# Trends, Directions for Further Research, and Some Open Problems of
Fractional Calculus

Kai Diethelm111Faculty of Applied Natural Sciences and Humanities (FANG),
University of Applied Sciences Würzburg-Schweinfurt, Ignaz-Schön-Str. 11,
97421 Schweinfurt, Germany<EMAIL_ADDRESS>ORCID: 0000-0002-7276-454X
222corresponding author Virginia Kiryakova333Institute of Mathematics and
Informatics, Bulgarian Academy of Sciences, Sofia – 1113, Bulgaria,
<EMAIL_ADDRESS>ORCID: 0000-0002-4591-6958 Yuri Luchko444Beuth Technical
University of Applied Sciences Berlin, Department of Mathematics, Physics, and
Chemistry, Luxemburger Str. 10, 13353 Berlin, Germany, luchko@beuth-
hochschule.de, ORCID: 0000-0002-9583-7377 J. A. Tenreiro Machado555Institute
of Engineering, Polytechnic of Porto, Dept. of Electrical Engineering, Rua Dr.
António Bernardino de Almeida, 431, 4249 – 015 Porto, Portugal,
<EMAIL_ADDRESS>ORCID: 0000-0003-4274-4879 Vasily E. Tarasov666Skobeltsyn
Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow
119991, Russia<EMAIL_ADDRESS>ORCID: 0000-0002-4718-6274
777Faculty “Information Technologies and Applied Mathematics”, Moscow Aviation
Institute (National Research University), Moscow 125993, Russia

(09 August 2021)

###### Abstract

The area of fractional calculus (FC) has been fast developing and is presently
being applied in all scientific fields. Therefore, it is of key relevance to
assess the present state of development and to foresee, if possible, the
future evolution, or, at least, the challenges identified in the scope of
advanced research works. This paper gives a vision about the directions for
further research as well as some open problems of FC. A number of topics in
mathematics, numerical algorithms and physics are analyzed, giving a
systematic perspective for future research.

_Keywords:_ Fractional calculus; Sonine kernels; general fractional integrals
and derivatives; fractional differential equations; numerical solution;
fractional dynamics

##  1 Introduction

In 1695 Gottfried Leibniz exchanged ideas with other mathematicians about what
he called Fractional Calculus (FC). FC appeared as a misname and generalized
integro-differentiation seems more adequate, but the name FC remained due to
historical reasons [1, 2, 3, 4]. The FC generalizes the standard differential
calculus to non-integer orders, real or complex. This scientific tool remained
mainly in the area of mathematics until the last two decades (see e.g. [5]),
when the research community recognized its superior performance for describing
many natural and artificial phenomena. For a comprehensive treatment of
present day knowledge in FC readers are referred to [6, 7] for mathematics,
[8] for numerical analysis, [9, 10] physics, [11] control, and [12, 13] for
applications. The FC became very popular in all branches of science and an
active area of development. We recall the list of 23 problems raised by
Hilbert by the beginning of the 20th century and its influence upon the
succeeding decades of research. While many problems were solved, efforts are
still on-going for answering the remaining ones. Therefore, formulating not
only possible challenges and problems, but also pointing towards future
directions of research may have a significant impact. This manuscript gives
the vision of the authors in what concerns the key upcoming issues of this
fast advancing area of knowledge.

We can estimate of the present day state of FC using present day publicly
available information, just to remind that until 1974 there were only 1 book
and 1 conference proceedings, devoted to FC as a topic, while by 2018 the FC
books were estimated to more than 240 (see [14] and Table 1 in [15]), and the
published FC articles hardly to be count as thousands. For that purpose we
selected the program VOSviewer [16, 17, 18, 19, 20, 21] as the software tool
for processing and clustering bibliographic information. We tackled data
available at Scopus database and collected papers published during year 2020.
Moreover, 8 search keywords where adopted, namely the set {Fractional
calculus, Fractional derivative, Fractional integration, Fractional dynamics,
Mittag-Leffler, Derivative of non-integer order, Integral of non-integer
order, Derivative of complex order, Integral of complex order} that led to
6,589 records. The VOSviewer allows a multitude of possible perspectives of
bibliographic analysis, but, for the sake of parsimony, we focus on 3 types of
scrutiny based on network plots:

  * •

Co-occurrence, with options Index keywords, Fractional counting, Minimum
number of occurrence of a keyword: 5, that reveals 1,116 keywords, as
portrayed in Fig. 1

  * •

Co-authorship, with options Countries, Fractional counting, Minimum number of
documents of a country: 5, and Minimum number of citations of a country: 2,
that includes 71 countries, represented in Fig. 2

  * •

Bibliographic coupling, with options Countries, Fractional counting, Minimum
number of documents of a country: 5, Minimum number of citations of a country:
2, that leads to 71 countries, shown in Fig. 3

Figure 1 shows that FC is presently applied in all fields of science.
Additionally, we observe two main clusters: one that includes the areas of
mathematics, physics, engineering and economy, and a more recent one that
covers medicine, biology and genetics. Figures 2 and 3 portrait also two main
clusters. One major cluster corresponding to China, USA, most European
countries, Russia, south America and several countries of Asia, such as,
Japan, South Korea, and Australia. A second cluster is formed by more recent
contributions mostly from Middle East and south Asia, such as India, Iran,
Turkey, Saudi Arabia and Pakistan.

Figure 1: Network plot using Co-occurrence, with options Index keywords,
Fractional counting. Figure 2: Network plot using Co-authorship, with options
Countries, Fractional counting. Figure 3: Network plot using Bibliographic
coupling, with options Countries, Fractional counting.

The emerging clusters are clearly correlated with the world state of FC. On
one hand, it is well known the applicability of FC is describing complex
phenomena including non-locality and memory effects, which accounts for the
results in Fig. 1 . On the other hand, the portraits in Figs. 2 and 3 are also
natural, since we have an historical development of FC in the countries
located in the first cluster, while we observe countries joining more recently
the area of FC that form the right cluster.

Given this state of affairs, we outline our vision for a future solid progress
of FC in the next sections. Section 2 discusses important analytical aspects
that need further work. Section 3 addresses numerical problems in the scope of
present day computational era. Section 4 analyses problems in physics and the
application of the FC tools. Finally, Section 5 outlines the main conclusions.
Due to space limitations, these sections cover a limited set of topics.
Nonetheless, the present work gives a synthetic view and stimulates not only a
sound scientific debate, but also gives room for a future continuation.

##  2 Analytical Aspects

The fractional integrals and derivatives of arbitrary order
$\alpha\in\mathbb{R}$ or even $\alpha\in\mathbb{C}$ can be roughly
characterized as an interpolation of the $n$-th order derivatives and the
$n$-fold definite integrals ($n\in\mathbb{N}$). Evidently, without posing some
additional conditions, this interpolation is not unique and thus, several
different definitions of the FC operators have been suggested starting from
the very origins of this theory. In particular, the Riemann-Liouville integral
and derivative, the Grünwald-Letnikov derivative, the Marchaud derivative, the
Weyl integral and derivative of periodic functions, the Erdélyi-Kober integral
and derivative, the Hadamard integral and derivative, the Riesz and the Feller
potentials and fractional derivatives, and the Dzherbashian-Caputo fractional
derivative have been introduced and studied to mention only some of the most
used FC fractional integrals and derivatives. However, these operators were
defined neither artificially nor arbitrary. All of them possess a strong
mathematical motivation and their definitions and properties were exactly
adjusted both to the spaces of functions and their domains and to the problems
that these operators were useful for. For a detailed discussion of the
classical FC operators including their history and applications we refer the
readers to the “bible of FC” - the encyclopedic monograph [22] and to the
recent books [6, 23, 24]. Later on, some other fractional integrals and
derivatives were introduced in form of integral and integro-differential
operators with the suitable special functions in the kernels, see e.g. [25,
26]. These operators proved to have also various useful applications in
several mathematical topics.

Within the last few years, the situation in FC essentially changed. Several
new definitions of the fractional derivatives were suggested and,
unfortunately, some of them cannot be called neither fractional nor
derivatives, see, e.g., [27], [28]. Thus, the questions like “What are the
fractional integrals and derivatives?”, “What are their decisive mathematical
properties?”, “What fractional operators make sense in applications and why?”
became extremely important and acute.

The aim of this section is in providing some partial answers to the questions
mentioned above (see [29] for further contributions to this topic). We start
with a discussion of the characteristic properties that the one-parameter FC
operators should possess. Then we consider an example of realization of this
schema in the case of the fractional integrals and derivatives acting on the
functions defined on a finite interval. Finally, we discuss a class of the
general fractional derivatives with the Sonine kernels defined for the
functions on the real positive semi-axis and formulate some open problems for
further research.

###  2.1 Desiderata for the FC operators

In this subsection, we closely follow the ideas recently presented in [30].
For the sake of brevity, we restrict ourselves to the case of the one-
parameter fractional integrals and derivatives of a real order $\alpha\geq 0$.
The properties formulated below should be valid on some suitable spaces of
functions that we do not specify here (see [30] for details).

The families of the operators $I^{\alpha},\ \alpha\geq 0$ and $D^{\alpha},\
\alpha\geq 0$ can be interpreted as fractional integrals and fractional
derivatives, respectively, if they satisfy the following six properties
(called desiderata in [30]):

HL1) The operators $I^{\alpha}$ and $D^{\alpha}$ are linear operators on the
suitable linear spaces of functions.

HL2) For the operators $I^{\alpha}$, the index law (semigroup property)

$I^{\alpha_{1}}\circ I^{\alpha_{2}}=I^{\alpha_{1}+\alpha_{2}},\ \
\alpha_{1},\alpha_{2}\geq 0$ (1)

holds true, where $\circ$ denotes composition of operators.

HL3) The operator $D^{\alpha}$ is left inverse to the operator $I^{\alpha}$:

$D^{\alpha}\circ I^{\alpha}=\mbox{Id},\ \ \alpha\geq 0.$ (2)

HL4) The following limits exist in some sense

$\lim_{\alpha\to 0}D^{\alpha}=D^{0},\ \ \lim_{\alpha\to 1}D^{\alpha}=D^{1}$
(3)

and define linear maps.

HL5) The map $D^{0}$ is the identity operator.

HL6) The map $D^{1}$ is the first order derivative.

We mention once again that the precise formulations of the properties
(desiderata) HL1)-HL6) presented above can be found in [30]. In particular, in
[30], a description of the spaces of functions, where these properties should
hold valid, is provided. Some examples of realization of the schema specified
by the properties HL1)-HL6) were discussed in the recent publications [31, 32,
33].

###  2.2 One-parameter FC operators on a final interval

This subsection is devoted to a description of a one-parameter family of the
fractional integrals defined for the functions on a finite interval and
several families of the corresponding fractional derivatives in the sense of
the desiderata presented in the previous subsection. For the details and the
proofs of the theorems we refer to the recent publication [34].

We start with an important result regarding a characterization of a one-
parameter family of the fractional integrals defined for the functions on a
finite interval.

######  Theorem 1 ([35]).

Let $E$ be the space $L_{p}(0,1),\,1\leq p<+\infty$, or $C[0,1]$. Then there
is precisely one family $I^{\alpha},\,\alpha>0$ of operators on $E$ satisfying
the following conditions:

CM1) $(I^{1}\,f)(x)=\int_{0}^{x}f(t)\,dt,\ f\in E$ (interpolation condition),

CM2) $(I^{\alpha}\,I^{\beta}\,f)(x)=(I^{\alpha+\beta}\,f)(x),\
\alpha,\beta>0,\ f\in E$ (index law),

CM3) $\alpha\to I^{\alpha}$ is a continuous map of $(0,\,+\infty)$ into
${\mathcal{L}}(E)$ for some Hausdorff topology on ${\mathcal{L}}(E)$, weaker
than the norm topology (continuity),

CM4) $f\in E$ and $f(x)\geq 0$ (a.e. for $E=L_{p}(0,1)$) $\Rightarrow$
$(I^{\alpha}\,f)(x)\geq 0$ (a.e. for $E=L_{p}(0,1)$) for all $\alpha>0$ (non-
negativity).

That family is given by the Riemann-Liouville fractional integrals with
$\alpha>0$:

$(I^{\alpha}\,f)(x)=\begin{cases}\frac{1}{\Gamma(\alpha)}\int_{0}^{x}(x-t)^{\alpha-1}\,f(t)\,dt,&\alpha>0,\\\
f(x),&\alpha=0.\end{cases}$ (4)

As we see, Theorem 1 does not cover the case $\alpha=0$. However, it is well
known (Theorem 2.6 in [22]) that the Riemann-Liouville fractional integrals
$I^{\alpha},\,\alpha\geq 0$ given by (4) form a semigroup in $L_{p}(0,\ 1),\
p\geq 1$, which is strongly continuous for all $\alpha\geq 0$, i.e., the
relation

$\lim_{\alpha\to\alpha_{0}}\|I^{\alpha}f-I^{\alpha_{0}}f\|_{L_{p}(0,1)}=0$ (5)

holds valid for any $\alpha_{0},\ \alpha_{0}\geq 0$ and for any $f\in
L_{p}(0,1)$. Thus, we can uniquely extend the family of the Riemann-Liouville
fractional integrals defined for $\alpha>0$ to a family defined for
$\alpha\geq 0$ by setting $I^{0}=\mbox{Id}$. Evidently, this extended family
of operators satisfies the properties CM1)-CM4) for all $\alpha\geq 0$.

The conditions formulated in Theorem 1 are much stronger than those presented
in the previous subsection. However, they are very natural and useful for
applications and thus, in a certain sense, the Riemann-Liouville fractional
integrals are the only “right” one-parameter family of the fractional
integrals defined for the functions on a finite interval.

In what follows, without loss of generality, we restrict ourselves to the
operators defined on the interval $[0,\,1]$.

In order to introduce the suitable fractional derivatives, the following
result regarding solvability of the Abel integral equation is employed:

######  Theorem 2 ([22]).

The Abel integral equation

$(I^{\alpha}\,\phi)(x)=\frac{1}{\Gamma(\alpha)}\int_{0}^{x}(x-t)^{\alpha-1}\,\phi(t)\,dt\
=f(x),\ \,0<\alpha<1,\ x\in[0,\,1]$ (6)

possesses a unique solution in $L_{1}(0,1)$ if and only if

$I^{1-\alpha}\,f\in\mbox{AC}([0,1])\ \ \mbox{and}\ \ (I^{1-\alpha}\,f)(0)=0,$
(7)

where the notation $\mbox{AC}([0,1])$ stands for the space of functions that
are absolutely continuous on the interval $[0,\,1]$:

$f\in\mbox{AC}([0,1])\,\Leftrightarrow\,\exists\phi\in
L_{1}(0,1):\,f(x)=f(0)+\int_{0}^{x}\phi(t)\,dt,\ x\in[0,\,1].$ (8)

If these conditions are satisfied, the solution is given by the formula

$\phi(x)=\frac{d}{dx}\,(I^{1-\alpha}\,f)(x)=\frac{d}{dx}\,\frac{1}{\Gamma(1-\alpha)}\int_{0}^{x}(x-t)^{-\alpha}\,f(t)\,dt,\
\,0<\alpha<1.$ (9)

Using the representation (8), we define a (weak) derivative of a function
$f\in\mbox{AC}([0,1])$:

$f(x)=f(0)+\int_{0}^{x}\phi(t)\,dt,\ x\in[0,\,1]\ \Rightarrow\
\frac{df}{dx}:=\phi\in L_{1}(0,1).$ (10)

In the further discussions, we interpret the first order derivative of the
absolutely continuous functions in the sense of the formula (10).

According to the property HL3) from the desiderata, a fractional derivative
for the functions defined on a finite interval is introduced as a linear
operator $D^{\alpha},\ \alpha\geq 0$ left inverse to the Riemann-Liouville
fractional integral $I^{\alpha}$, i.e., as an operator that satisfies the
relation (1st Fundamental Theorem of FC)

$(D^{\alpha}\,I^{\alpha}\,\phi)(x)=\phi(x),\ x\in[0,\,1]$ (11)

on appropriate nontrivial space of functions.

It turns out that there exist infinitely many different families of the
fractional derivatives in the sense of relation (11). In the rest of this
subsection, we discuss some known and new families of the fractional
derivatives on the interval $[0,\,1]$ (see [34] for details).

######  Remark 1.

The formula (11) and the relation $(I^{0}\,f)(x)=f(x)$ uniquely define the
fractional derivative of order $\alpha=0$ as the identity operator:
$(D^{0}\,f)(x)=f(x)$. In what follows, we mainly restrict ourselves to the
case $0<\alpha\leq 1$ and to the space $L_{1}(0,1)$ and its subspaces (a
similar theory can be developed for, say, $L_{p}(0,1),\ 1<p<+\infty$ and its
subspaces).

The formula (11) can be rewritten in equivalent form of two equations:

$(I^{\alpha}\ \phi)(x)=f(x),\ \ (D^{\alpha}\,f)(x)=\phi(x),\ \,x\in[0,\,1].$
(12)

As we see, the second of equations (12) defines the fractional derivative
$D^{\alpha}$ of a function $f$ as the solution $\phi$ of the Abel integral
equation with the right-hand side $f$. Now we recall Theorem 2 and reformulate
it as follows:

######  Theorem 3.

On the space of functions $I^{\alpha}(L_{1}(0,1))$, the unique fractional
derivative $D^{\alpha}$ of order $\alpha$, $0<\alpha<1$ is the Riemann-
Liouville fractional derivative given by the formula

$(D^{\alpha}_{RL}\,f)(x)=\frac{d}{dx}\,(I^{1-\alpha}\,f)(x)=\frac{d}{dx}\,\frac{1}{\Gamma(1-\alpha)}\int_{0}^{x}(x-t)^{-\alpha}\,f(t)\,dt.$
(13)

Evidently, the formula (13) makes sense for a space of functions larger than
$I^{\alpha}(L_{1}(0,1))$, namely, for the space
$\\{f:\,I^{1-\alpha}\,f\in\mbox{AC}([0,\,1])\\}$.

Moreover, the 1st Fundamental Theorem of FC (relation (11)) for the Riemann-
Liouville fractional derivative is valid on the whole space $L_{1}(0,1)$,
i.e., the formula

$(D^{\alpha}_{RL}\,I^{\alpha}\,f)(x)=f(x),\ x\in[0,\,1],\ f\in L_{1}(0,1)$
(14)

holds true ([22]).

For the Riemann-Liouville fractional derivatives, the properties HL4)-HL6)
from the desiderata formulated in the previous subsection are valid on the
suitable spaces of functions (see, e.g., [22]). Thus, the families of the
Riemann-Liouville fractional integrals and derivatives are the fractional
integrals and derivatives in the sense of the desiderata.

However, there exist infinitely many other one-parameter families of the
fractional derivatives associated to the Riemann-Liouville fractional
integrals. They are introduced in the rest of this subsection (see [34] for
details).

On the space $\mbox{AC}([0,\,1])$, the Caputo fractional derivative
$D^{\alpha}_{C}$ of order $\alpha$, $0<\alpha\leq 1$ is defined as follows:

$(D^{\alpha}_{C}\,f)(x)=(I^{1-\alpha}\frac{d}{dx}f)(x).$ (15)

As in the case of the Riemann-Liouville fractional derivative, the 1st
Fundamental Theorem of FC (relation (11)) for the Caputo fractional derivative
is valid on even larger space of functions

$X_{FT}=\left\\{f:\,I^{\alpha}f\in\mbox{AC}([0,\,1])\ \mbox{and}\
(I^{\alpha}f)(0)=0\right\\},$ (16)

i.e., the formula

$(D^{\alpha}_{C}\,I^{\alpha}\,f)(x)=f(x),\ \,x\in[0,\,1],\ f\in X_{FT}$ (17)

holds true ([34]). It is worth mentioning that the space $X_{FT}$ can be also
characterized as follows (Theorem 2.3 in [22]):

$X_{FT}=I^{1-\alpha}(L_{1}(0,1))\,(\forall f\in X_{FT}\,\exists\phi\in
L_{1}(0,1):\,f(x)=(I^{1-\alpha}\,\phi)(x)).$ (18)

Another family of the fractional derivatives associated to the Riemann-
Liouville fractional integral is the Hilfer fractional derivatives of order
$\alpha,\ 0<\alpha\leq 1$ and type $\gamma_{1}$, $0\leq\gamma_{1}\leq
1-\alpha$:

$(D^{\alpha,\gamma_{1}}_{H}\,f)(x)=(I^{\gamma_{1}}\,\frac{d}{dx}\,I^{1-\alpha-\gamma_{1}}\,f)(x).$
(19)

This derivative is well defined on the space
$\left\\{f:\,I^{1-\alpha-\gamma_{1}}\,f\in\mbox{AC}([0,\,1])\right\\}$.

The 1st Fundamental Theorem of FC (relation (11)) for the Hilfer fractional
derivative is valid on the space $X_{FT}$ defined by (16) ([34]):

$(D^{\alpha,\gamma_{1}}_{H}\,I^{\alpha}\,f)(x)=f(x),\ x\in[0,\,1],\ f\in
X_{FT},\ 0<\alpha\leq 1,\ 0\leq\gamma_{1}\leq 1-\alpha.$ (20)

For every type $\gamma_{1}$, $0\leq\gamma_{1}\leq 1-\alpha$, the Hilfer
fractional derivatives $D^{\alpha,\gamma_{1}}_{H}$ of order $\alpha,\
0<\alpha\leq 1$ form a one-parameter family of the fractional derivatives in
the sense of the 1st Fundamental Theorem of FC (relation (11)). For
$\gamma_{1}=0$, this family coincides with the Riemann-Liouville fractional
derivatives, while for $\gamma_{1}=1-\alpha$ we get the Caputo fractional
derivatives.

The construction of the Hilfer fractional derivative can be extended to the
case of $n$ compositions of the first order derivatives and appropriate
Riemann-Liouville fractional integrals. As a result, we get infinitely many
different families of the one-parameter fractional derivatives that were
called in [34] the $n$th level fractional derivatives of order $\alpha,\
0<\alpha\leq 1$ and type $\gamma=(\gamma_{1},\ \gamma_{2},\dots,\gamma_{n})$.

To define these fractional derivatives, we introduce the parameters
$\gamma_{1}\mathchar 44\relax\penalty 0\gamma_{2}\mathchar 44\relax\penalty
0\dots\mathchar 44\relax\penalty 0\gamma_{n}\in\mathbb{R}$ that satisfy the
conditions

$0\leq\gamma_{k}\ \mbox{and}\ \alpha+s_{k}\leq k,\ \ k=1,2,\dots,n$ (21)

with $s_{k}$ in form

$s_{k}:=\sum_{i=1}^{k}\,\gamma_{i},\ k=1,2,\dots,n.$ (22)

The $n$th level fractional derivative of order $\alpha,\ 0<\alpha\leq 1$ and
type $\gamma=(\gamma_{1},\ \gamma_{2},\dots,\gamma_{n})$ is defined as follows
([34]:

$(D^{\alpha,(\gamma)}_{nL}\,f)(x)=\left(\prod_{k=1}^{n}(I^{\gamma_{k}}\,\frac{d}{dx})\right)\,(I^{n-\alpha-
s_{n}}\,f)(x).$ (23)

The right-hand side of (23) is well defined for the functions from the space
$\\{f:\left(\prod_{k=i}^{n}(I^{\gamma_{k}}\,\frac{d}{dx})\right)\,I^{n-\alpha-
s_{n}}\,f\in\mbox{AC}([0,\,1]),\ \ i=2,\dots n+1\\}$.

The $n$th level fractional derivative of order $\alpha,\ 0<\alpha\leq 1$ and
any type $\gamma=(\gamma_{1},\ \gamma_{2},\dots,\gamma_{n})$ is associated to
the Riemann-Liouville fractional integral via the 1st Fundamental Theorem of
FC.

######  Theorem 4 ([34]).

The $n$th level fractional derivative is a left-inverse operator to the
Riemann-Lioville fractional integral on the space $X_{FT}$ defined by (16):

$(D^{\alpha,(\gamma)}_{nL}\,I^{\alpha}\,f)(x)=f(x),\ f\in X_{FT},\
x\in[0,\,1].$ (24)

######  Remark 2.

In [36], the Cauchy problems for the fractional differential equations with
the operators similar to the $n$th level fractional derivatives
$D^{\alpha,(\gamma)}_{nL}$ (in other notations and with other restrictions on
the parameters) were treated. However, the connection to the Riemann-Liouville
fractional integrals in form of Theorem 4 was not discussed in [36].

For further properties of the $n$th level fractional derivatives and the
fractional relaxation equations with these derivatives we refer the interested
reader to [34] and [37].

###  2.3 General FC operators with the Sonine kernels

The previous subsections were devoted to the one-parameter families of the
fractional integrals and derivatives. Without any doubts, these FC operators
can be interpreted as direct generalizations of the $n$-folds integrals and
integer order derivatives and thus they are objects important and useful both
for analytical and numerical treatment and for applications. However, within
the last years, some more general fractional operators were introduced and
employed as models in several applications including the linear
viscoelastisity and anomalous diffusion and wave propagation. In particular,
much attention was given to the multi-term fractional differential operators
that are the finite sums of the one-parameter fractional derivatives with
different orders and their generalizations in form of the distributed order
fractional derivatives that can be interpreted as sums of infinitely many
fractional derivatives with different orders that belong to a certain interval
(usually, to the intervals $[0,1]$, or $[1,2]$, or $[0,2]$). These more
general fractional derivatives are determined either by finite (multi-term
case) or infinite (distributed order case) many orders of the fractional
derivatives and thus they require a different procedure for their
characterization compared to the schema we presented in the previous
subsections.

One of the possible approaches for a description of both the single-term, the
multi-term, and the distributed order fractional derivatives is in the
framework of the so called general fractional calculus suggested in [38].

In this subsection, we present some results derived in [38] and in the recent
publications [39]-[40] regarding the general fractional derivatives (GFD) and
the general fractional integrals (GFI). We start with the case of the
operators with the “generalized order” restricted to the interval $(0,\ 1)$.
The GFDs of the Riemann-Liouville and Caputo types, respectively, are defined
as the following integro-differential operators of the convolution type:

$(\mathbb{D}_{(k)}f)(t)=\frac{d}{dt}\int_{0}^{t}k(t-\tau)f(\tau)\,d\tau,$ (25)
$(_{*}\mathbb{D}_{(k)}f)(t)=\int_{0}^{t}k(t-\tau)f^{\prime}(\tau)\,d\tau,$
(26)

where $k$ is a nonnegative locally integrable function. For an absolutely
continuous function $f$ satisfying $f^{\prime}\in
L^{1}_{loc}(\mathbb{R}_{+})$, the relation

$(_{*}\mathbb{D}_{(k)}f)(t)\ =\ (\mathbb{D}_{(k)}(f-f(0)))(t)\ =\
(\mathbb{D}_{(k)}f)(t)-k(t)f(0)$ (27)

between the Caputo and Riemann-Liouville types of GFDs holds true. In [38],
the Caputo type GFD was introduced in form of the right-hand side of (27) that
is well defined for a lager class of functions (in particular, for absolutely
continuous functions) compared to the definition (26) that requires the
inclusion $f^{\prime}\in L^{1}_{loc}(\mathbb{R}_{+})$. In what follows, we
mainly address the GFD ot the Caputo type in the sence of the right-hand side
of the formula (27).

The Riemann-Liouville and Caputo fractional derivatives defined by (13) and
(15), respectively, are particular cases of the GFDs (25) and (26) with the
kernel

$k(t)=h_{1-\alpha}(t),\ 0<\alpha<1$ (28)

with the power function $h_{\beta}(t):=t^{\beta-1}/\Gamma(\beta),\ t>0,\
\beta>0$. Other important particular cases of (25) and (26) are the multi-term
fractional derivatives and the fractional derivatives of the distributed
order. They are generated by (25) and (26) with the kernels

$k(t)=\sum_{k=1}^{n}a_{k}\,h_{1-\alpha_{k}}(t),\ \
0<\alpha_{1}<\dots<\alpha_{n}<1,\ a_{k}\in\mathbb{R},\ k=1,\dots,n$ (29)

and

$k(t)=\int_{0}^{1}h_{1-\alpha}(t)\,d\rho(\alpha),$ (30)

respectively, where $\rho$ is a Borel measure on $[0,\,1]$.

In [38], the case of the GFDs (25) and (27) with the kernels that satisfy the
following properties was considered:

K1) The Laplace transform $\tilde{k}$ of $k$,

$\tilde{k}(p)=({\mathcal{L}}\,k)(p)\ =\ \int_{0}^{\infty}k(t)\,e^{-pt}\,dt$

exists for all $p>0$,

K2) $\tilde{k}(p)$ is a Stieltjes function,

K3) $\tilde{k}(p)\to 0$ and $p\tilde{k}(p)\to\infty$ as $p\to\infty$,

K4) $\tilde{k}(p)\to\infty$ and $p\tilde{k}(p)\to 0$ as $p\to 0$.

In what follows, we denote the set of the kernels that satisfy the conditions
K1)-K4) by $\mathcal{K}$. As shown in [38], for each $k\in\mathcal{K}$, there
exists a completely monotone function $\kappa$ such that the Sonine condition
holds true:

$(k*\kappa)(t)=\int_{0}^{t}k(t-\tau)\kappa(\tau)\,d\tau\ =\ \\{1\\},\ t>0,$
(31)

where by $\\{1\\}$ we denote the function that takes the constant value $1$
for $t>0$. The kernels that satisfy the Sonine condition (31) are called the
Sonine kernels. The set of all Sonine kernels is denoted by $\mathcal{S}$
($\mathcal{K}\subset\mathcal{S}$). For the properties of the Sonine kernels
and their examples we refer to [28, 39, 41, 40, 42, 43] and other related
publications.

A GFI with the kernel $\kappa$ associated to the kernel $k$ of the GFD by
means of the relation (31) is introduced as follows:

$(\mathbb{I}_{(k)}f)(t)=\int_{0}^{t}\kappa(t-\tau)f(\tau)\,d\tau.$ (32)

The notions of GFD and GFI are justified by the following relations ([38]):

$(_{*}\mathbb{D}_{(k)}\,\mathbb{I}_{(k)}\,f)(t)=f(t),$ (33)

for any locally bounded measurable function $f$ on $\mathbb{R}_{+}$ and

$(\mathbb{I}_{(k)}\,{}_{*}\mathbb{D}_{(k)}\,f)(t)=f(t)-f(0)$ (34)

for any absolutely continuous function $f$ on $\mathbb{R}_{+}$.

The formula (33) and the relation (27) between the GFDs of the Caputo and the
Riemann-Liouville types lead to the identity

$(\mathbb{D}_{(k)}\,\mathbb{I}_{(k)}\,f)(t)=f(t),$ (35)

i.e., the Riemann-Liouville GFD is also a left inverse operator to the GFI
defined by (32).

As mentioned in [28, 42], the functions that satisfy the Sonine condition (31)
cannot be continuous at the point $t=0$ and thus the “new fractional
derivatives” with the continuous kernels introduced recently in the FC
literature do not belong to the class of the GFDs that we discuss in this
subsection.

Recently, in the papers [39]-[40], another important class of the Sonine
kernels was treated.

######  Definition 1 ([39]).

Let $\kappa,\,k\in C_{-1,0}(0,+\infty)$, where

$C_{\alpha,\beta}(0,+\infty)\,=\,\\{f:\ f(t)=t^{p}f_{1}(t),\ t>0,\
\alpha<p<\beta,\ f_{1}\in C[0,+\infty)\\},$ (36)

be a pair of the Sonine kernels, i.e., the Sonine condition (31) be fulfilled.
The set of such Sonine kernels is denoted by $\mathcal{L}_{1}$:

$(\kappa,\,k\in\mathcal{L}_{1})\ \Leftrightarrow\ (\kappa,\,k\in
C_{-1,0}(0,+\infty))\wedge((\kappa\,*\,k)(t)\,=\,\\{1\\}).$ (37)

The properties of the GFI and GFD with the kernels from $\mathcal{L}_{1}$
including the 1st and the 2nd Fundamental Theorems of FC were investigated in
[39]-[40] on the space $C_{-1}(0,+\infty):=C_{-1,+\infty}(0,+\infty)$. In
particular, we mention the mapping property

$\mathbb{I}_{(\kappa)}:\,C_{-1}(0,+\infty)\,\rightarrow C_{-1}(0,+\infty),$
(38)

the commutativity law

$\mathbb{I}_{(\kappa_{1})}\,\mathbb{I}_{(\kappa_{2})}=\mathbb{I}_{(\kappa_{2})}\,\mathbb{I}_{(\kappa_{1})},\
\kappa_{1},\,\kappa_{2}\in\mathcal{L}_{1},$ (39)

and the index law

$\mathbb{I}_{(\kappa_{1})}\,\mathbb{I}_{(\kappa_{2})}=\mathbb{I}_{(\kappa_{1}*\kappa_{2})},\
\kappa_{1},\,\kappa_{2}\in\mathcal{L}_{1}$ (40)

that are valid on the space $C_{-1}(0,+\infty)$.

######  Theorem 5 ([39]).

Let $\kappa\in\mathcal{L}_{1}$ and $k$ be its associated Sonine kernel.

Then, the GFD (25) is a left inverse operator to the GFI (32) on the space
$C_{-1}(0,+\infty)$:

$(\mathbb{D}_{(k)}\,\mathbb{I}_{(\kappa)}\,f)(t)=f(t),\ f\in
C_{-1}(0,+\infty),\ t>0,$ (41)

and the GFD (26) is a left inverse operator to the GFI (32) on the space
$C_{-1,(k)}(0,+\infty)$:

$(_{*}\mathbb{D}_{(k)}\,\mathbb{I}_{(\kappa)}\,f)(t)=f(t),\ f\in
C_{-1,(k)}(0,+\infty),\ t>0,$ (42)

where $C_{-1,(k)}(0,+\infty):=\\{f:\ f(t)=(\mathbb{I}_{(k)}\,\phi)(t),\
\phi\in C_{-1}(0,+\infty)\\}$.

######  Theorem 6 ([39]).

Let $\kappa\in\mathcal{L}_{1}$ and $k$ be its associated Sonine kernel.

Then, the relations

$(\mathbb{I}_{(\kappa)}\,{}_{*}\mathbb{D}_{(k)}\,f)(t)=f(t)-f(0),\ t>0,$ (43)
$(\mathbb{I}_{(\kappa)}\,\mathbb{D}_{(k)}\,f)(t)=f(t),\ t>0$ (44)

hold valid for the functions $f\in C_{-1}^{1}(0,+\infty):=\\{f:\ f^{\prime}\in
C_{-1}(0,+\infty)\\}$.

In [39, 41], the $n$-fold GFIs and GFDs with the Sonine kernels from
$\mathcal{L}_{1}$ were defined and studied. For details, we refer the
interested readers to these publications. In the rest of this subsection, we
discuss the GFI and GFD of arbitrary order that were recently introduced in
[40].

To define these operators, we first formulate a condition on their kernels
that generalizes the Sonine condition (31):

$(\kappa\,*\,k)(t)=\\{1\\}^{<n>}(t),\ n\in\mathbb{N},\ t>0,$ (45)

where

$f^{<n>}(t):=(\underbrace{f*\ldots\ *f}_{\mbox{$n$ times}})(t).$

Thus, the relation

$\\{1\\}^{<n>}(t)=h_{n}(t)=\frac{t^{n-1}}{(n-1)!}$

holds valid. Evidently, the Sonine condition corresponds to the case $n=1$ of
the more general condition (45).

Then we proceed with defining the kernels that satisfy the condition (45) and
belong to the suitable spaces of functions.

######  Definition 2 ([40]).

Let the functions $\kappa$ and $k$ satisfy the condition (45) and the
inclusions $\kappa\in C_{-1}(0,+\infty)$, $k\in C_{-1,0}(0,+\infty)$ hold
true.

The set of pairs $(\kappa,\,k)$ of such kernels will be denoted by
$\mathcal{L}_{n}$.

As mentioned in [40], there are at least two reasonable possibilities to
construct a pair $(\kappa_{n}\,k_{n})$ of the kernels from $\mathcal{L}_{n},\
n>1$ based on the Sonine kernels $\kappa,\,k$ from $\mathcal{L}_{1}$. The
first strategy consists in building the kernels $\kappa_{n}=\kappa^{<n>}$ and
$k_{n}=k^{<n>}$. The kernels $\kappa_{n}$ and $k_{n}$ satisfy the relation
(45) because $\kappa$ and $k$ are the Sonine kernels. However, the pair
$(\kappa_{n},\ k_{n})$ does not always belong to the set $\mathcal{L}_{n}$.
This is the case only when the inclusion $k^{<n>}\in C_{-1,0}(0,+\infty)$
holds true (of course, $\kappa^{<n>}\in C_{-1}(0,+\infty)$ for any
$n\in\mathbb{N}$). A more general and important possibility for construction a
pair $(\kappa_{n},\,k_{n})$ of the kernels from $\mathcal{L}_{n},\ n>1$ based
on the Sonine kernels $\kappa,\,k$ from $\mathcal{L}_{1}$ is presented in the
following theorem:

######  Theorem 7 ([40]).

Let $(\kappa,\ k)$ be a pair of the Sonine kernels from $\mathcal{L}_{1}$.

Then the pair $(\kappa_{n},\ k_{n})$ of the kernels given by the formula

$\kappa_{n}(t)=(\\{1\\}^{<n-1>}\,*\,\kappa)(t),\ \ k_{n}(t)=k(t)$ (46)

belongs to the set $\mathcal{L}_{n}$.

It is worth mentioning that in [44], some important sub-classes of the kernels
$\kappa,\,k$ from $\mathcal{L}_{n}$ in form of convolutions of several
different kernels from $\mathcal{L}_{1}$ were suggested.

Now we introduce the general fractional integrals and derivatives of arbitrary
(non-integer) order and discuss some of their basic properties (for more
properties and examples see [40]).

######  Definition 3 ([40]).

Let $(\kappa,\ k)$ be a pair of the kernels from $\mathcal{L}_{n}$.

The GFI with the kernel $\kappa$ has the same form as before

$(\mathbb{I}_{(\kappa)}\,f)(t):=\int_{0}^{t}\kappa(t-\tau)f(\tau)\,d\tau,\
t>0,$ (47)

whereas the GFDs of the Riemann-Liouville and Caputo types with the kernel $k$
are defined as follows:

$(\mathbb{D}_{(k)}\,f)(t):=\frac{d^{n}}{dt^{n}}\,\int_{0}^{t}k(t-\tau)f(\tau)\,d\tau,\
t>0,$ (48)
$(_{*}\mathbb{D}_{(k)}\,f)(t):=\left(\mathbb{D}_{(k)}\,\left(f(\cdot)-\sum_{j=0}^{n-1}f^{(j)}(0)h_{j+1}(\cdot)\right)\right)(t),\
t>0.$ (49)

Evidently, the GFI (47) with the kernel $\kappa(t)=h_{\alpha}(t),\ \alpha>0$
is reduced to the Riemann-Liouville fractional integral (4) and the Riemann-
Liouville and Caputo fractional derivatives of the order $\alpha,\
n-1<\alpha<n,\ n\in\mathbb{N}$ are particular cases of the GFDs (48) and (49)
with the kernel $k(t)=h_{n-\alpha}(t)$.

To justify the denotation GFI and GFD, we provide formulations of the 1st and
2nd fundamental theorems of FC for the GFDs (48) and (49) of the Riemann-
Liouville and Caputo types (for the proofs see [40]).

######  Theorem 8 ([40]).

Let $(\kappa,\ k)$ be a pair of the kernels from $\mathcal{L}_{n}$.

Then, the GFD (48) is a left inverse operator to the GFI (47) on the space
$C_{-1}(0,+\infty)$:

$(\mathbb{D}_{(k)}\,\mathbb{I}_{(\kappa)}\,f)(t)=f(t),\ f\in
C_{-1}(0,+\infty),\ t>0,$ (50)

and the GFD (49) is a left inverse operator to the GFI (47) on the space
$C_{-1,(k)}(0,+\infty)$:

$(_{*}\mathbb{D}_{(k)}\,\mathbb{I}_{(\kappa)}\,f)(t)=f(t),\ f\in
C_{-1,(k)}(0,+\infty),\ t>0,$ (51)

where the space $C_{-1,(k)}(0,+\infty)$ is defined as in Theorem 5.

######  Theorem 9 ([40]).

Let $(\kappa,\ k)$ be a pair of the kernels from $\mathcal{L}_{n}$.

Then, the relation

$(\mathbb{I}_{(\kappa)}\,{}_{*}\mathbb{D}_{(k)}\,f)(t)=f(t)-\sum_{j=0}^{n-1}f^{(j)}(0)\,h_{j+1}(t)$
(52)

holds valid on the space $C_{-1}^{n}(0,+\infty):=\\{f:\,f^{(n)}\in
C_{-1}(0,+\infty)\\}$ and the formula

$(\mathbb{I}_{(\kappa)}\,\mathbb{D}_{(k)}\,f)(t)=f(t),\ t>0$ (53)

is valid for the functions $f\in C_{-1,(\kappa)}(0,+\infty)$.

Further properties and examples of the GFIs and GFDs of arbitrary order were
presented in the recent papers [40, 44]. For the results regarding the
fractional ODEs and PDEs with the GFDs we refer the readers to [38, 41] and to
the recent survey [45] (see also the references therein).

###  2.4 Open problems

In this subsection, we formulate some open problems regarding the one-
parameter families of the fractional integrals and derivatives and the GFIs
and GFDs.

  1. 1.

Deduce a characterization of the fractional integrals for the functions
defined on a finite interval that satisfy the desiderata presented in the
previous subsection (the Riemann-Liouville fractional integrals are one of
such families that satisfies essentially stronger conditions compared to those
from desiderata, see Theorem 1).

  2. 2.

Characterize all families of the fractional derivatives for the functions
defined on a finite interval that are associated to the Riemann-Liouville
fractional integrals through the 1st Fundamental Theorem of FC (the $n$th
level fractional derivatives including the Riemann-Liouville, Caputo, and
Hilfer derivatives are examples of such families).

  3. 3.

Deduce a general form of the fractional integrals and derivatives for the
functions defined on a finite interval in the sense of the desiderata
formulated in the previous section.

  4. 4.

Consider the problems formulated above in the case of the functions defined on
the positive real semi-axis and on $\mathbb{R}$.

  5. 5.

Consider the problems formulated above in the multi-dimensional case, i.e.,
for the functions defined on $\mathbb{R}^{n}$.

  6. 6.

For a function $\kappa\in C_{-1}(0,+\infty)$, determine the conditions for the
inclusion $\kappa\in\mathcal{L}_{1}$ and derive a representation for the
associated kernel $k$.

  7. 7.

For a function $\kappa\in C_{-1}(0,+\infty)$, determine the conditions for the
inclusion $\kappa\in\mathcal{L}_{n}$ and derive a representation for the
associated kernel $k$.

  8. 8.

Adjust the notions of the GFIs and GFDs to the case of a finite interval and
$\mathbb{R}$.

  9. 9.

Develop a theory of the GFIs and GFDs on other conventional spaces of
functions including $L_{p}$ and Hölder weighted spaces.

  10. 10.

Consider the time-fractional differential equations with the GFDs of arbitrary
order to model, for instance, the processes intermediate between diffusion and
wave propagation.

##  3 Numerical Aspects

For the numerical handling of fractional differential and integral operators,
and in particular for the numerical solution of initial value problems
associated to fractional ordinary differential equations, i.e. problems of the
form

${}^{\text{C}}\\!D_{a+}^{\alpha}y(t)=f(t,y(t)),\qquad
y^{(k)}(a)=y_{0}^{(k)}\quad(k=0,1,\ldots\lceil\alpha\rceil-1)$ (54)

(where the differential operator is chosen to be of Caputo’s type [24, Chapter
3] because this is the type that is most important in the mathematical
modeling of processes in physics, engineering, economics, etc.), quite a lot
of approaches and techniques have been proposed. Recent surveys about the
state of the art in this respect may be found, e.g., in [23, 46, 47, 48, 49].
In addition to these well known results, we can witness the publication of a
large number of new papers dealing with such aspects. Unfortunately, many of
these papers merely describe the application of well known concepts to special
types of equations, thus not really providing any innovative steps in this
area of research, or—as explained in detail in [50]—they do not take into
account the analytic properties of the exact solutions to the differential
equations in an appropriate way, thus stating misleading results.

For the most important numerical approaches, a detailed understanding of their
properties is available (cf., e.g., [51, 52, 53, 54]). This includes, in
particular, thorough discussions of the weaknesses and limitations of these
algorithms [55, 56, 57]. In this context, we believe that the following
fundamental issues require attention in the near future and that the
corresponding questions will provide ample opportunities for further research:

  * •

_The non-locality of fractional differential operators and its implications_ :
This includes, in particular, the observation that traditional numerical
algorithms require an $O(N^{2})$ amount of run time and an $O(N)$ amount of
memory when taking $N$ time steps, both of which may be prohibitively large in
practical application scenarios. Regarding the run time aspect, a number of
approaches have been developed that provide a remedy. In particular, a general
FFT based technique for numerically dealing with convolution integrals has
been developed in [58, 59]. Its concrete implementation in combination with
the traditional Adams method from [51, 52] has been described in [48], thus
reducing the computational cost to $O(N\log^{2}N)$ (but not lowering the
memory requirement). Almost the same can be said about the nested mesh
technique of [56] which has an $O(N\log N)$ complexity but does not reduce the
memory demands either. The approaches of [60, 61, 62] address both matters;
they have an $O(N\log N)$ computational complexity and require an $O(\log N)$
amount of memory. Here now (to be precise: in Subsections 3.1 and 3.2,
respectively), we shall indicate two completely different potential approaches
to deal with this challenge, neither of which is based on using traditional
one-step or multistep methods. Specifically, the approach of Subsection 3.1
has the goal of reducing the complexity even further to the best possible
values, i.e. $O(N)$ for the computational cost and $O(1)$ for the memory,
whereas the methods of Subsection 3.2 have the goal of providing the
approximate solution over the entire interval of interest in one single (but
large) operation rather than progressing through the interval step by step.

  * •

_Terminal value problems_ : For initial value problems like (54) and their
numerical handling, a well developed theory is available. However, the
application of such mathematical models in practice requires the user to be
able to provide the initial values $y^{(k)}(a)$, i.e. to have access to
information about the exact solution at time $t=a$, the starting point of the
process (indicated by the fact that $a$ is also the starting point of the
differential operator ${}^{\text{C}}\\!D_{a+}^{\alpha}$ in the differential
equation). In concrete technical applications, this information may not
necessarily be available; rather, one can sometimes only measure $y(t)$ (and,
if necessary, its derivatives) at some point $t=b>a$. This problem requires
numerical methods that are completely different from those for (54); it will
be discussed in Subsection 3.3.

###  3.1 Algorithms Based on Infinite State Representations

In view of the observations recalled in the introduction to Section 3, a
number of novel numerical methods for solving fractional differential
equations have been developed; see, e.g., [63, 64, 65, 66, 67, 68, 69, 70, 71,
72, 73, 74, 75]. Although there are various differences in the details, all
these methods share the common feature that they are based on some non-
classical representation of the Caputo-type fractional differential operator
${}^{\text{C}}\\!D_{a+}^{\alpha}$ of order $\alpha>0$ that appears in the
differential equations under consideration, i.e. instead of one of the
traditional forms

${}^{\text{C}}\\!D_{a+}^{\alpha}y(t)=\frac{1}{\Gamma(\lceil\alpha\rceil-\alpha)}\int_{a}^{t}(t-\tau)^{\lceil\alpha\rceil-\alpha-1}y^{(\lceil\alpha\rceil)}(\tau)\,\mathrm{d}\tau$
(55a) or
${}^{\text{C}}\\!D_{a+}^{\alpha}y(t)=\frac{1}{\Gamma(\lceil\alpha\rceil-\alpha)}\frac{\mathrm{d}^{\lceil\alpha\rceil}}{\mathrm{d}t^{\lceil\alpha\rceil}}\int_{a}^{t}(t-\tau)^{\lceil\alpha\rceil-\alpha-1}\Bigg{[}y(\tau)-\sum_{k=0}^{\lceil\alpha\rceil-1}\frac{y^{(k)}(a)}{k!}\tau^{k}\Bigg{]}\,\mathrm{d}\tau,$
(55b)

one uses a relation of the type [24, §3.2]

${}^{\text{C}}\\!D_{a+}^{\alpha}y(t)=\int_{0}^{\infty}\phi(w,t)\,\mathrm{d}w,$
(56a) the so-called infinite state representation (or diffusive
representation, cf. [76]) of the fractional derivative of $y$. Here the
integrand $\phi$ (whose values $\phi(w,t)$ for $w\in(0,\infty)$ are known as
the infinite states of the observed system at time $t$) solves an
inhomogeneous linear first order initial value problem of the form
$\frac{\partial}{\partial
t}\phi(w,t)=h_{1}(w)\phi(w,t)+h_{2}(w)y^{(\lceil\alpha\rceil)}(t),\qquad\phi(w,0)=0,$
(56b) with certain functions $h_{1},h_{2}:(0,\infty)\to\mathbb{R}$ for which
many different specific choices have been suggested in the papers mentioned
above.

Using this representation, the task of numerically evaluating the fractional
differential operator (which is the core of every numerical method for solving
such differential equations) amounts to

  1. 1.

approximately computing the solution $\phi(w,t)$ to the initial value problem
(56b) at the time point $t=t_{n}$ currently under consideration for certain
suitably chosen values of $w$, and

  2. 2.

based on this information, numerically evaluating the integral on the right-
hand side of eq. (56a) with a properly designed quadrature formula.

In this context, the “suitable choice” of the values $w$ mentioned in the
first of the two items above is essentially determined by the location of the
nodes of the quadrature formula used in the second item.

From an algorithmic and software engineering point of view, this is an
excellent approach because it is both fast—having a computational complexity
of $O(N)$ when $N$ time steps need to be taken rather than the $O(N^{2})$ of
the original fractional Adams method from [51] or the at best $O(N\log N)$
cost of its modified version mentioned above—and memory efficient in the sense
that a computation over $N$ time steps requires only a memory amount of $O(1)$
instead of $O(\log N)$ as in the method from [60, 61, 62] or even $O(N)$ in
all other versions of the Adams method (and, in this comparison, replacing the
Adams method by a different classical approach would not make any difference
in these respects [47]). This observation is an immediate consequence of the
fact that one has to solve the differential equation (56b) which is a
differential equation of order 1 and hence does not exhibit any non-locality
or any other memory effects.

However, from a numerical analysis perspective, the situation is highly
unsatisfactory for a number of reasons:

  * •

Almost all the arguments on which the specific versions of the algorithm
indicated above are based are heuristic in nature. In particular, the
quadrature formulas for the integral from eq. (56a) required in step 2 are
constructed in an ad hoc manner; the analytical properties of the integrand
that are well known to strongly influence the accuracy of the approximate
results [77] are taken into account only partially in the proposal from [64]
and are completely ignored in most of the other schemes.

  * •

There is hardly any investigation regarding the interaction between the
numerical ODE solver required in step 1 of the algorithm and the quadrature
formula from step 2. The only information usually provided is the observation
that the nodes of the quadrature formulas that have been proposed have such
magnitudes that the solution of the ODE requires an A-stable method. Then,
according to Dahlquist’s well known second barrier [78, §V.1], the order of
the ODE solver cannot be higher than 2 if it is chosen from the set of linear
multistep methods; a higher order can however be achieved by using suitable
implicit Runge-Kutta methods [78, §IV.6]. In practice, to avoid the
intricacies connected to higher order Runge-Kutta methods, all the proposals
found in the literature suggest to use a linear multistep method (often, the
backward Euler method). The low order seems to be acceptable because, as
mentioned above, no strong effort has been invested into the selection of the
quadrature scheme, so using a high order but complex ODE solver does not seem
to be worth while anyway.

  * •

The typically chosen building blocks for the overall algorithm contain a
number of parameters (number and location of quadrature nodes, quadrature
weights, choice of the ODE solver, step size of ODE solver) whose influence on
the accuracy of the final result is essentially unknown. None of the papers
mentioned above contains any comprehensive error analysis.

  * •

For many of the approaches mentioned above, it is not immediately clear how
the implied constants in the error bounds (if such bounds are known at all)
depend on the order $\alpha$ of the differential operators in question; in
particular, it is often unknown whether the bounds hold uniformly for all
$\alpha\in(0,1)$ or whether the constants blow up as $\alpha\to 0$ or
$\alpha\to 1$.

Therefore, there exists a strong demand for a thorough investigation of the
non-classical algorithms for the numerical solution of fractional differential
equations.

###  3.2 Spectral Methods and Related Techniques

A frequent objection against the use of one-step or multistep methods for
solving fractional differential equations is that these methods are of some
kind of a local nature which does not appear to be a natural concept for
dealing with operators like fractional derivatives that are not local. One
potential and promising approach in this direction is the use of spectral
methods.

Traditionally, such spectral methods are based on classical polynomials [79,
80], but clearly this is not ideal because polynomials cannot accurately
capture the singular behaviour that the exact solutions to fractional
differential equation exhibit near the starting point of the associated
differential operator [81]. A more suitable alternative is to use generalized
polynomials with a combination of integer and non-integer exponents as
suggested, e.g., in [82, 83, 84]. This alternative comes with two significant
advantages:

  1. 1.

The exponents can be chosen to exactly match those present in the analytical
solution to the differential equation, thus eliminating the error component
induced by these (generalized) monomials and leading to a significantly better
convergence rate.

  2. 2.

If the basis of the set of generalized polynomials is chosen in an appropriate
way [83] then, for certain classes of differential equations, the coefficient
matrices of the discretized differential equations become sparse, thus greatly
reducing the computational cost and the memory requirements of the resulting
algorithms.

One of the key outcomes of [84] is the recommendation to use a Petrov-Galerkin
strategy for the construction of the approximate solution, i.e. one should
choose the test function space for the Galerkin method to be different from
the space of trial functions in which the approximate solution is sought.
Following the development of [84], we shall summarize this construction by
means of the simplest possible example application, namely the initial value
problem

${}^{\mathrm{C}}\\!D_{-1+}^{\alpha}y(t)=f(t),\qquad y(-1)=y_{0},\quad
t\in[-1,1]$ (57)

i.e. a differential equation whose right-hand side is independent of the
unknown function $y$ so that the analytical solution process essentially
amounts to a fractional integration of the forcing function. (The interval
$[-1,1]$ has been chosen for convenience and without loss of generality.) In
this special case, one starts with the construction of an associated Sturm-
Liouville eigenvalue problem of the first kind [83],

$\displaystyle{}^{\mathrm{RL}}\\!D_{1-}^{\alpha/2}K\,{}^{\mathrm{C}}\\!D_{-1+}^{\alpha/2}\Phi_{1,\lambda}(t)+\lambda(1-t)^{-\alpha/2}(1+t)^{-\alpha/2}\Phi_{1,\lambda}(t)$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\Phi_{1,\lambda}(-1)$
$\displaystyle=$ $\displaystyle 0,$ (58)
$\displaystyle{}^{\mathrm{RL}}\\!I_{1-}^{1-\alpha/2}K\,{}^{\mathrm{C}}\\!D_{-1+}^{\alpha/2}\Phi_{1,\lambda}(t)\Big{|}_{t=1}$
$\displaystyle=$ $\displaystyle 0$

with some constant $K$, and computes the eigenfunctions of this problem. It
turns out that these eigenfunctions can be written as

$\mathcal{P}_{1,n}(t)=(1+t)^{\alpha/2}P_{n-1}^{-\alpha/2,\alpha/2}(t)\qquad(n=1,2,3,\ldots)$
(59)

where $P_{k}^{(-\alpha/2,\alpha/2)}$ denotes the standard $k$-th degree Jacobi
polynomial for the weight function defined by the superscript in the usual way
[85, Chapter IV]. The eigenfunctions $\mathcal{P}_{1,n}$ for
$n\in\\{1,2,\ldots,N\\}$ with some predefined natural number $N$ are chosen as
the basis functions of the space in which the approximate solution is sought.

Moreover, a similar Sturm-Liouville eigenvalue problem of the second kind [83]

$\displaystyle{}^{\mathrm{RL}}\\!D_{-1+}^{\alpha/2}K\,{}^{\mathrm{C}}\\!D_{1-}^{\alpha/2}\Phi_{2,\lambda}(t)+\lambda(1-t)^{-\alpha/2}(1+t)^{-\alpha/2}\Phi_{2,\lambda}(t)$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\Phi_{2,\lambda}(1)$
$\displaystyle=$ $\displaystyle 0,$ (60)
$\displaystyle{}^{\mathrm{RL}}\\!I_{-1+}^{1-\alpha/2}K\,{}^{\mathrm{C}}\\!D_{1-}^{\alpha/2}\Phi_{2,\lambda}(t)\Big{|}_{t=-1}$
$\displaystyle=$ $\displaystyle 0$

is constructed, and the associated eigenfunctions of this problem are
determined too. In this case, they can be written in the form

$\mathcal{P}_{2,n}(t)=(1-t)^{\alpha/2}P_{n-1}^{\alpha/2,-\alpha/2}(t)\qquad(n=1,2,3,\ldots),$
(61)

and these functions (for $n=1,2,\ldots,N$) are used as the test functions.

Based on these definitions of trial and test functions, one can then apply the
standard Petrov-Galerkin scheme to compute the approximate solution $Y_{N,0}$
of the differential equation from the given problem (57) augmented with the
initial condition $y(-1)=0$. The fact that we can only work with this initial
condition is due to the observation that, as can be seen from eq. (59), all
basis functions satisfy $\mathcal{P}_{1,n}(-1)=0$, and so the approximate
solution—being a linear combination of these basis functions—must also satisfy
$Y_{N,0}(-1)=0$. The case of the general initial condition given in eq. (57)
then leads to the approximate solution $Y_{N}(t)=y_{0}+Y_{N,0}(t)$ with the
function $Y_{n,0}$ from above being the solution of the differential equation
with a homogeneous initial condition. This follows from the fact that the
additive constant $y_{0}$ is annihilated by the Caputo differential operator
in the differential equation (57).

A particularly relavant feature of this process is that the stiffness matrix
of the resulting linear system that determines the coefficients of the
approximate solution $Y_{N,0}$ is actually a diagonal matrix, so the system
can be solved in an extremely simple way.

From a more general perspective, one starts from the given differential
equation and constructs associated Sturm-Liouville eigenvalue problems of the
first kind and of the second kind, respectively. The eigenfunctions of the
eigenvalue problem of the first kind are then used the basis functions for the
space of approximate solutions, and the eigenfunctions of the eigenvalue
problem of the second kind are chosen as the test functions of the Petrov-
Galerkin algorithm.

A detailed and recent survey describing the state of the art in this area of
research may be found in [81].

In a similar way, one can handle so-called spectral element methods where the
domain is first decomposed in a finite element like manner, and afterwards a
spectral kind of approximation is used on each element; cf., e.g., the survey
in [86].

The description given above indicates that the strategy proposed in [84]
depends on the precise form of the given differential equation in various
respects. This applies, e.g., to the construction of the two Sturm-Liouville
eigenvalue problems and the determination of their eigenfunctions (i.e. the
trial and test functions), but also to the specific way in which a potential
inhomogeneous initial condition can be incorporated (although the latter can
be generalized rather easily). From a practical perspective, it would be
advantageous to have a set of test functions and a set of trial functions that
could be used not just for one type of differential equations but for a broad
class of such problems. The search for sets of such basis functions (that, of
course, should be able to guarantee a rapid convergence of the sequence of
approximate solutions towards the exact solution) is a topic of great
interest. Moreover, one might raise the question whether a different
construction of the basis and test functions could lead to an even faster
convergence.

A notable weakness of this approach that might also be addressed in future
research projects is that one is naturally lead to consider the covergence
behaviour and the error estimates in an $L_{2}$ norm only whereas other
solvers for fractional ordinary differential equations are usually analyzed
with respect to the stronger $L_{\infty}$ norm.

###  3.3 Terminal Value Problems

A completely different but also potentially very relevant question with
respect to which many open questions exist is also connected to the fractional
differential equation

${}^{\text{C}}\\!D_{a+}^{\alpha}y(t)=f(t,y(t))$ (62a) that is discussed with
some given function $f$ on the interval $[a,a+T]$ for some $T>0$. We
concentrate on the practically most important case $\alpha\in(0,1)$. Keeping
in mind the well kown properties of the operator
${}^{\text{C}}\\!D_{a+}^{\alpha}$ and, in particular, the classical existence
and uniqueness result for initial value problems [24, Theorem 6.5], it is
natural to believe that this problem requires exactly one additional condition
of the form $y(b)=y^{*}$ (62b)

with some given numbers $y_{0}\in\mathbb{R}$ and $b\in[a,a+T]$ to assert the
uniqueness of the solution. Such a condition with $b>a$ may be necessary for
example because measurements of the state of the modeled system are available
only at time $t=b$ but not at time $t=a$ (the starting point of the
differential operator in eq. (62a)). It turns out in the case $b>a$, however,
that the uniqueness can only be guaranteed in the case where the problem is
scalar, i.e. when we are looking for a function $y:[a,a+T]\to\mathbb{R}$, but
not in the vector-valued case where $y:[a,a+T]\to\mathbb{R}^{d}$ with $d\geq
2$, cf. [87]. We thus impose the additional restriction to consider only the
scalar case. In this situation, the problem can be decomposed into two steps:

  1. 1.

Solve the problem on the interval $[a,b]$.

  2. 2.

Solve the problem on the interval $[a,a+T]$.

Once the first step has been completed, the initial value $y(a)$ is known, and
hence it can be added to the differential equation in the form of an initial
condition, replacing the terminal condition (62b) and thus creating an initial
value problem whose solution can be computed with standard and well
established methods to find the solution of the second step and hence the
overall solution. Therefore, the only really novel task is step 1. Since the
auxiliary condition is formulated at the end point of the interval of
interest, this problem is called a _terminal value problem_.

From an analytical point of view, the fundamental difference between initial
and terminal value problems is highlighted when the two problems are rewritten
as equivalent integral equations: While, as is well known, an initial value
problem is equivalent to a _Volterra_ integral equation [24, Lemma 6.2], the
terminal value problem (62) can be reformulated on $[a,b]$ as

$y(t)=y^{*}+\frac{1}{\Gamma(\alpha)}\int_{a}^{b}G(t,s)f(s,y(s))\,{\mathop{\text{d}}}s$
(63a) with $G(t,s)=\begin{cases}-(b-s)^{\alpha-1}&\mbox{ for }s>t,\\\
(t-s)^{\alpha-1}-(b-s)^{\alpha-1}&\mbox{ for }s\leq t,\end{cases}$ (63b)

which is a _Fredholm_ integral equation [24, Theorem 6.18]. From this
difference in the types of the integral equations, it is already clear that
the terminal value problem needs completely different numerical methods than
the initial value problem; in particular, one can see that the nature of a
fractional order terminal value problem is much closer to an integer order
boundary value problem than to an integer order initial value problem.
Therefore, it is natural to solve fractional order terminal value problems
with the help of methods based on the construction principles coming from the
area of integer order boundary value problems.

Following the line of thought outlined above, a first obvious strategy to
solve terminal value problems is the class of shooting methods (cf., e.g.,
[88, Chapter 2] for a very detailed discussion of such methods in the case of
integer order equations): One starts with an initial guess for the initial
value $y(a)$, solves the associated initial value problem, and compares the
terminal value, i.e. the approximation for $y(b)$, obtained in this way with
the desired value $y^{*}$. If these two values are sufficiently close to each
other, the approximate solution is accepted; otherwise, a new attempt is
started with a suitably modified initial value.

Such methods have been proposed for fractional terminal value problems in [89,
§6] where the classical Adams method from [51, 52] has been suggested for
solving the initial value problems. Various improvements of this algorithm
with the aim of increasing its efficiency have been discussed in [90]. One of
the main observations from that paper is that in the early stages of the
iterative process, i.e. when the selected initial value is still far away from
the correct one, it is not necessary to use a very small step size for the
initial value problem solver. However, that proposal still leaves room for
improvement in this direction, and numerous other questions remain open as
well, most importantly the question what the best algorithm for selecting the
initial value in the next iteration step is. This question is currently under
active consideration [91].

A case study conducted by Ford and Morgado [92] provides a comparison of this
shooting method based on the Adams scheme with corresponding approaches using
other solvers for the initial value problems, essentially based on linear
multistep methods, and comes to the conclusion that, among the methods
investigated, the one using Adams algorithm provides the best balance between
accuracy and computational effort. Obviously, this investigation can only
cover a limited number of alternatives to the Adams method and might be
extended to other schemes. A different replacement for the Adams scheme,
namely a collocation method using a combination of classical piecewise
polynomials and generalized polynomials (with non-integer exponents) has been
shown to also provide relatively good results in [93].

Shooting methods are, of course, not the only possible approach to numerically
handle fractional terminal value problems. In view of the integral equation
represtentation (63a) of the problem, all the known standard techniques for
the numerical solution of weakly singular Fredholm equations are available
too, cf., e.g., [94]. It seems, however, that among the many classes of
methods that can be used in principle such as, e.g., Nyström methods,
collocation methods, or Galerkin methods, only a very small number has
actually been attempted for fractional terminal value problems. A notable
instance is a collocation scheme, again based on a combination of classical
polynomials and polynomials with non-integer exponents (which, in view of the
well known properties of the exact solutions to fractional differential
equations, seems to be a quite natural idea) discussed in [95]. The
convergence results shown in that paper indicate the method to be quite
attractive and rapidly convergent. A comparison of the computational cost of
this method to the cost of a shooting method would be useful to get a
comprehensive impression. Similar studies for other approaches to the Fredholm
equation (Galerkin, Nyström, etc.) would be most welcome. Such investigations
should in particular include a discussion about a suitable set of basis
functions that combines the ability to accurately model the asymptotic
behaviour of the exact solution as $t\to a$, thus exactly capturing the
singular behaviour of its derivatives, with a rapid convergence of the
remaining part of the approximate solution to the smooth part of the exact
solution. This latter aspect in particular includes the task of applying the
spectral or spectral element methods discussed in Subsection 3.2 to the
general class of terminal value problems described in the present subsection.

##  4 Applications in Physics

For the first time, FC was applied by Niels Henrik Abel in 1823 [96, 97] to
the tautochrone problem, which is the kinematic problem of finding the
trajectory of point mass sliding without friction to its lowest point, when
time does not depend on its starting point of the trajectory. Then
applications of FC to problems of physics were proposed in the works of Joseph
Liouville, beginning in 1832 [98, 99, 100, 101]. For Liouville, the
applications of FC were not easy illustration to awaken the interest of other
mathematicians. The applicability of FC was the meaning of the existence of
this mathematical theory. This relationship to FC significantly distinguishes
Liouville from his predecessors, whose main goal was a purely mathematical
generalization of ordinary calculus [101, p. 307]. Liouville’s contribution to
the application of FC is described by J. Lützen in [99, 100] and [101, pp.
307–320]. The contribution of the first pioneers of the application of FC in
physics is described in [4].

Fractional calculus, equations with fractional integrals, derivatives and
differences are a powerful tool for describing local processes in time and
space with different types of nonlocality. Conventionally, all applications
can be divided into five directions: nonlocality in space and time in the
framework of continuous and discrete approaches, and their interrelationships,
which can be called (CT) Nonlocality in continuous-time; (DT) Nonlocality in
discrete-time; (CS) Nonlocality in continuous-space; (DS) Nonlocality in
discrete-time; (R) Relationship between these directions.

Moreover, there is a need to classify the types of nonlocality and to link
these types with the description of the types of phenomena. For example, it is
necessary to distinguish nonlocalities in time that describe (1) memory, (2)
distributed lag (time delay); (3) distributed scaling (dilatation, dilation),
and others. Note that these different phenomena and types of nonlocality must
be described by different types of kernels of fractional operators.

One of the most important trends in the modern stage of applications of FC is
the urgent need to describe general types of nonlocal phenomena and the
corresponding sets of kernels of fractional operators that form FC. We
emphasize here the importance of not only, and not so much specific examples
from various fields of science, since there can be an infinite number of such
examples. It is important to solve general questions, it is important to
answer the questions of what types of operator kernels (and therefore
fractional operators) what types of phenomena can describe [102]. For example,
(1) the Riemann-Liouville and Caputo fractional operators can describe memory;
(2) the fractional operators of Hadamard type and Erdelyi-Kober can describe
scaling and dilation; (3) Fractional operators in the form of a Laplace
convolution, whose kernel satisfies the normalization condition, can describe
a distributed lag (time delay), and so on [102].

The very important trend in application of FC is investigations and results
concerning of general form of nonlocality, which can be described by general
form of operator kernels, and not its particular implementations and
representations [103].

Due to page limits for review, we will highlight only applications of FC that
seem very promising, important, and interesting. Unfortunately, some of the
important and promising applications of FC will remain out of consideration.
Some of these areas are described in volumes 4 and 5 of Handbook of Fractional
Calculus with Applications [9, 10], which contain 25 reviews on various areas
of applications of FC in physics. Other areas of application of FC are
described in reviews [104, 105, 106, 107] and in books on fractional dynamics
[108, 109, 110].

###  4.1 Nonlocal Continuum and Lattice Mechanics

The classical area of application of FC is the continuum mechanics, which
remains in the trend of future research. At the same time, the interest is
currently shifting from nonlocalities in time to nonlocalities in space, and
in the direction of nonequilibrium thermodynamics, and electrodynamics of
media with space dispersion and frequency dispersion.

Continuum mechanics with nonlocality in time has a long history dating back to
the work of Ludwig Boltzmann in 1874. The first physical model of media with
nonlocaliy in time has been proposed by Ludwig Boltzmann in 1874 and 1876
[111, 112] for isotropic viscoelastic media. In these works, te fading and
superposition principles were suggested. Then this approach is used in the
works of Vito Volterra [113]. The first application of FC to continuum
mechanics of viscoelastic as media with nonlocality in time was proposed in
work of Andrey N. Gerasimov [114] in 1948. At present time, FC is actively
used to describe continuum mechanics with nonlocality in time (for example,
see [115, 116, 117, 118] and references therein).

Continuous mechanics with nonlocality in space, in contrast to mechanics with
nonlocality of time, has a shorter history. Apparently the first application
of FC to continuum mechanics with nonlocality in space was proposed by V.S.
Gubenko in 1957 [119, 120]. The theory of nonlocal continuum mechanics was
formally initiated by the papers of Kröner [121], and Eringen [122, 123, 124].
Nonlocal theory is based on the assumption that the forces between material
points are a long-range type, thus reflecting the long-range character of
inter-atomic forces [125, 126].

In electrodynamics and mechanics of media the nonlocality in space is
interpreted as special form of spatial dispersion (SD) [127, 128, 129]. For
the first time, FC was used to describe SD in electrodynamics and mechanics in
works [130, 131] and [132, 133, 134, 135], where fractional form of SD are
described.

For continuous models of media with nonlocality in space, the limitation is
the absence of a fractional analogue of usual tensor and vector calculus
[136], [109, pp. 241–264]. This is due to the fact that nonlinear coordinate
transformations transform fractional operators of non-integer order into
pseudo-differential operators, accompanied by a change in the form of
nonlocality. For the first time, a consistent mathematical formulation of
fractional vector calculus was proposed in work [136], which includes
fractional generalizations of the differential operations (gradient,
divergence, curl), the integral operations (flux, circulation), and the
relationship of these operators in the form the generalized Gauss, Stokes and
Green theorems (see also [109, pp. 241–264]).

It is well known in solid state physics that long-range interactions between
particles can be considered as a source of nonlocality in space, which largely
determines the properties of the medium [125]. Because of this, it seems
natural to construct mechanics of continuum and lattice with nonlocality in
space [125, 137, 138]. Long-range interactions in lattice models of media is
important in physics, starting with Dyson’s work in 1969 [139, 140] about
models of spin chains and lattices with long-range interactions. However, the
active use of FC in this direction begins only now.

The use of FC for constructing lattice models with nonlocality in space (with
long-range interactions) and consideration of the continuous (continuum) limit
of these models, which is described by fractional differential equations,
actually began with the works [141, 142, 143, 144] in 2006 (see also [109, pp.
153–214]). Then the proposed approach was generalized from one-dimensional
case for $n$-dimensional case, and was represented as a lattice FC [145, 146,
147, 148, 149]. This representation allows to formulate concept of exact
differences of integer and fractional orders in [150, 151, 152] and [153,
154]. Unfortunately, for exact fractional differences, which are proposed in
[153, 154, 147], the fundamental theorems of FC have not been proven for non-
integer orders of operators at present time.

###  4.2 Economics with Nonlocality in Time: Memory and Distributed Lag

The first application of fractional calculus in economics can be considered
the work of the Nobel laureate C.W.J. Granger, R. Joyeux, in 1980 [155], in
which the Grünwald-Letnikov fractional differences were actually rediscovered.
In this work, a fractional generalization of ARIMA models was proposed. In
recent years, various attempts to apply FC in economics and finance are
suggested. The history of these applications of FC is given in [156] and [106,
pp. 5–32], [107]. Many of these attempts have various disadvantages that
described in the review [157].

FC is used to construct the basic concepts of economic theory and to form
mathematical models of economic processes on their basis [107]. The directions
of further applications of FC in this area are described in [156] (or [106,
pp. 5–32]), and in Section 30 [107, pp. 532–539]. Additional references on
application of FC in economics and finance can be found in the list of
references in [107, 156].

Note that, in contrast to mechanics and physics, where relaxation equations
are often used, in economics, growth equations are more often applied. The
fractional differential equations describing relaxation and growth have a
qualitative difference in the behavior of the solution due to their difference
in asymptotic behavior. This is one of the distinctive features of the novelty
of the behavior of economic processes with memory from processes with memory
in physics. The appearance of memory or memory changes can lead to a
significant acceleration or slowdown of the process, that is, the
characteristic time of the process can be substantially increased or decreased
by memory.

Application of FC is important to describe nonlocality in the form of
distributed lag (time delay). The distributed lag has been considered starting
with the models: (1) the model with uniform distributed lag that was suggested
by Michal A. Kalecki [158] in 1935 to described business cycles [159, pp.
251–254]; (2) the models with the exponential distribution of delay time was
proposed by Alban W.H. Phillips [160] in 1954; (3) models and operators with
distributed lag that were considered by Roy G.D. Allen [159] in 1956. The
distributed lag is described by the kernels that are probability density
functions (p.d.f), when the operators are defined by the Laplace convolution
with these kernels. The distributed lag is caused by finite speeds of
processes, and cannot interpreted as a memory. In physics, the distributed lag
as a form of nonlocality in time is interpreted as hysteresis.

Simultaneous presence of memory effects and distributed lag is important for
economic models. However, for fractional operators with distributed lag, which
are proposed in [161] and used in [162, 163], [107, Ch.20-23], the fundamental
theorems of FC have not been proved in general. Note that p.d.f of the gamma
distribution is the Sonin kernel, and we can use the general FC [164, 165, 39,
41, 40, 44] in this case and some other distributions on the positive
semiaxis. This is an important direction of future research in FC and its
applications in economics and physics.

###  4.3 Non-Markovian Dynamics of Open Quantum Systems

The theory of open quantum systems is the most general type of modern quantum
mechanics as fundamental theory [166, 167]. Moreover, this theory has great
practical importance for the creation of quantum computers and quantum
informatics. The influence of the environment changes the quantum computation,
which is realired by dynamics of quantum systems of qubits. Note that the
Schrödinger equation describes only pure quantum states and cannot be used for
general description of quantum dynamics.

In recent decades, the theory of open quantum systems has been actively
developing (for example, see basic papers [168, 169, 170], books [166, 171,
167], and reviews [172]). The most general form of Markovian equations, which
describe quantum observables and quantum states of quantum systems, was
suggested by Gorini, Kossakowski, Sudarshan and Lindblad in [168, 169, 170].
The most general explicit form of equations for the quantum mechanical systems
is the Lindblad equations, which describe Markovian dynamics of quantum
observables and quantum states.

We should emphasize that the openness of classical and quantum systems (the
interaction with the environment) can lead to nonlocality in time [173].

Currently, modern quantum mechanics is faced with the question of the most
general form of the equation describing the non-Markovian dynamics of quantum
systems with memory. Attempts to construct a non-Markovian theory of open
quantum systems with memory have been actively pursued recently (for example,
[174, 175, 176, 177, 178, 179], and references therein). The non-Markovian
character of quantum processes was often interpreted as nonlocality in time.
Note that exact solutions of equations describing non-Markovian dynamics with
memory have practically not been proposed in these works. All these attempts
were not associated with the use of FC. In construction of a consistent non-
Markvian theory of open quantum systems, the FC gives new possibilities for
the development of this direction.

For the first time, the use of FC to take into account memory effects (non-
Markovity) in open quantum systems was proposed in work [109] (see Chapter 20
in book [109, pp. 477–482] and [180, 181, 182]). The solutions of non-
Markovian equations describing quantum systems with memory, which is based on
FC, were proposed in these works.

For the first time, fractional powers of Lindblad superoperators were defined
and used to describe non-Markovian dynamics in 2008 [167] (see Chapter 20 in
book [167, pp. 433–444], Chapter 20 in book [109, pp. 458–464, 468–477], [183,
184]). Solutions of the proposed generaized Lindblad equations, which describe
non-Markovian quantum dynamics, were derived in [167, 183, 184, 109].

For construction of a theory of non-Markovian quantum dynamics, it is
important to consider time-dependent parameters [185, 181] and the
relativistic quantum systems with memory [186]. It is important for the non-
Markovian quantum dynamics the generalization in the directions: the path
integral for open quantum systems [187], [167, pp. 475–485]; uncertainty
relation for open quantum systems [188]; the pure stationary states of open
quantum systems [189, 190], [167, pp. 453–462]. The non-Markovian quantum
dynamics can also be considered in the framework of fractional generalizations
of Heisenberg equations [191, 167], [109, pp. 457–466], in contrast to the
time-fractional Schrödinger equation. For further reference, it can be used
the list of references in [182].

Quantum mechanics with nonlocality in space and quantum field theory with non-
locality in space-time by using models of lattices with long-range
interactions and its continuum limits [145, 146, 147, 148, 149, 150, 151, 152,
153, 154] are important future direction of applications of FC to quantum
physics. This is due to the fact that at present various nonlocal models are
being actively investigated within the framework of quantum field theory at
present time. In addition, lattice models with long-range action are being
actively investigated within the framework of quantum physics of solids and
condensed matter.

###  4.4 Nonlocal Discrete Maps

In nonlinear dynamics, maps with discrete time are derived from integer-order
differential equations with periodic kicks (Sec. 5.2, 5.3 in [108, pp. 60–68],
and Ch. 18 in [109, pp. 409–453]). These maps define the next step only by the
previous step (or a fixed number of previous steps. In discrete maps with
nonlocality in time means that the next step depends on all past steps.

Discrete maps from fractional differential and integral equations of non-
integer orders can be also derived. For the first time such discrete maps were
obtained in article [192, 193, 194] in 2008. The proposed nonlocal maps can be
derived by using the equivalence of the FDE and the Volterra integral
equations in [193, 194, 109]. Then, this approach has been applied in [195,
196, 182, 197, 107, 103, 198, 199]. The first computer simulations of the
suggested nonlocal maps were made in [200, 201], and then numerical
simulations have proved the existence of new types of attractors and new types
of chaotic behavior for these maps (see [202, 203, 204, 205, 206] and
references therein).

A new general approach to the study of the chaotic behavior of discrete maps
with nonlocality was proposed in [103, 206]. In the framework of this
approach, new types of attractors and chaotic nonlocal dynamics can be
discovered for nonlocal discrete maps with discrete convolution [103, 206].

###  4.5 Self-Organization with Memory

Self-organization is a process of formation of ordered spatial or temporal
structures that can be realized without external influences. Self-organization
can be realized in physical, chemical, biological and economic processes [207,
208, 209, 210]. For the first time, FC was used to describe the processes of
self-organization, where effects of self-organization with memory is
considered, in 2018 [211], (see also Chapter 19 in [107, pp. 364–382]).

The emergence and change of memory can leads to the appearance of hierarchy of
relaxation times and qualitatively change the behavior of the systems at the
remaining other parameters unchanged. This can lead to the self-organization,
which is generated only by memory.

We should note that the discrete maps with memory and nonlocality in time,
which are derived from fractional differential equations [192, 193, 194, 109,
195, 196, 182, 197, 199, 107], can demonstrate qualitatively new types of
chaotic and regular behavior. Similarly, fundamentally new types of self-
organization are expected in processes with nonlocality in time and memory.
This direction of future researches is very important for nonlinear fractional
dynamics.

##  5 Conclusions

This paper started by presenting the present day trends of research in FC.
Stemming from the state of scientific knowledge, possible directions for
further research were pointed. Associated with this analysis some open
problems in FC were also identified. The work was organized in three main sub-
areas namely, aspects in mathematical analysis, numerical processing, and
applications in physics. Given the limitation of space, this work is
necessarily parsimonious, but gives a synthetic vision that can somehow guide
researchers in the future.

## Declarations

#### Funding.

The authors have not received any external funding in connection with this
manuscript.

#### Conflicts of interest.

The authors declare that they have no conflict of interest.

#### Code availability (software application or custom code).

Not applicable.

#### Data availability.

No datasets were generated during the current study. The data analysed during
the study are available from Elsevier’s Scopus database but restrictions apply
to the availability of these data, which were used under license for the
current study, and so are not publicly available. Data are however available
from the authors upon reasonable request and with permission of Elsevier B.V.

#### Authors’ contributions.

All authors contributed to the design and to the writing of the paper. All
authors read and approved the final manuscript.

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