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The stretched exponential behavior and its underlying dynamics. The phenomenological approach

  • Katarzyna Górska EMAIL logo , Andrzej Horzela , Karol A. Penson , Giuseppe Dattoli and Gerard H. E. Duchamp
Published/Copyright: February 19, 2017
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 20 Issue 1

Abstract

We show that the anomalous diffusion equations with a fractional spatial derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the Lévy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution is conceived as integral transform whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and for different initial conditions.

MSC 2010: Primary 35R11; Secondary 26A33; 60G18; 60G52; 49M20
Key Words and Phrases: fractional calculus; stretched or compressed exponential function; fractional ordinary and partial differential equations; Green functions

Appendix A The derivations of Eqs. (2.13) and (2.14)

We start with the evolution property written for the stretched exponential function for t0 ≤ t1 ≤ t2:

(8.1)e−(t2−t0τ0)α=e−(t2−t1τ0)α∘e−(t1−t0τ0)α.

The symbol ’o’ denotes a composition of functions which should be understood as the multiplication of stretched exponential functions defined in Eq. (2.3). Thus, the r.h.s. of Eq. (8.1) is equal to

(8.2)RHS of Eq. (8.1)=∫0∞dye−ygα((t2−t1τ0)α,y)∫0∞dze−zgα((t1−t0τ0)α,z)
(8.3)=∫0∞dy∫y∞dxe−xgα((t2−t1τ0)α,y)gα((t1−t0τ0)α,x−y)=∫0∞dxe−x∫0xdygα((t2−t1τ0)α,y)gα((t1−t0τ0)α,x−y),

where in Eq. (8.2) the new variable x = y + z is employed. After applying Eq. (2.3) for the l.h.s. of Eq. (8.1) and comparing with Eq. (8.3) we obtain Eq. (2.13).

The proof of Eq. (2.14) begins with the evolution property for the compressed exponential:

(8.4)e−|t2−t0τ0|α=e−|t2−t1τ0|α∘e−|t1−t0τ0|α,

where ’o’ denotes now the composition of two functions given in Eq. (2.9). Substituting Eq. (2.9) into Eq. (8.4) and making similar changes of variable like in the case of Eq. (8.2) we get Eq. (2.14).

Appendix B Calculation of the second integral in Eq. (3.6)

The second integral in Eq. (3.6) has the simple singularity at y = x, where 0 < x < ∞. Taking the contour of integration which is the upper right one fourth of the circle, denoted by QR, with the pole at y = x, see Fig. 3, and using the Cauchy’s integral theorem [8], we find that

Figure 3 The contour of integration used in Eq. (9.1).
Figure 3

The contour of integration used in Eq. (9.1).

(9.1)∫0xe−yξx−ydy=−limR→∞∫QRe−zξx−zdz−limε→0∫Sεe−zξx−zdz−limR→∞∫iR0ie−irξx−irdr−limR→∞∫xRe−yξx−ydy.

The imaginary part of the integral over the quadrant QR vanishes. This can be shown by setting z = Reiθ and studying the integral

(9.2)ImlimR→∞∫QRe−zξx−zdz=limR→∞∫0π/2Im[ψ(Reiθ)]dθ,

where ψ(z) = ize-ξz/(x – z) and Im[ψ(Reiθ)] is an even function of R and θ equals to

(9.3)Im[ψ(Reiθ)]=Re−Rξcos⁡θx2−2Rxcos⁡θ+R2[(xcos⁡θ−R)×cos⁡(Rξsin⁡θ)+xsin⁡θsin⁡(Rξsin⁡θ)].

Observe that Eq. (9.3) for θ = π/2 can be estimated by Im[ψ(iR)] ≤ R(x–R)/(x2 +R2) which it is smaller or equal to Im[ψ(R)] for 0 < x < R. That leads to

Im[ψ(Reiθ)]≤Im[ψ(R)]=Re−Rξx−R.

After substituting it into Eq. (9.2) it can be shown that the imaginary part of integral over the quadrant of radius R goes to zero by choosing R sufficiently large.

We set x – z = εeiϕ in the second integral in the right hand side of Eq. (9.1), so that

limε→0∫Sεe−zξx−zdz=−ie−xξlimε→0∫0πeεexp⁡(iϕ)ξdϕ=−iπe−xξ.

The third integral in the r.h.s. of Eq. (9.1) after changing the variable of integration ir = y and using [23, Eq. (3.352.1) on p. 340] gives the real function

∫−R0e−yξx−ydy=e−xξEi(Rξ+xξ)−Ei(xξ),

where Ei(x) is the exponential integral, see [23, Section 8.2]. In the forth integral in the r.h.s. of Eq. (9.1) we change the variable z as follows x – z = u. That gives

(9.4)∫xRe−yξx−ydy=e−xξ∫x−R0euξu.

Using the series representation of the exponential it can be shown that Eq. (9.4) is a real function which goes to infinity for R → ∞. Concluding the considerations we find

(9.5)Im∫0xe−yξx−ydy=πe−xξ.

Acknowledgements

We thank the anonymous referees for constructive remarks. We thank Dr. Ł. Bratek and Prof. K. Weron for the important discussions.

K. G., A. H. and K. A. P. were supported by the PAN-CNRS program for French-Polish collaboration and the BGF scholarship founded by French Embassy in Warsaw, Poland. Moreover, K. G. thanks for support from MNiSW (Warsaw, Poland), “Iuventus Plus 2015-2016”, program no IP2014 013073 and the Laboratoire d’Informatique de l’Université Paris-Nord in Villetaneuse (France) whose warm hospitality is greatly appreciated.

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Received: 2016-5-10
Revised: 2016-12-6
Published Online: 2017-2-19
Published in Print: 2017-2-1

© 2017 Diogenes Co., Sofia

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https://doi.org/10.1515/fca-2017-0014
Keywords for this article
fractional calculus; stretched or compressed exponential function; fractional ordinary and partial differential equations; Green functions

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  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 20–1–2017)
  4. Survey paper
  5. Ten equivalent definitions of the fractional laplace operator
  6. Research paper
  7. Consensus of fractional-order multi-agent systems with input time delay
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  9. Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-Type Hadamard derivatives
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  23. A foundational approach to the Lie theory for fractional order partial differential equations
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  27. New results in stability analysis for LTI SISO systems modeled by GL-discretized fractional-order transfer functions
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