Distributed Order Derivatives and Relaxation Patterns

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1 Distributed Order Derivatives and Relaxation Patterns arxiv:95.66v [math-ph] 5 May 29 Anatoly N. Kochubei Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska 3, Kiev, 6 Ukraine Running head: Distributed Order Derivatives Abstract We consider equations of the form Dρ) u ) t) = λut), t >, where λ >, D ρ) is a distributed order derivative, that is D ρ) ϕt) = D α) ϕ)t)dρα), D α) is the Caputo-Dzhrbashyan fractional derivative of order α, ρ is a positive measure. The above equation is used for modeling anomalous, non-exponential relaxation processes. In this work we study asymptotic behavior of solutions of the above equation, depending on properties of the measure ρ. Key words: fractional derivative; distributed order derivative; anomalous relaxation PACS numbers: 2.3.-f; 5.9.+m; 87.Ed.

2 INTRODUCTION Anomalous, non-exponential, relaxation processes occur in various branches of physics; see e.g. [3, 8, 2] and references therein. Just as the exponential function e λt λ > ) appearing in the description of classical relaxation is a solution of the simplest differential equation u = λu with the initial condition u) = ), new kinds of derivatives are used to obtain models of slow relaxation. It is generally accepted now that the power law of relaxation corresponds to the Cauchy problem D α) u ) t) = λut), t > ; u) =, ) where D α) u ) t) = d Γ α) dt t t τ) α uτ)dτ t α u) 2) isthecaputo-dzhrbashyanfractionalderivativeoforderα,); wereferto[7,3]forvarious notions and results regarding fractional differential equations. The solution of the problem ) has the form ut) = E α λt α ) where E α is the Mittag-Leffler function, and ut) Ct α here and below we denote various positive constants by the same letter C), as t. This kind of evolution describes the temporal behavior related to α-fractional diffusion typical for fractal media. A still slower, logarithmic relaxation [, 6] is described by the Cauchy problem D µ) u ) t) = λut), t > ; u) =, 3) where D µ) u ) t) = D α) u ) t)µα)dα, 4) µ is a non-negative continuous function on [, ]. A rigorous mathematical treatment of the problem 3) for general classes of the weights µ was given in [4, 5]. Some nonlinear equations with distributed order derivatives were studied in []. As above, this kind of relaxation models is connected with models of ultraslow diffusion [4,, 2, 4, 7, 22]; these papers contain further references on related subjects. Let u λ t) be the solution of the problem 3) the notation is changed slightly, compared to [4]). If µ), then u λ t) Clogt), t. 5) If µα) aα ν, α a >, ν > ), then u λ t) Clogt) ν, t. 6) Itwasassumedeverywhere in[4]thatµ C 3 [,]andµ) ; infact, intheinvestigation of the asymptotic behavior of u λ we use only that µ L,). Therefore the arguments in [4] cover the case where µα) aα ν, α, 2

3 with a >, < ν <, and yield the asymptotics u λ t) Clogt) +ν, t. 7) In all the above cases, the exponential function e λt, the Mittag-Leffler powerlike evolution E α λt α ), and all the logarithmic evolutions 5)-7), the resulting functions are completely monotone, that is ) j u j) λ t), j =,,2,..., for all t. In this paper we look for other possible relaxation patterns corresponding to the weight functions µ tending to at the origin) faster than the power function Section 2) or to the definition of the distributed order derivative not by the formula 4) but by the expression Dρ) u ) t) = D α) u ) t)dρα), 8) where ρ is a jump measuresection 3). In these cases the solutions remain completely monotone while their asymptotic behavior can be quite diverse, from the iterated logarithmic decay to a function decaying faster than any power of logarithm but slower than any power function. 2 THE MAIN CONSTRUCTIONS Suppose that ρ is a positive finite measure on [,] not concentrated at, and consider the distributed order derivative 8). Substituting 2) into 8) we find that where Dρ) u ) t) = d dt ks) = t kt τ)uτ)dτ kt)u) 9) s α dρα), s >. ) Γ α) The right-hand side of 9) makes sense for a continuous function u, for which the derivative d t kt τ)uτ)dτ exists. dt It is clear from ) that k L loc, ), and k is decreasing. Therefore the function k possesses the Laplace transform Kp) = ks)e ps ds = p α dρα), Rep >. The holomorphic function Kp) can be extended analytically onto the whole complex plane cut along the half-axis R = {Imp =,Rep }. Obviously, Kp), as p a precise asymptotics is found for some cases in [4] and Section 3 below). 3

4 Note that if we consider the moments of the measure ρ, and introduce their generating function Mz) = M n = x n dρx), z n M n n! = e xz dρx), then Kp) = p Mlogp). The moment generating functions were studied by a number of authors for example, [5, 2]). Considering the relaxation equation Dρ) u ) t) = λut), t >, ) with λ >, we apply formally the Laplace transform which is justified post factum, using the smoothness properties and the asymptotic behavior of the solution). For the Laplace transform ũ λ p) of the solution u λ t) of the equation ) satisfying the initial condition u λ ) = we get the expression ũ λ p) = Kp) pkp)+λ. 2) Since pkp), as p, we have ũ λ p) p, p = σ +iτ, σ,τ R, τ. Therefore [6], ũ λ is the Laplace transform of some function u λ t), and for almost all t, u λ t) = d dt2πi Note that for p C\R we have ImpKp) = γ+i γ i e pt p Kp) dp. 3) pkp)+λ p α sinαargp)dρα), so that ImpKp) = only for argp =. This means that pkp)+λ, and the representation 3) is valid for an arbitrary γ >. As in [4], in the present more general situation we deform the contour of integration, and then differentiate under the integral, so that u λ t) = Kp) dp 4) 2πi pkp)+λ S γ,ω e pt 4

5 where the contour S γ,ω consists of the arc and two rays T γ,ω = {p C : p = γ, argp ωπ}, Γ ± γ,ω = {p C : argp = ±ωπ, p γ}. 2 < ω <, Just as in the case of a measure ρ with a smooth density considered in [4], it is easy to show that the function u λ belongs to C, ) and is continuous at the origin; from its construction and the formula 9), it follows that the initial condition u λ ) = is indeed satisfied. Let us consider first the case of a measure ρ with a continuous density, dρα) = µα)dα. This case was investigated in [4], and it was proved that u λ is completely monotone various additional assumptions made in [4] and needed for other problems studied in that paper, were not actually used here). In this paper we consider the case of a different behavior of the density µ near the origin, implying a different asymptotics of u λ t), as t. Theorem. If µ C[,], and where a >, γ >, β >, then Proof. Let us write Kp) as µα) aα γ e β α, as α, u λ t) Clogt) γ e 2 βlogt) 2, t. 5) Kp) = p e αz µ α)dα, z = log p, where µ is the extension of µ by zero onto R +, and use an asymptotic result for Laplace integrals from [9] Theorem 3., Case 9). We get Kp) 2ap β z )γ+ 2 K γ+ 2 βz), p +, where, as before, z = log p ), K m is the McDonald function. It is well known that K m t) π 2) /2 t /2 e t, t, so that Kp) Cp L ), p +, 6) p where Ls) = logs) γ e 2 βlogs) 2. 5

6 It follows from 6) or directly from the definition of Kp)) that pkp), as p +. Therefore by 2), ũ λ p) λ Kp), as p +. Since we already know that the function u λ is monotone, we may apply the Karamata-Feller Tauberian theorem see Chapter XIII in [9]) which implies the desired asymptotics of u λ t), t. In the case under consideration, the function u λ t) decreases at infinity slower than any negative power of t, but faster than any negative power of logt. It is also seen from 5) that the decrease is accelerated if β > becomes bigger, so that the less the weight function µ is near, the faster is the relaxation for large times. 3 THE STEP STIELTJES WEIGHT Let us consider the case where the integral in 8) is a Stieltjes integral corresponding to a right continuous non-decreasing step function ρα). In order to investigate a sufficiently general situation, we assume that the function ρ has two sequences of jump points, β n and ν n, n =,,2,..., where β n, ν n, β = ν,). We may assume that the sequence {β n } is strictly decreasing while {ν n } is strictly increasing. Denote ρt) = ρt) ρt ), = ρβ n ) n ), η n = ρν n ) n ); we have,η n > for all n. It will be convenient to assume that β < e and to write η =. Since ρ is a finite measure, we have also By ), so that ks) = <, Γ β n ) s βn + Kp) = η n <. 7) n= p βn + η n Γ ν n ) s νn, s >, η n p νn. 8) As before, we denote by u λ t) the solution of the relaxation equation ) with the initial condition u λ ) =. The symbol f g will, as usual, mean that f = Og) and g = Of). n= Theorem 2. i) The function u λ is completely monotone. iii) If ii) u λ x) [ Γ2 β n ) x βn + loglog β n ) b < b > ), then u λ x) = O loglogx) b 6 ] η n, x ; 9) Γ2 ν n ) x νn ), x. 2)

7 iv) If β b n < b > ), then ) u λ x) = O, x. 2) logx) b Proof. Using the representation 4) and following [4] we can write u λ t) = 2πi T γ,ω e pt Kp) pkp)+λ dp+ π Im γ r e treiωπ dr λ π Im γ e treiωπ rre iωπ Kre iωπ )+λ) Let us substantiate passing to the limit as γ. As p, pkp), so that Kp) pkp)+λ C ξn p βn +η n p νn ), whence as γ. It was shown in [4] that J Ce γt ξn γ βn +η n γ νn), J 2 π s e s sinstanωπ)ds, dr def = J +J 2 J 3. as γ. The integral J 3 is the sum of I = λ π γ Im e treiωπ r ) ) Re re iωπ Kre iωπ )+λ dr and We have I 2 = λ π γ Im Re e treiωπ r e treiωπ r ) ) ) Im re iωπ Kre iωπ )+λ = r e trcosωπ sintrsinωπ), and this expression has a finite limit, as r. Since also pkp), as p, we see that we may pass to the limit in I, as γ. 7 dr.

8 Let Substituting 8) and denoting Φr,ω) = Im re iωπ Kre iωπ )+λ. { [ Gr,ω) = ξn r βn cosωπβ n )+η n r νn cosωπν n ) ] +λ } 2 { [ + ξn r βn sinωπβ n )+η n r νn sinωπν n ) ]} 2 we find that [ ξn r βn sinωπβ n )+η n r νn sinωπν n ) ] Φr,ω) = Gr, ω) The denominator tends to λ 2, as r. Noting that ) Re e treiωπ r = r e trcosωπ costrsinωπ). and using 7) we find that the integrand in I 2 belongs to L, ). Passing to the limit γ we obtain the representation u λ t) = π s e s sinstanωπ)ds λ π r e trcosωπ sintrsinωπ)ψr,ω)dr λ r e trcosωπ costrsinωπ)φr,ω)dr π where [ ξn r βn cosωπβ n )+η n r νn cosωπν n ) ] +λ Ψr,ω) = Gr, ω) Next, let us pass to the limit, as ω. It follows from the Lebesgue theorem that the first two integrals tend to zero, and we get u λ t) = λ π r e tr [ ξn r βn sinπβ n )+η n r νn sinπν n ) ] Gr, ) that is, up to a positive factor, u λ is the Laplace transform of a locally integrable non-negative function bounded at infinity. Therefore u λ is completely monotone.. dr, 8

9 ii) It will be convenient to turn to the Laplace-Stieltjes transform instead of the Laplace transform. Set κx) = x ks)ds, v λ x) = so that the Laplace-Stieltjes transforms are as follows: κp) = By 2), v λ p) = lp) κp) where lp) = We have κx) = x u λ s)ds, e px dκx) = Kp), v λ p) = ũ λ p). pkp)+λ Γ2 β n ) x βn + The function κ is monotone increasing. Denote Since β n,ν n,), we find that κ ζ) = limsup x n= is a slowly varying function near the origin. η n Γ2 ν n ) x νn, x >. 22) κζx) κx), ζ >. κζx) ζκx), 23) so that κ ζ) ζ. Thus, κ is an O-regularly varying function see [2], especially Corollary 2..6). On the other hand, it is seen from 22) that κ ζ) for ζ >, so that κ +) =. Thus we are within the conditions of the Ratio Tauberian Theorem [2], Theorem 2..), which yields the relation v λ x) l )κx), x, so that x v λ x) Cκx), x. 24) In order to pass from 24) to 9), we have to check further Tauberian conditions. Since u λ is completely monotone, it is, in particular, non-increasing, thus belonging to the class BI see Section 2.2 in [2] for the definitions of this class and the class PI used below). It follows from 23) that also κ BI. Next, κx) Since ζ βn ζ ν for ζ >, we have κζx) κx) ζ ν Γ2 β n ) x βn + Γ2 β n) x βn + η Γ2 ν ) x ν Γ2 β n) x βn + ηn Γ2 ν n) x νn ) = ζ ν 9 η Γ2 ν ) x ν. n=2 η n Γ2 ν n) x νn Γ2 β n) x βn + ηn Γ2 ν n) x νn )

10 where n=2 η n Γ2 ν n) x νn Γ2 β n) x βn + ηn Γ2 ν n) x νn ) Cx +ν n=2 η n Γ2 ν n ) x νn, as x. Thus, κ PI. Now we are within the conditions of the O-version of Monotone Density Theorem [2], Proposition 2..3), which implies the required asymptotic relation 9). iii) iv) Let us prove 2). The proof of 2) is similar and simpler, and we leave it to a reader. It is obviously sufficient to deal with the first summand in each element of the series in 9). Let us consider the function where a,b >. We have ϕx) = x a loglogx) b, x e, 25) ϕ x) = x a loglogx) b ψx) where ψx) = b aloglogx. It is easy to check that ψ decreases on [e, ), ψe) = b, logx ψx), as x. The maximal value of the function ϕ is attained at a single point x where ψx ) =, that is b = aloglogx. logx Denote y = logx. Then b = alogy, so that y logy = b. We will in fact need an y a asymptotic behavior of y as a function of a, as a. It is known see Section I.5.2 in [8]) that Therefore where the constants do not depend on a,b. We have ϕx) ϕx ), so that, by 25), y = log b a loglog b a +O loglog b a log b a b C log b ) b x C 2 log b 26) a a a a) x a ϕx ) loglogx) b. We need an estimate of ϕx ) making explicit its dependence on a. By 26), ) a b x a C 3 log b ) a a a ). where ) a b = e alog b a and ) log b a a = e aloglog b a, as a. Next, x C 4, so that a a loglogx loglogc 4 +log a ) logc 5log a ) C 6loglog a

11 whence ϕx ) C 7 loglog a )b. As a result, Γ2 β n ) x βn C 8 [ loglog ) ] b β n loglogx) b, and we have proved 2). References [] T. M. Atanacković, L. Oparnica, and S. Pilipović, J. Math. Anal. Appl ), 59 68; ), 749. [2] N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular Variation. Cambridge University Press, 989. [3] A. Blumen, A. A. Gurtovenko, and S. Jespersen, J. Non-Crystalline Solids 35 22), 7 8. [4] A. V. Chechkin, R. Gorenflo, I. M. Sokolov and V. Yu. Gonchar, Fract. Calc. Appl. Anal. 6 23), [5] J. H. Curtiss, Ann. Math. Statistics 3 942), [6] V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus, Pergamon Press, Oxford, 965. [7] S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei. Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkhäuser, Basel, 24. [8] M. V. Fedoryuk, Asymptotics. Integrals and Series, Nauka, Moscow, 987 Russian). [9] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York, 97. [] R. Gorenflo and F. Mainardi, J. Phys.: Conf. Ser. 7 25), 6. [] R. Gorenflo and F. Mainardi, In: M. M. Novak Ed.), Complex Mundi. Emergent Patterns in Nature, World Scientific, Singapore, 26, pp [2] A. Hanyga, J. Phys. A 4 27), [3] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 26. [4] A. N. Kochubei, J. Math. Anal. Appl ), [5] A. N. Kochubei, Ukrainian Math. J. 6 28),

12 [6] F. Mainardi, A. Mura, R. Gorenflo, and M. Stojanović, J. Vib. Control 3 27), [7] M. M. Meerschaert and H.-P. Scheffler, Stoch. Proc. Appl. 6 26), [8] R. Metzler and J. Klafter, J. Phys. A 37 24), R6 R28. [9] E. Ya. Riekstynsh Riekstiņš), Asymptotic Expansions of Integrals, Vol., Zinatne, Riga, 974 Russian). [2] O. Roth, S. Ruscheweyh, and L. Salinas, Proc. Amer. Math. Soc ), [2] P. Talkner, Phys. Rev. E 64 2), art. 6. [22] S. Umarov and R. Gorenflo, Z. Anal. Anwend ),

DETERMINATION OF AN UNKNOWN SOURCE TERM IN A SPACE-TIME FRACTIONAL DIFFUSION EQUATION

Journal of Fractional Calculus and Applications, Vol. 6(1) Jan. 2015, pp. 83-90. ISSN: 2090-5858. http://fcag-egypt.com/journals/jfca/ DETERMINATION OF AN UNKNOWN SOURCE TERM IN A SPACE-TIME FRACTIONAL