–29 (In Russian). [10] R. Gorenflo, A. Iskenderov, Y. Luchko, Mapping between solutions of fractional diffusion– wave equations, Fract. Calculus Appl. Anal. 3 (1) (2000) 75–86. [11] R. Gorenflo, Y. Luchko, F. Mainardi, Analytical properties and applications of the Wright function, Fract. Calculus Appl. Anal. 2 (1999) 383–414. [12] R. Gorenflo, Y. Luchko, F. Mainardi, Wright functions and scale-invariate solutions of the diffusion–wave equation, J. Comput. Appl. Math. 118 (2000) 175–191. [13] A.N. Kochubei, Fractional order diffusion, J. Diff. Eqs. 26 (4) (1990) 485–492. [14] N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum, Kluwer, 1999. [15] Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Mathematica Vietnamica 24 (1999) 207–233. [16] F. Mainardi, The fundamental solutions for the fractional diffusion–wave equation, Appl. Math. Lett. 9 (6) (1996) 23–28. [17] F. Mainardi, R. Gorenflo, On Mittag–Leffler type functions in fractional evolution processes, J. Comput Appl. Math. 118 (2000) 283–299. [18] F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calculus Appl. Anal. 4 (2) (2001) 153–192. [19] A. Millet, P.-L. Morien, On a stochastic wave equation in two space dimensions: regularity of the solution and its density, Stoch. Proc. Appl. 86 (2000) 141–162. [20] E. Orsingher, Damped vibrations excited by white noise, Adv. Appl. Prob. 16 (3) (1984) 562– 584. [21] S. Peszat, J. Zabczyk, Nonlinear stochastic wave and heat equations, Prob. Theory Relat. Fields 116 (2000) 421–443. [22] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999. [23] A.I. Saichev, G.M. Zaslawsky, Fractional kinetic equations: solutions and applications, Chaos 7 (4) (1997) 753–764. [24] W.R. Schneider, Fractional diffusion, in: R. Lima, L. Streit, D.V. Mendes (Eds.), Dynamics and Stochastic Processes, Theory and Applications, Volume 355 of Lecture Notes in Physics, Springer, Heidelberg, 1990, pp. 276–286. [25] W.R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1) (1989) 134–144. [26] H.M. Srivastava, K.C. Gupta, S.P. Goyal, The H-functions of one and two variables with applications, South Asia Publishers, New Delhi, Madras, 1982.
Harmonic analysis of random fractional diffusion–wave equations q V.V. Anh a
a,*
, N.N. Leonenko
b,1
School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane Qld 4001, Australia b School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH, UK
Abstract This paper presents the Green functions and spectral representations of the meansquare solutions of the fractional diffusion–wave equations with random initial conditions. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Fractional diffusion equation; Diffusion–wave equation with random data; Spectral representation; Mittag–Leffler function; Volterra-type fractional integral equation
1. Introduction We consider the n-dimensional fractional diffusion–wave equation ob u c=2 a=2 ¼ lðI DÞ ðDÞ u; otb
l > 0;
ð1:1Þ
where u ¼ uðt; xÞ; t 2 R1 , x 2 Rn is assumed to be a causal function in time, i.e., vanishing for t < 0, and the fractional derivative in time is taken in the Caputo–Djrbashian sense:
q Partially supported by the Australian Research Council grants A89601825, C19600199, and NATO grant PST.CLG.976361. * Corresponding author. E-mail addresses: [email protected] (V.V. Anh), leonenkon@Cardiff.ac.uk (N.N. Leonenko). 1 Address: Department of Mathematics, Kyiv University (National), Kyiv 252601, Ukraine.
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. doi:10.1016/S0096-3003(02)00322-3
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V.V. Anh, N.N. Leonenko / Appl. Math. Comput. 141 (2003) 77–85
8 m o uðt; xÞ > > ; ob u < otm Z t ¼ m 1 otb > mb1 o uðs; xÞ > : ðt sÞ ds; Cðm bÞ 0 osm
b ¼ m 2 N; ð1:2Þ m 1 < b < m;
(see Caputo [6], Djrbashian and Nersesian [9], Djrbashian [8]). Here, D is the n-dimensional Laplace operator, and the operators ðI DÞc=2 , c P 0, and ðDÞa=2 ; a > 0, are interpreted as inverses of the Bessel and Riesz potentials respectively (see Appendix B of Anh and Leonenko [4]). Both Bessel and Riesz potentials are considered to be defined in a weak sense in the frequency domain in terms of fractional Sobolev spaces. We refer to Eq. (1.1) as a fractional diffusion equation when 0 < b 6 1 and as a fractional wave equation when 1 < b 6 2. In the stochastic situation, linear and non-linear heat and wave equations have been studied by Kampe de Feriet [7], Orsingher [20], Anh and Leonenko [2], Millet and Morien [19], Peszat and Zabczyk [21] (see also the references therein). Beghin et al. [5] considered linear Korteweg–de Vries equation with random data. Mathematical aspects of the initial-value problems and boundary-value problems for Eq. (1.1) with the fractional derivative in the sense of Caputo– Djrbashian, Riemann–Liouville or inverse Riesz potential, etc. (see Podlubry [22]) and for other equations of this type (with c ¼ 0) have been treated by several authors including Schneider and Wyss [25], Schneider [24], Kochubei [13], Mainardi [16], Saichev and Zaslavsky [23], Gorenflo et al. [11,12], Gorenflo et al. [10], Angulo et al. [1], Anh and Leonenko [3,4]. On the other hand, some partial differential equations of fractional order of type (1.1) were successfully used for modelling relevant physical processes (see for example, Podlubny [22], Anh and Leonenko [3,4]). We shall concentrate on the case of random initial conditions, that is, for 0 < b 6 1, uðt; xÞjt¼0 ¼ u0 ðxÞ ¼ nðxÞ;
x 2 Rn ;
ð1:3Þ
while for 1 < b 6 2, uðt; xÞjt¼0 ¼ u0 ðxÞ ¼ nðxÞ;
o uðt; xÞ ¼ u1 ðxÞ ¼ gðxÞ; ot t¼0
x 2 Rn ;
ð1:4Þ
where nðxÞ and gðxÞ are real measurable random fields defined on a complete probability space ðX; F; P Þ. The Green function of Eq. (1.1) is given in Section 2 together with its special cases. The initial-value problems (1.1), (1.3) and (1.1), (1.4) and the spectral representations of their mean-square solutions are then addressed in Section 3.
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2. Green functions We shall extensively use the following entire function of order q ¼ 1=a and type 1: Ea;b ðzÞ ¼
1 X k¼0
zk ; Cðak þ bÞ
a > 0; b 2 C; z 2 C:
ð2:1Þ
The function (2.1) is known as the generalized Mittag–Leffler function (see Djrbashian [8], Mainardi and Gorenflo [17]). Special simple cases are E1;1 ðzÞ ¼ ez ;
ez 1 ; E2;1 ðþz2 Þ ¼ coshðzÞ; z E2;2 ðzÞ ¼ sinhðz1=2 Þ=z1=2 :
E1;2 ðzÞ ¼
E2;1 ðz2 Þ ¼ cos z;
Consider now Eq. (1.1) subject to the initial condition uðt; xÞjt¼0 ¼ dðxÞ;
x 2 Rn ;
ð2:2Þ
where dðxÞ is the Dirac delta function. Let S ¼ SðRn Þ be the Schwartz space of rapidly decreasing C 1 ðRn Þ functions with its dual S0 ¼ S0 ðRn Þ, which is the space of tempered distributions. We shall denote by b u ¼ Fx ½u the Fourier transform of a distribution u 2 S0 b ¼G b ðt; nÞ; with respect to the space variable x 2 Rn : In particular, let G n n t > 0; n 2 R being the dual variable of x 2 R , be the Fourier transform of the fundamental solution (i.e., the Green function) of the Cauchy problem (1.1) and (1.4). By the Fourier transform with respect to x, problem (1.1) and (1.4) is equivalent to the Cauchy problem b db G a 2 c=2 b ¼ ljnj ð1 þ jnj Þ G ; dtb
b ð0; nÞ ¼ 1: G
ð2:3Þ
(see Anh and Leonenko [4]). To solve (2.3), we use some important results of Djrbashian and Nersesian [9, Theorems 5 and 6], and Luchko and Gorenflo [15] (see also Kochubei [13], Gorenflo et al. [10] or Anh and Leonenko [4]). The following result is obtained: Theorem 1. The Cauchy problem (2.3) has a unique solution given by
b ðt; nÞ ¼ Eb;1 ltb jnja ð1 þ jnj2 Þc=2 ; G
ð2:4Þ
where 0 < b 6 2; a > 0; c > 0 and Eb;1 ðxÞ; x P 0, is the Mittag–Leffler function (2.1) of the negative real argument. Moreover, for u0 ðxÞ 2 S (or u0 ðxÞ 2 S0 with compact support), the initial value problem (1.1), (1.3) for the fractional diffusion equation ð0 < b 6 1Þ or the initial value problem (1.1), (1.4) with gðxÞ 0 for the fractional wave equation ð1 < b 6 2Þ has the unique solution.
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uðt; xÞ ¼
Z
Gðt; x yÞu0 ðyÞ dy;
Rn
where the Green function G has Fourier transform (2.4). Remark 1. This theorem was formulated by Anh and Leonenko [4] for the case 0 < b 6 1. Let Gðt; xÞ; t > 0; x 2 Rn , be the fundamental solution of the Eq. (1.1), whose Fourier transform is given by (2.4). The inverse Fourier transform can be written as Z n Gðt; xÞ ¼ ð2pÞ eihk;xi Eb;1 ðltb jkja ð1 þ jkj2 Þc=2 Þ dk: ð2:5Þ Rn
For a > 0; b 2 ð0; 2 ; c P 0 such that c=2
Eb;1 ðltb jkja ð1 þ jkj2 Þ Þ 2 L1 ðRn Þ;
ð2:6Þ n
the Green function (2.5) Gðt; xÞ belongs to L1 ðR Þ for a fixed t > 0. Using the Hankel method, this inverse Fourier transform can be represented as Z 1 n=2 c=2 Gðt; xÞ ¼ ð2pÞ j xjð2nÞ=2 qn=2 Jðn2Þ=2 ðj xjqÞEb;1 ðqa ð1 þ q2 Þ tb lÞ dq; 0
ð2:7Þ where Jm ðzÞ is the Bessel function of the first kind of order m. The Hankel transform (2.7) exists for c P 0; b 2 ½0; 2 ; a > 0 such that
c=2 ð2:8Þ qðn1Þ=2 Eb;1 qa ð1 þ q2 Þ tb l 2 L1 ð½0; 1ÞÞ: For a fixed t, the condition (2.6) holds, for example, for every b 2 ð0; 1 if a þ c > n, while condition (2.8) is satisfied for a þ c > ðn þ 1Þ=2. From these ranges we see the important role of the parameter c in Eq. (1.1) (see Anh and Leonenko [4] for details). To the best of our knowledge, the Green function (2.5) or (2.7) of the fractional diffusion–wave equation (1.1) is the most general in the existing literature. If b ¼ 1, the Mittag–Leffler function E1;1 ðxÞ ¼ exp fxg; x P 0, and from (2.5) we obtain an explicit expression for the Green function Z n o n Gðt; xÞ ¼ ð2pÞ exp ihk; xi ltjkja ð1 þ jkj2 Þc=2 dk; a > 0; c P 0: Rn
ð2:9Þ For c ¼ 0; a ¼ 2; the Green function (2.9) reduces to the n-dimensional isotropic Gaussian density. For c ¼ 0; a ¼ 1, the Green function (2.9) is the
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density function of the n-dimensional isotropic Cauchy distribution, while for c ¼ 0, a 2 ð0; 2 , the function (2.9) reduces to the density function of the n-dimensional symmetric stable distribution. Note that for a > 2, the function (2.9) may become negative for some value of x (see Beghin et al. [5]). For c ¼ 0, we are able to give a new explicit expression for the Green function (2.5) in terms of H-functions of Fox (see Anh and Leonenko [4]). For example, if c ¼ 0, a ¼ 2; b 2 ð0; 1 , the Green function (2.5) is reduced to ! 2 j xj ð1; 1Þ ð1; bÞ n 2;1 n=2 ; ð2:10Þ Gðt; xÞ ¼ p j xj H2;3 4tb l ðn=2; 1Þ ð1; 1Þ ð1; 1Þ 2;1 where H2;3 are the Fox H-functions (see Srivastava et al. [26]). The Green function (2.10) can be reduced to the Green functions of Schneider and Wyss [25] or Kochubei [13] (see Anh and Leonenko [25] for details). Note that for n ¼ 1; 0 < b 6 2, c ¼ 0, a ¼ 2, our Green function coincides with the Green function of Mainardi [16], which was obtained in terms of the entire function of order 1=ð1 þ kÞ
Wa;b ðzÞ ¼
1 X k¼0
zj ; j!Cðak þ bÞ
a > 1; b 2 C; z 2 C;
which is known as WrightÕs function (see Gorenflo et al. [11]) or generalized Bessel function due to the relation z m 1 2 Jm ðzÞ ¼ W1;mþ1 z : 2 4 The Wright function is also a special case of H-function (see Srivastava et al. [26] or Gorenflo et al. [11]). In this case ðn ¼ 1; 0 < b 6 2; c ¼ 0; a ¼ 2Þ 1 j xj b Gðt; xÞ ¼ b=2 pffiffiffi M b=2 pffiffiffi ; ; x 2 R1 ; t > 0; 2t t l l 2 where
b M u; 2
¼ Wb;1b ðuÞ; 2
u P 0:
2
In the simplest cases, 1 1 u2 ¼ pffiffiffi exp M u; ; 2 4 p
u
1 ¼ 32=3 Ai 1=3 M u; 3 3
with the Airy function AiðzÞ. For n ¼ 1, b ¼ a, c ¼ 0, we obtain the neutral– fractional diffusion with the Green function of a fractional Cauchy type (see Gorenflo et al. [10]):
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Gðt; xÞ ¼
1 j xja1 ta sin ap=2 : p t2a þ 2j xja ta cos ap=2 þ j xj2a
Some other explicit expressions of the Green function of the fractional diffusion–wave equation can be found in recent papers by Anh and Leonenko [4] or Mainardi et al. [18].
3. Spectral representations We use some elements of the spectral theory of random fields (see, for example, Leonenko [14]. Consider now the Eq. (1.1) with 0 < b 6 1, the initial condition Eq. (1.3), where nðxÞ ¼ nðx; xÞ : X Rn ! R1 , is a real, mean-square continuous homogeneous (in the wide sense) random field with mean EnðxÞ ¼ 0, and covariance function Z Bn ðxÞ ¼ covðnð0Þ; nðxÞÞ ¼ eihk;xi Fn ðdkÞ; ð3:1Þ Rn
where F is the spectral measure, that is, a bounded non-negative measure on ðRn ; BðRn ÞÞ, BðRn Þ being the r-field of Borel sets on Rn . In view of KarhunenÕs Theorem, there exists a complex-valued orthogonally scattered random measure Zn , such that, for every x 2 Rn , the random field itself has the spectral representation (P ––a:s) Z 2 nðxÞ ¼ eihk;xi Zn ðdkÞ; EjZn ðAÞj ¼ Fn ðAÞ; A 2 BðRn Þ: ð3:2Þ Rn
From (2.4), (2.5) and (3.2), we obtain the solution of the initial-value problem (1.1) and (1.3) for 0 < b 6 1 or the initial-value problem (1.1) and (1.4) with gðxÞ ¼ 0 for 1 < b 6 2. This can be written as the convolution Z uðt; xÞ ¼ Gðt; x yÞnðyÞ dy n ZR ¼ eihk;xi Eb;1 ðltb jkja ð1 þ jkj2 Þc=2 ÞZn ðdkÞ; ð3:3Þ Rn
where Eb;1 is the Mittag–Leffler function (2.1), and the stochastic integrals (3.2) and (3.3) are interpreted in the mean-square sense. In addition Z ob u ¼ l eihk;xi jkja ð1 þ jkj2 Þc=2 Eb;1 ðltb jkja ð1 þ jkj2 Þc=2 ÞZn ðdkÞ otb Rn c=2
¼ lðI DÞ ðDÞ
a=2
u;
where the fractional derivatives in space are interpreted in the mean-square sense in the frequency domain (see Anh and Leonenko [4]). Thus, we can in-
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terpret (3.3) as the mean-square solution or L2 ðXÞ-solution. We arrive at the following result: Theorem 2. The mean-square solution of the initial-value problem (1.1) and (1.3) for 0 < b 6 1 or (1.1), (1.4) with gðxÞ 0 for 1 < b 6 2 is given by (3.3). Moreover, the covariance function of the random field (3.3) is of the form Z c=2 covðuðt; xÞ; uðs; yÞÞ ¼ eihk;xy i Eb;1 ðltb jkja ð1 þ jkj2 Þ Þ Rn c=2
Eb;1 ðlsb jkja ð1 þ jkj2 Þ ÞFn ðdkÞ; where Fn is the spectral measure of the initial-condition field (3.2). In general, we are not able to solve the initial-value problem (1.1) and (1.4) for 1 < b 6 2 in which the random field gðxÞ; x 2 Rn is not zero. But for 1 < b 6 2; a ¼ 2; c ¼ 0, we propose to consider, instead of the Cauchy problem (1.1) and (1.4) the following fractional integral equation of Volterra type: Z t 1 uðt; xÞ ¼ u0 ðxÞ þ tu1 ðxÞ þ ðt sÞb1 Duðs; xÞ ds; 1 < b 6 2; CðbÞ 0 ð3:4Þ which reduces to the integrated wave equation for b ¼ 2. The solution of (3.4) can be expressed in term of the initial condition (1.4) as Z Z uðt; xÞ ¼ Gðt; x yÞu0 ðyÞ dy þ Gð1Þ ðt; x yÞu1 ðyÞ dy; ð3:5Þ Rn
Rn
where Gðt; xÞ is defined as Z eihk;xi Gðt; xÞ dx ¼ Eb;1 ðjkj2 tb Þ
ð3:6Þ
Rn
and ð1Þ
G ðt; xÞ ¼
Z
t
Gðs; xÞ ds: 0
In particular, for n ¼ 1 and the limiting case b ¼ 2 Gðt; xÞ ¼
dðt xÞ þ dðt þ xÞ ; 2
ð3:7Þ
and if u1 ðxÞ 0, we have dÕAlembert formula uðt; xÞ ¼ 12½u0 ðx tÞ þ u0 ðx þ tÞ : From (3.1)–(3.3) and (3.6)–(3.8) we arrive at the following result:
ð3:8Þ
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Theorem 3. Consider the solution uðt; xÞ of the fractional integral equation of Volterra type (3.4) for 1 < b 6 2 ða ¼ 2; c ¼ 0Þ, in which u0 ðxÞ ¼ nðxÞ; u1 ðxÞ ¼ gðxÞ, where nðxÞ and gðxÞ; x 2 Rn , are two uncorrelated measurable mean-square continuous homogeneous random fields with zero means and covariance functions Bn ðxÞ and Bg ðxÞ of the form (3.1) with spectral measures Fn and Fg having the spectral representations of the form (3.2) with complex-valued random spectral measures Zn and Zg respectively. Then the random field uðt; xÞ has the following spectral representation: Z Z uðt; xÞ ¼ eihk;xi Eb;1 ðtb jkj2 ÞZn ðdkÞ þ t eihk;xi Eb;2 ðtb jkj2 ÞZg ðdkÞ; Rn
Rn
1 < b 6 2; t > 0; x 2 Rn ; with the covariance structure Z eihk;xy i Eb;1 ðtb jkj2 ÞEb;1 ðsb jkj2 ÞFn ðdkÞ þ ts covðuðt; xÞ; uðs; yÞÞ ¼ Rn Z eihk;xy i Eb;2 ðtb jkj2 ÞEb;2 ðsb jkj2 ÞFg ðdkÞ; Rn
where Eb;1 ðxÞ; x P 0 and Eb;2 ðxÞ; x P 0 are the Mittag–Leffler functions (2.1). Remark 2. Note that for 1 < b 6 2; the Mittag–Leffler functions Eb;1 ðxÞ; x P 0 and Eb;2 ðxÞ; x P 0 have zeroes in contrast to the case 0 < b 6 1, in which the Mittag–Leffler function Eb;1 ðxÞ; x P 0 is completely monotonic (see, for example, Schneider [24]). Now, let n ¼ 1; b ¼ 2 and consider the dÕAlembert formula (3.8), that is, the case gðxÞ ¼ 0. Then, for the wave equation ðb ¼ 2; a ¼ 2; c ¼ 0; l ¼ 1Þ with the initial random field (3.2), we obtain the following spectral representation Z uðt; xÞ ¼ eikx cosðktÞZn ðdkÞ ð3:9Þ R1
with covariance function covðuðt; xÞ; uðs; yÞÞ ¼
Z
cosðkðx yÞÞ cos2 ðktÞFn ðdkÞ: R1
We call (3.9) the dÕAlembert random field.
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