# Fractional derivatives Fractional derivatives arise as a generalization of integer order derivatives and have a long history: its origin could be found in the work of G. W. Leibniz and L. Euler. Shortly after being introduced, the new theory turned out to be very attractive for many famous mathematicians and scientists (e.g. P. S. Laplace, B. Riemann, J. Liouville, N. H. Abel, J. B. J. Fourier, et al.) due to the numerous possibilities it offered for applications. Fractional calculus, the field of mathematics dealing with operators of differentiation and integration of arbitrary real or even complex order, extends many of the modeling capabilities of conventional calculus and integer-order differential equations and finds its application in various scientific areas, such as physics, mechanics, engineering, economy, finance, biology, chemistry, etc.

Basic definitions and properties of integrals and derivatives of an arbitrary order could be seen in the following presentation: Our group is active in working on the development of the fractional calculus and our work is influenced by the requirements coming from applications but also includes extensive studies of certain aspects of the theory.

## Analysis of equations involving fractional derivatives

Equations involving fractional differential operators are called Fractional Differential Equations (FDE). One can consider various types of FDEs, such as heat and diffusion type FDE, including fractional derivatives instead of integer order ones in time, or any evolution system with fractional derivatives in time and with fractional operators in space variable, as for example fractional (sub-)Laplacian. In particular, second-order partial differential equations with Caputo fractional derivatives in the time-variable and Bessel operator in the space-variable, or multi-term generalisation of the time-fractional diffusion-wave equation in abstract settings, and their corresponding initial and boundary value, direct and inverse type problems have been in the focus of our research recently. In order to study regularity properties of solutions for some cases of the fractional wave equation, microlocal analysis of solutions has been conducted, and we are interested in improving existing results, as well as in further similar applications of pseudo-differential and microlocal techniques in the analysis of various FDE. ## Fractional derivatives and the wave equation

One of the first employment of real-order derivatives were in linear viscoelasticity, for the purpose of a more precise description of the properties and behavior of viscoelastic materials. Models that describe waves occurring in the viscoelastic media are given in the form of fractional differential equations supplied with certain initial and/or boundary conditions and are derived by the rheological analogy from equations that postulate basic physical laws. The presence of fractional derivatives in these models is natural in a sense, due to the viscoelastic character of the media or material under consideration. The latter has been described by different constitutive equations involving derivatives of real and also complex order, such as the fractional generalized Zener, Maxwell, or Kelvin-Voigt model. Wave propagation phenomena has been studied on finite and infinite spatial domain. We are concerned with the questions of solvability and regularity of solutions, thermodynamical restrictions, the impact of the initial data and boundary conditions, as well as in numerical verifications of results.

## Analysis of FDE on groups

For a homogeneous group of homogeneous dimension Q, for a (Haar) measurable and compactly supported function, one can define fractional sub-Laplacian (−∆s) on the group. Then, the multi-term generalizations of the time-fractional diffusion-wave equation for general operators with a discrete spectrum are considered and applied to the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of groups. Blow-up results to fractional heat equation with logarithmic nonlinearity on homogeneous groups are obtained due to recently established fractional logarithmic Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups. Fractional integral operators in anisotropic Morrey and Campanato spaces have also been considered. We are interested in further analysis of such equations as well as for possible applications of those results.

## Direct and inverse problems

We consider both direct and inverse problems for fractional derivative equations of different orders (sub-parabolic, pseudo-parabolic, parabolic-hyperbolic, etc.) In space, we considered elliptic, subelliptic, and hypoelliptic operators, as well as operators acting in abstract Hilbert spaces.

## Functional inequalities

### Van der Croput lemma for Mittag- Leffler functions.

Results on this topic are collected in papers [RT20a, RT20b] and presentation by profesor Ruzhansky is given within the Workshop on Fractional Calculus:

Click to access 2020-06-ghent-vd-corput-ml-web.pdf

Video of the talk Van der Corput lemma for Mittag-Leffler functions

Video of the talk Van der Corput, Mittlag-Leffler and tsunamis Further possible topics within fractional calculus that concern us include the calculus of variations where the focus is on the optimization of a function whose Lagrangian depends also on its fractional derivatives, applications in the image processing where fractional calculus has resulted in superior methods for the detection of edges, investigation of invariance of FDE under the group transformations, analysis of FDE on lattices, fractional stochastic FDE with application in the theory of optimal control, fractional derivatives and FDE in generalized functions settings, etc.

On arXiv:

[RRS20] Restrepo J., Ruzhansky M., Suragan D., Explicit representations of solutions for linear fractional differential equations with variable coefficients. arxiv

[RT20b] Ruzhansky M., Torebek B., Van der Corput lemmas for Mittag-Leffler functions. II. α-directions. arxiv

[RT20a] Ruzhansky M., Torebek B., Van der Corput lemmas for Mittag-Leffler functions. arxiv

Published:

1. Ruzhansky M., Tokmagambetov N., Torebek B., On a non-local problem for a multi-term fractional diffusion-wave equation. Fract. Calc. Appl. Anal., 23 (2020), 324-355. link, arxiv
2. Oparnica, Lj., Süli, E., Well-posedness of the fractional Zener wave equation for heterogenous viscoelastic materials, Fract. Calc. Appl. Anal.Vol. 23, No 1, pp. 23(1), 126-166, 2020.
3. Kassymov A., Ruzhansky M., Suragan D., Fractional logarithmic inequalities and blow-up results with logarithmic nonlinearity on homogeneous groups. NoDEA Nonlinear Differential Equations Appl., 27, no. 1, Paper No. 7, 2020.
4. Konjik, S., Oparnica, Lj., Zorica, D., Distributed order fractional constitutive stress-strain relation in wave propagation modeling, Z. Angew. Math. Phys., 70:51, 10.1007/s00033-019-1097-z, 2019.
5. Ruzhansky M., Suragan D., Yessirkegenov N., Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces, Fract. Calc. Appl. Anal., 21, 577-612, 2018.
6. Karimov E., Mamchuev M., Ruzhansky M., Non-local initial problem for second order time- fractional and space-singular equation, 2017, Hokkaido Math. J., to appear .
7. Agarwal P., Karimov E., Mamchuev M., Ruzhansky M., On boundary-value problems for a partial differential equation with Caputo and Bessel operators, in Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science: Novel Methods in Harmonic Analysis, Vol 2, Appl. Numer. Harmon. Anal., 707-718, Birkhauser/Springer, 2017.
8. Kassymov A., Ruzhansky M., Suragan D., Anisotropic fractional Gagliardo-Nirenberg, weighted Caffarelli-Kohn-Nirenberg and Lyapunov-type inequalities, and applications to Riesz potentials and p-sub-Laplacian systems, arXiv, 2018.
9. Ruzhansky M., Tokmagambetov N., Torebek B., On a non-local problem for a multi-term fractional diffusion-wave equation, arXiv, 2018.
10. Hörmann, G., Oparnica, Lj., Zorica, D., Solvability and microlocal analysis of the fractional Eringen wave equation, Mathematics and Mechanics of Solids, 23(10): 1420–1430, 2018.
11. Hörmann, G., Oparnica, Lj., Zorica, D., Microlocal analysis of fractional wave equations, Z. Angew. Math. Mech., 97(2):217-225, 2017. Atanackovic, T. M., Janev, M., Oparnica, Lj., Pilipovic, S. and Zorica, D. Space-time fractional Zener wave equation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471:20140614-1-25, 2015
12. Ruzhansky M., Serikbaev D., Tokmagambetov N., Torebek B., Direct and inverse problems for time-fractional pseudo-parabolic equationsarxiv
13. Ruzhansky M., Tokmagambetov N., Torebek B., Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations. Journal of Inverse and Ill-Posed Problems, 27 (2019), 891-911. linkarxiv We organised Workshop on Fractional Calculus, 9-10 June 2020