MAPPING BETWEEN SOLUTIONS OF FRACTIONAL DIFFUSION-WAVE EQUATIONS. Abstract

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1 MAPPING BETWEEN SOLUTIONS OF FRACTIONAL DIFFUSION-WAVE EQUATIONS Rudolf Gorenflo ), Asaf Iskenderov 2) and Yuri Luchko 3) Abstract We deal with a partial differential equation of fractional order where the time derivative of order β (; 2] is defined in the Caputo sense and the space derivative of order (; 2] is given as a pseudo differential operator with the Fourier symbol κ, κ IR. This equation contains as particular cases the diffusion and the wave equations and it has already appeared both in mathematical papers and in applications. The main result of the paper consists in giving a mapping in the form of a linear integral operator between solutions of the equation with different parameters, β and in presenting an explicit formula for the Green function of the Cauchy problem for the fractional diffusion-wave equation. Mathematics Subject Classification: 26A33 (main), 33C4, 44A, 45K5 Key Words and Phrases: fractional derivative; Mittag-Leffler function; generalized Wright function; Laplace transform; Fourier transform; Cauchy problem; Green function. Introduction A time-fractional diffusion equation obtained by replacing the first order timederivative in the diffusion equation by a generalized derivative of order β>(in Caputo or Riemann-Liouville sense) has been treated among others by Buckwar and Luchko [2], Engler [4], Fujita [6], Gorenflo, Luchko and Mainardi [7], Luchko ), 3) Partially supported by the Research Commission of Free University of Berlin (Project Convolutions )

2 2 R. Gorenflo, A. Iskenderov and Yu. Luchko and Gorenflo [], Mainardi [3] [4], Podlubny [7], Prüss [8], Samko, Kilbas and Marichev [2], Schneider and Wyss [22], and by Wyss [24]. A space-fractional diffusion equation obtained by replacing the second order space-derivative in the diffusion equation by the inverse of the Riesz potential of order > has been also considered (see, for example, Gorenflo and Mainardi [9], [], Mainardi, Paradisi and Gorenflo [5], Saichev and Zaslavsky [9], Scalas, Gorenflo and Mainardi [2] and references there). In [9] and [2] fractional derivatives appear in space as well as in time, as here in equation (). In this paper, stimulated by the above mentioned studies, we consider the Cauchy problem for the time- and space-fractional partial differential equation β u(x, t) t β = u(x, t) x, < 2, <β 2, () where the time-fractional derivative is defined in the Caputo sense, see [8] or [2] (t >, n <β n IN): d β f(t) dt β =(D β f)(t) = d n f(t) dt n, β = n IN, t Γ(n β) (t d n f(τ) τ)n β dτ n dτ, n <β<n (2) and the space-fractional derivative of order (; 2] is given as a pseudodifferential operator with the Fourier symbol κ, κ IR. Since the solutions (at least in form of integral representations) of this problem for some particular values of the parameters are well known (say, =2,<β 2), our attention will be first given to developing a method of constructing a solution of () with the parameters, β on the base of a known solution of () with the parameters γ, δ. More precisely, we present a mapping in the form of a linear integral operator between solutions of the equation () with the parameters, β and the same equation with the parameters γ, δ, the initial conditions at t =beingthe same for both equations. Similar results for the abstract Cauchy problem for the fractional evolution equation can be found in []. We give also an explicit formula for the Green function of the Cauchy problem for the fractional diffusion-wave equation (). In dependence on the relation between the parameters and β the Green function is given in terms of the generalized Wright function. 2. Preliminary facts and notations Let ˆf(κ) =(Ff)(κ) = + e +iκx f(x) dx, κ IR,

3 MAPPING BETWEEN SOLUTIONS... 3 be the Fourier transform of a sufficiently well-behaved function f(x), and let f(x) =(F ˆf)(x) = 2π be the inverse Fourier transform. Let f (s) =(Mf)(s) = + be the Mellin transform of a function f(τ), and let f(τ) =(M f )(τ) = 2πi γ+i γ i e iκx ˆf(κ) dκ, if ˆf L (IR), f(τ) τ s dτ, γ < Rs <γ 2, f (s) τ s ds, τ >, Rs = γ, γ <γ<γ 2, be the inverse Mellin transform. For a sufficiently well-behaved function f we define the space-fractional derivative of the order, < 2 for the equation () as a pseudo-differential operator with the symbol κ : (F d d x f)(κ) = κ ˆf(κ). (3) We remark that the notation for the space-fractional derivative used here was introduced by Saichev and Zaslavsky [9]. This operator is closely connected with the inversion of the Riesz potential (see [2]) and can be rewritten in the form ( <<2, ) d d x f(x) = 2Γ( )cos(π) For = the relation (3) can be interpreted as d d x f(x) = π f(x + ξ) 2f(ξ)+f(x ξ) ξ + dξ. (4) d + dx f(ξ) x ξ dξ. Let, F M denote the juxtapositions of a function f with its Fourier transform ˆf and its Mellin transform f, respectively. We then have for the corresponding convolutions: + g(x ξ)f(ξ)dξ F ĝ(κ) ˆf(κ), (5) g(ξ)f(τ/ξ) dξ ξ M g (s)f (s). (6)

4 4 R. Gorenflo, A. Iskenderov and Yu. Luchko Let us consider the initial value problem for the fractional differential equation of the order <β 2: (D β u)(t)+λu(t) =, u() = c, c, λ IC. (7) If β (; 2] we add the condition u () = to the initial conditions (7). It is known (see, for example, [8] or [2]) that the solution of this problem is given as u(t) =ce β ( λt β ), (8) where z k E β (z) = Γ( + βk),β>, is the Mittag-Leffler function. For the elements of the theory of this function we refer, for example, to [3]. 3. Main results For the equation () we consider the Cauchy problem u(x, ) = ϕ(x), x IR, u(±,t)=, t >. (9) If β (; 2] we add to (9) the condition u t (x, ) =, where u t (x, t) = tu(x, t). Let ϕ L (IR). By solution of the Cauchy problem for the equation () we mean a function u β (x, t) which satisfies the conditions (9). First we give the relation between the solutions of the Cauchy problem for the equation () with the parameters, β and γ, δ, respectively. Theorem 3.. Let u β (x, t) and u γδ (x, t) be solutions of the Cauchy problem for the equation () with the parameters (, 2], β (; 2] and γ, δ (; ], respectively and let γ +<. Assume u β (x, ) = u γδ (x, ) = ϕ(x), ϕ L (IR) and define Then S βγδ (x, t) := 2π u β (x, t) = + + E β ( κ t β ) E δ ( κ γ t δ ) e iκx dκ. S βγδ (x ξ,t)u γδ (ξ,t)dξ. () P r o o f. By applying the Fourier transform with respect to x to the equation () with the conditions (9) we get ( D β + κ ) û β (κ, t) =, ()

5 MAPPING BETWEEN SOLUTIONS... 5 { û(κ, ) = ˆϕ(κ), û t (κ, ) =, if <β 2. In accordance with (7), (8), the solution of (), (2) is given by (2) Analogously, û β (κ, t) = ˆϕ(κ)E β ( κ t β ). (3) û γδ (κ, t) = ˆϕ(κ)E δ ( κ γ t δ ). (4) If δ (; ] the Mittag-Leffler function E δ (z) has no zeros on the negative semiaxis (there are such zeros in the case δ (, 2], see [8] or [3]). The relations (3) and (4) give us û β (κ, t) = E β( κ t β ) E δ ( κ γ t δ ) ûγδ(κ, t). (5) Using the asymptotics of the Mittag-Leffler function (see [3], [8]) E (z) m k= z k Γ( k) + O( z m ), m IN, π/2 < arg z<2π π/2 and the condition γ +<of the theorem we arrive at the fact that for fixed t> the function Ŝ βγδ (κ, t) := E β( κ t β ) E δ ( κ γ t δ ) is in the space L (IR). Applying the convolution relation (5) to (5) we get the mapping (). Now we give an explicit formula for the Green function of the Cauchy problem for the equation () in terms of special functions of the hypergeometric type. Theorem 3.2. The solution of the Cauchy problem (9) for the equation () with <β<2, β, < 2 or β =, < 2 is given by the formula u β (x, t) = + G β (x ξ,t)ϕ(ξ)dξ, x IR, t >. (6) Here the Green function G β (x, t) is defined for x =by for x, β>by G β (,t)= { t β/ β, sin(π/)γ( β ), Γ(/), β =, πt / G β (x, t) = [ ( 2 2Ψ, 2 ), (, ) t β 2 π x (,β),(, 2 ); 2 x (7) ], (8)

6 6 R. Gorenflo, A. Iskenderov and Yu. Luchko for x, β<by G β (x, t) = π x 2 t β 2Ψ 2 + πt β and for x, β = by [ ( 2 2, ] 2 ), (, ) x ( β, β), ( 2, 2 ); 2 t β [ ( 2Ψ, 2 ), (, 2 ) ] x2 2 ( 2, ), ( β, 2β ); 4t 2β (9) G β (x, t) = π x t sin(π/2) t 2 +2 x t. (2) cos(π/2) + x 2 For the notation p Ψ q we refer to [23], in longhand writing we have [ ] (a,a ),...,(a p,a p ) pψ q (b,b )...(b q,b q ) ; z = This function is called the generalized Wright function. P r o o f. It follows from the formula (3) that G β (x, t) = 2π + p i= Γ(a i + A i k) z k q i= Γ(b i + B i k) k!. (2) e iκx E β ( κ t β ) dκ, x IR, t >. The last relation can be rewritten in terms of the cos-fourier transform G β (x, t) = π cos(κx) E β ( κ t β ) dκ, x IR, t >. (22) If x = the integral in the right-hand side is reduced to the Mellin integral transform of the Mittag-Leffler function at the point s =. It converges under the conditions < β < 2, β, > orβ =, >. Its value for <β<2, β is given by (see, for example, [6]) π E β ( κ t β ) dκ = πt β/ E β ( u)u du (23) = If β =weget Remarking now that Γ( )Γ( ) πt β/ Γ( β ) = π t β/ sin(π/)γ( β ). e κt dκ = Γ(/) πt /. G β (x, t) =G β ( x, t) (24)

7 MAPPING BETWEEN SOLUTIONS... 7 we can restrict attention to the case x = x > in the integral (22). This integral can be interpreted as a Mellin convolution (6) of the functions g(ξ) = E β ( ξ t β )andf(ξ) = π x ξ cos(/ξ) withτ =/ x. Using the known Mellin integral transforms (see, for example, [6]) g (s) = g(τ)τ s dτ = s Γ( )Γ( s ) t β s Γ( β s), < R(s) <, f Γ( (s) = 2 s 2 ) π x 2 s Γ( s 2 ), < R(s) <, the relation (6) and the inverse Mellin integral transform we arrive at the representation γ+i Γ( s G β (x, t) = )Γ( s ) Γ( 2 s 2 ) ( ) t β s π x 2πi γ i Γ( β s) 2 s Γ( s 2 ) ds, <γ<, x (25) the right-hand side of which is a particular case of the general Fox H-function (see, for example, [6], [2], [22]). Our next step is to represent it in terms of the special functions of the hypergeometric type. According to the general theory of the Mellin-Barnes integrals (see [6]) the integral in (25) is convergent if <β<2. The form of its series representation depends on the relation between the parameters and β. We consider three cases: (a) β>,(b)β<,(c)β =. (a) β>: In this case the contour of integration in the integral (25) can be transformed to the loop L starting and ending at infinity and encircling all the poles s k = k, k =,, 2,... of the function Γ(s/). The residue theorem gives us the result G β (x, t) = π x k= Γ( k) Γ( + βk)γ( 2 k) ( 2 t β ) k. (26) As can be seen from the asymptotics of the gamma function the series on the right-hand side converges absolutely in the whole complex plane. (b) β<: This situation is more complicated because we have to transform the contour of integration in (25) to the loop L + encircling all the poles s k = ( + k), k=,, 2,... and s m =2m+,m=,, 2,... of the functions Γ( s )andγ( 2 s 2 ), respectively. Applying the residue theorem we arrive at the representation G β (x, t) = π x 2 t β x Γ( k) ) k ( x Γ( β βk)γ( k) 2 t β (27)

8 8 R. Gorenflo, A. Iskenderov and Yu. Luchko ( x2 + Γ( + 2 m)γ( 2 m) ) m β πt m= m!γ( 2 + m)γ( β 2β m). 4t 2β Finally, we consider the case (c) β = : In this case the series representation of the Green function is given by the formula (26) if <t< x and by the formula (27) if < x <t. Let us simplify the corresponding expressions. Using the duplication and reflection formulae for the gamma function we get for <t< x : G (x, t) = Γ( k) ( 2 t ) k π x Γ( + k)γ( 2 k) x = π x k= k= ) k sin(πk/2) ( t x = x t sin(π/2) π t 2 +2 x t cos(π/2) + x 2. To get the last relation we used the formula r k sin(ka) = I r k e ika = I reia re ia = r sin a,a IR, r <. 2r cos a + r2 k= k= As to the representation (27) in the case β = we see that the second sum in the right-hand side is identically equal to zero because /Γ( β 2β m)= /Γ( 2m) =, m =,, 2,...Thenwehavefor< x <t: G (x, t) = π x 2 t = sin(πk/2) π x k= The theorem is proved. ( x t Γ( k) Γ( k)γ( k) ( x 2 t ) k ) k = x t sin(π/2) π t 2 +2 x t cos(π/2) + x 2. Remark 3.. Using the duplication formula for the gamma function and the substitution s = τ we can transform the representation (25) to the form G β (x, t) = γ+i ( ) Γ( τ)γ(τ)γ( τ) t β τ x 2πi γ i Γ( 2 τ)γ( 2 τ)γ( βτ) x dτ, <γ<. (28) Correspondingly, the Green function G β (x, t) can be represented in terms of the generalized Wright function 2 Ψ 3 with the argument t β / x : G β (x, t) = [ ] (,),(, ) tβ x 2 Ψ 3 (,β),(, 2 ), (, 2 ); x (29)

9 MAPPING BETWEEN SOLUTIONS... 9 in the case β>, G β (x, t) = x t β 2Ψ 3 + t β in the case β<. [ (, ), (, ) ( β, β), ( 2, 2 ), ( 2, 2 [ ( 2Ψ, ), (, ) ( x ) ] / 3 ( 2, 2 ), ( 2, 2 ), ( β, β ); t β x ); t β Remark 3.2. For β = = our method gives the well-known Cauchy kernel G (x, t) = t π t 2 + x 2. Therefore, the Green function (2) in the case = β, <2givenby G β (x, t) = π x t sin(π/2) t 2 +2 x t cos(π/2) + x 2 can be considered as a fractional generalization of the Cauchy kernel. Remark 3.3. In the case β = the first term of the representation (9) is identically equal to zero because /Γ( β βk) =/Γ( k) =,,, 2,... The formulae (8), (9) can be rewritten in this case in the form G (x, t) = π x k= Γ( + k) k! ( sin(πk/2) G (x, t) = Γ((2k +)/) πt (2k)! in accordance with the results presented in [5]. Remark 3.4. ( x2 t 2 t ) k x for <<, ] ) k, for < 2, If = 2 the formula (9) can be rewritten in the form G 2β (x, t) = [ x π + ( ) k Γ( 2 2 (2k +2)) Γ( + k)γ( β 2 (2k +2)) ( ) k Γ( 2 (2k +)) Γ( + k)γ( β 2 (2k +)) ( ) x 2k+ ] 2t β/2. ( ) x 2k+2 2t β/2 Applying the duplication and reflection formulae for the gamma function we get G 2β (x, t) = [ 2 x Γ(2 + 2k)Γ( β 2 (2k +2)) ( ) x 2k+2 t β/2 (3)

10 R. Gorenflo, A. Iskenderov and Yu. Luchko where = 2 x k= + Γ( + 2k)Γ( β 2 (2k +)) ( ) k+ ( ) x k t β/2 = Γ(k)Γ( β 2 k) φ(; β; z) = ( ) x 2k+ ] t β/2 2t β/2 φ( β 2 ; β 2 ; x ), tβ/2 z k k!γ(β + k) is the Wright function. The last formula is in accordance with the known results (see, for example, [4]). Acknowledgement: The authors are grateful to Professor F. Mainardi for discussions and for helpful comments. References [] E. B a z h l e k o v a, The abstract Cauchy problem for the fractional evolution equation. Fractional Calculus & Applied Analysis (998), [2] E. B u c k w a r, Yu. L u c h k o, Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. J. Math. Anal. Appl. 227 (998), [3] M.M. D j r b a s h i a n, Integral Transforms and Representations of Functions in the Complex Plane. Nauka, Moscow (966). In Russian. [4] H. E n g l e r, Similarity solutions for a class of hyperbolic integrodifferential equations. Differential Integral Eqn-s, No 5 (997), [5] A. E r d élyi,w.magnus,f.oberhettingerandf.g.tr i c o m i Higher Transcendental Functions, Vol. 3, McGraw-Hill Book Co., New York (954). [6] Y. F u j i t a, Integrodifferential equation which interpolates the heat and the wave equations. Osaka J. Math. 27 (99), 39-32, [7]R.Gorenflo,Yu.LuchkoandF.Mainardi,Analytical properties and applications of the Wright function. Fractional Calculus & Applied Analysis 2 (999), [8] R. G o r e n f l o and F. M a i n a r d i, Fractional calculus: integral and differential equations of fractional order. In: Fractals and Fractional Calculus in Continuum Mechanics (Eds. A. Carpinteri, F. Mainardi). Wien and New York, Springer Verlag (997),

11 MAPPING BETWEEN SOLUTIONS... [9] R. G o r e n f l o and F. Ma i n a r d i, Approximation of Lévy-Feller diffusion by random walk. ZAA 8 (999), [] R. G o r e n f l o and F. M a i n a r d i, Random walk models for space-fractional diffusion processes. Fractional Calculus & Applied Analysis (998), [] Yu. L u c h k o and R. G o r e n f l o, Scale-invariant solutions of a partial differential equation of fractional order. Fractional Calculus & Applied Analysis (998), [2] Yu. L u c h k o and R. G o r e n f l o, An operational method for solving fractional differential equations with the Caputo derivatives. Acta Mathematica Vietnamica 24 (999), [3] F. M a i n a r d i, Fractional relaxation-oscillation and fractional diffusionwave phenomena. Chaos, Solitons and Fractals 7 (996), [4] F. M a i n a r d i, Fractional calculus: some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics (Eds. A. Carpinteri and F. Mainardi). Wien and New York, Springer Verlag (997), [5] F. M a i n a r d i, P. P a r a d i s i and R. G o r e n f l o, Probability distributions generated by fractional diffusion equations. In: (eds. J. Kertesz, I. Kondor) Econophysics: an Emerging Science, Kluwer, Academic Publishers, Dordrecht, 999, in press. [6] O.I. M a r i c h e v, Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables. Chichester, Ellis Horwood (983). [7] I.Podlubny,Fractional Differential Equations, Mathematics in Science and Engineering, Vol. 98. New York, Academic Press (999). [8] J. P r ü s s, Evolutionary Integral Equations and Applications. Basel, Birkhäuser (993). [9] A. S a i c h e v, G. Z a s l a v s k y, Fractional kinetic equations: solutions and applications. Chaos 7, No 4 (997), [2]S.G.Samko,A.A.KilbasandO.I.Marichev,Fractional Integrals and Derivatives: Theory and Applications. New York and London, Gordon and Breach Science Publishers (993). [2] E.Scalas,R.GorenfloandF.Mainardi,Fractionalcalculus and continuous-time finance. Submitted.

12 2 R. Gorenflo, A. Iskenderov and Yu. Luchko [22] W.R.Schneider,W.Wyss,Fractionaldiffusionandwaveequations. J. Math. Phys. 3 (989), [23] E. M. W r i g h t, The asymptotic expansion of the generalized hypergeometric function. Journal London Math. Soc. (935), [24] W. W y s s, Fractional diffusion equation. J. Math. Phys. 27 (986), ) Dept. of Mathematics and Computer Science Received: Arnimalle 2 6 Free University of Berlin D-495 Berlin GERMANY fax: gorenflo@math.fu-berlin.de 2) Dept. of Applied Mathematics, Baku State University, 7-th Binagadi Str., 3/2, 37, Baku-, AZERBAIJAN tel/fax : (+9942) asaf@azdata.net 3 ) Dept. of Mathematics and Computer Science Arnimallee 2 6 Free University of Berlin D-495 Berlin GERMANY fax: luchko@math.fu-berlin.de Prof. Dr. Rudolf GORENFLO is a member of the Editorial Board of the FCAA Journal. Professor emeritus at Free Univ. of Berlin, there Full Professor Former positions: researcher at Max-Planck-Institute for Plasma Physics in Garching near München; Doc. and Prof. at Aachen Technical Univ.; Guest Prof. at Heidelberg and Tokyo Universities; Director of Third Math. Institute of Free Univ. of Berlin; President of Berlin Math. Soc.; Head of Free Univ. Research Projects Modeling and Discretization, Regularization, Convolutions ; Leading member of NATO Collaborative Res. Project Fractional Order Systems, etc.

13 MAPPING BETWEEN SOLUTIONS... 3 His main research interests are in Numerical Analysis, Integral Equations, Fractional Calculus, with more than research articles, co-editorship of 7 international conference proceedings. Author of the book: R. Gorenflo and S. Vessella. Abel Integral Equations: Analysis and Applications. LNM No 46, Springer- Verlag, 99. Co-editor of Berliner Studienreihe zur Mathematik, member of the advisory board of the journal Acta Mathematica Vietnamica, memberof the editorial board of Journal of Inverse and Ill-Posed Problems. Prof. Dr. Asaf ISKENDEROV - Full Professor, Doctor of Sciences (978) from M.Lomonosov s Moscow State University, Head of Department Optimization and Control and Head of Scientific Laboratory Mathematical Modeling at Baku State University. Specialist in fields: Mathematical Physics, Numerical Analysis and Optimal Control. Research interests: Inverse and Ill-posed problems, numerical methods of mathematical modeling, and optimal control of nonlinear processes. Author of more than papers and 2 books, participant in more than 4 International Conferences, guest researcher at Free University of Berlin in 993, 994, 998, had invited reports at prestige Universities in Republics of former Soviet Union, Germany, Czech Republic, Turkey, etc. Dr. Yuri LUCHKO is graduated (985) and Ph.D. (993) from the Belarussian State University, Minsk, now a participant of the Research Project Convolutions at the Free University of Berlin. Main research interests: Operational Calculus, Integral Transforms, Fractional Calculus, Special Functions. About 4 research articles on above subjects. Coauthor of the book: The Hypergeometric Approach to Integral Transforms and Convolutions. Mathematics and its Applications, 287, Kluwer Acad. Publ., 994. DAAD-grant at Free University of Berlin (Sept Sept. 995).