form of the sought values $v_{j}$ is obtained with the aid of the Z-transform,
which maps a sequence $(a_{i})_{i\in{\mathbb{N}}_{0}}$ to a formal power
series ${\mathcal{A}}(z):=\sum_{i=0}^{\infty}a_{i}z^{i}$. With the transforms
$\alpha(z)=\sum_{i=0}^{k}\alpha_{i}z^{i}$, $\beta(z)=\sum_{i=0}^{k}\beta
z^{i}$, ${\mathcal{V}}(z):=\sum_{i=0}^{\infty}v_{i}z^{i}$,
${\mathcal{U}}(z)=\sum_{i=0}^{\infty}u(t_{i})z^{i}$, one arrives at

$\displaystyle\frac{1}{{\Delta t}}\alpha(z){\mathcal{V}}(z)$
$\displaystyle=\beta(z)\left(s{\mathcal{V}}(z)+{\mathcal{U}}(z)\right).$

Solving for ${\mathcal{V}}$ gives with the generating function
$\delta(z):=\alpha(z)/\beta(z)$ of the multistep method

$\displaystyle{\mathcal{V}}(z)$ $\displaystyle=(\delta(z)/{\Delta
t}-s)^{-1}{\mathcal{U}}(z).$

Hence, the Z-transform ${\mathcal{Y}}(z)=\sum_{i=0}^{\infty}y_{i}z^{i}$
satisfies

$\displaystyle{\mathcal{Y}}(z)$ $\displaystyle=\frac{1}{2\pi{\rm
i}}\int_{s\in\sigma+{\rm i}{\mathbb{R}}}K(s)(\delta(z)/{\Delta
t}-s)^{-1}\,ds\,{\mathcal{U}}(z)=K(\delta(z)/{\Delta t}){\mathcal{U}}(z).$

Expanding

$\displaystyle K(\delta(z)/{\Delta t})$
$\displaystyle=\sum_{j=0}^{\infty}\omega_{j}z^{j}$

and recognizing that the power series of $K(\delta(z)/{\Delta
t}){\mathcal{U}}(z)$ involves the convolution of the sequences
$(\omega_{j})_{j\in{\mathbb{N}}_{0}}$ and $(u(t_{j}))_{j\in{\mathbb{N}}_{0}}$
gives the desired relation

$\displaystyle y_{n}=\sum_{i=0}^{n}\omega_{n-i}u(t_{i}),\qquad n=0,1,\ldots$
(81)

for the Convolution Quadrature.

The formal calculation in Example 7.1 shows how to design different CQ methods
by choosing the function $\delta$, which in turn is given by the choice of the
underlying ODE solver. To formulate the method, only the Laplace transform $K$
of the kernel $k$ is required. Even for rather general $K$, the weights
$\omega_{j}$ can be obtained numerically by applying the trapezoidal rule to
the Cauchy integral formula for derivatives, [213, Sec. 3.1]. The underlying
ODE solver (e.g., the multistep method in Example 7.1) is often required to be
A-stable. This stipulation derives from the observation that the domain of
holomorphy of $K$ often contains the whole right half-plane (e.g., the
function $K(s)=s^{\alpha}$, $\alpha\not\in{\mathbb{N}}_{0}$, which is only
defined on the slit plane ${\mathbb{C}}\setminus(-\infty,0]$) and the
observation that for A-stable ODE solvers, the function $\delta$ maps the
complex unit ball $B_{1}(0)\subset{\mathbb{C}}$ to a subset of the right half-
plane $\\{z\in{\mathbb{C}}\,|\,\operatorname{Re}z>0\\}$. However, the
requirement of $A$-stability of the underlying ODE solver depends on the
problem under consideration; for example, for (76) with $\alpha\in(0,1)$ and
an elliptic operator $A$ weaker stability requirements suffice, [211], [206,
Sec. 3.1], [213, Sec. 8].

For an application of CQ to fractional derivatives, one notes that the Laplace
transform $K_{\alpha}(s)=\int_{t=0}^{\infty}t^{\alpha}e^{-st}\,dt$ of the
kernel $k(t)=t^{\alpha}$ is explicitly given by
$K_{\alpha}(s)=\Gamma(1+\alpha)s^{-(1+\alpha)}$. The Laplace transform of the
differentiation operator is simply $s$. Hence, for $u$ with $u(0)=0$, the
Laplace transform of $\mathfrak{\tilde{D}}^{\alpha}_{0,t}u$ is given by
$s^{\alpha}{\mathfrak{L}}u(s)$. That is, for $\alpha\in(0,1)$ the CQ
realization of the Caputo derivative for functions vanishing at $t=0$ can be
done with the kernel $K_{\alpha}(s)$. Since the Riemann-Liouville and Caputo
derivatives coincide for such functions, the same CQ techniques can be
employed. There is a extensive convergence theory for CQ based on uniform
meshes, see, e.g., [213, Secs. 2.5, 5.3]. Generally speaking, for sufficiently
smooth function $u$, the difference $y(t_{n})-y_{n}$ of the values (80) and
(81) is $O({\Delta t}^{p})$ if  a) the order of the underlying ODE solver is
$p$ and  b) $u$vanishes to sufficient order at $t=0$.  Concerning the
requirement a): if one has to base CQ on an A-stable ODE solver, then
multistep methods are limited to order $p=2$ by Dahlquist’s barrier. However,
CQ can also be based on Runge-Kutta methods, [214] and arbitrary order
A-stable Runge-Kutta methods such as the Radau IIA formulas are available,
[213, Sec. 5]. In the context of FODEs and FPDEs, requirement b) is typically
a more limiting restriction as the solution $u$ of the FODE/FPDE cannot be
expected to vanish to high order at $t=0$. Order reduction is then visible.
One remedy is to resort to a “corrected” CQ akin to that discussed above for
the L1 method. For general CQ methods, see, e.g, [215] and [206, Sec. 3.2] for
the case (76) of an elliptic $A$. An alternative remedy is to base the CQ on
non-uniform meshes that are refined towards $t=0$, see [216] for an recent
account on such CQ methods. Finally, we point to [217] for a posteriori error
estimates for CQ on general meshes.

####  7.1.3 Fast Methods

At first sight, the evaluation of fractional differential operators in the
nodes $t_{0},\ldots,t_{n}$ leads to complexity $O(n^{2})$ due to the time-
stepping nature; alternatively, the computation can be viewed as a matrix-
vector multiplication with a lower triangular matrix. For uniform point
distributions, the convolution structure leads to Toeplitz matrices and
algorithms with $O(n\log n)$ complexity using FFT techniques are available for
Toeplitz matrices, see [213, Sec. 3.3] for such algorithms in the context of
CQ. Especially in the context of FPDEs, which involves additionally
discretizations of spatial operators, various “fast” methodologies have
emerged that exploit the fact that the pertinent kernel $k(s)=s^{\alpha}$ is
smooth with a singularity only at the origin. As such, compression techniques
developed for integral operators (IO) such as fast multipole techniques [218],
wavelet compression, [219] and ${\mathcal{H}}$-matrices, [220] that lead to
polylogarithmic-linear complexity $O(n\log^{\beta}n)$ (for some $\beta\geq 0$)
for storage and matrix-vector multiplication are, in principle, available. The
broad ${\mathcal{H}}$-matrix format consists of blockwise low-rank matrices,
and, in the context of IOs, blockwise low-rank approximation are obtained by
approximating the kernel function by, e.g., piecewise polynomials. Therefore,
${\mathcal{H}}$-matrix compression is well-suited for Galerkin or collocation
discretizations of IOs on both uniform and nonuniform meshes; see, e.g., [220,
Sec.10.6] for a discussion of the case of Volterra IOs.
${\mathcal{H}}$-matrices come with an (approximate) arithmetic that includes
addition, multiplication, and inverse of matrices, [220].

In the context of fractional differential operators, a more popular technique
to effectively realize a compression is to approximate the kernel by a (short)
sum of exponentials (SOE) instead of piecewise polynomials. A detailed
analysis is given in [206, Sec. 4.2] for the L1; see also the survey [221] for
related schemes. To illustrate the main point, consider the L1 scheme, i.e.,
the realization of $\int_{s=0}^{t_{n}}k(t_{n}-s)(\Pi_{1}u)(s)\,ds$, where
$\Pi_{1}u$ is the piecewise linear interpolant of $u$ on a uniform mesh with
nodes $t_{j}=j{\Delta t}$. One assumes that the kernel $k(s)$ admits the
approximation $k(s)\approx\sum_{\ell=0}^{N_{k}}\omega_{\ell}e^{-s_{\ell}s}$
for some small $N_{k}$ and suitable $\omega_{\ell}$,
$s_{\ell}\in{\mathbb{R}}$; the smoothness of $k$ away from $s=0$ ensures that
this can be achieved for the argument $s$ bounded away from $0$. Since this
approximation is poor for $s$ close to $0$, the following approximation is
based on a decomposition into a “history” part and a “local” part:

$\displaystyle\int_{0}^{t_{n}}k(t_{n}-s)\Pi_{1}u(s)\,ds$
$\displaystyle=\int_{0}^{t_{n-1}}k(t_{n}-s)\Pi_{1}u(s)\,ds+\int_{t_{n-1}}^{t_{n}}k(t_{n}-s)\Pi_{1}u(s)\,ds.$

The “local” part $\int_{t_{n-1}}^{t_{n}}k(t_{n}-s)\Pi_{1}u(s)\,ds$ is
evaluated exactly and requires only knowledge of the values $u(t_{n})$ and
$u(t_{n-1})$. The “history” part $\int_{0}^{t_{n-1}}k(t_{n}-s)\Pi_{1}u(s)\,ds$
is realized efficiently with a recursion for the $N_{k}$ terms
$H_{\ell}(t_{n}):=\int_{0}^{t_{n}}e^{-s_{\ell}(t_{n}-s)}\Pi_{1}u(s)\,ds$:

$\displaystyle\int_{0}^{t_{n-1}}k(t_{n}-s)\Pi_{1}u(s)\,ds$
$\displaystyle\approx\sum_{\ell=0}^{N_{k}}\omega_{\ell}\int_{0}^{t_{n-1}}e^{-s_{\ell}(t_{n}-s)}\Pi_{1}u(s)\,ds=\sum_{\ell=0}^{N_{k}}\omega_{\ell}e^{-{\Delta
t}s_{\ell}}H_{\ell}(t_{n-1}),$

and the update formula for the $N_{k}$ terms $H_{\ell}(t_{n})$ is given by

$\displaystyle H_{\ell}(t_{n})$
$\displaystyle=H_{\ell}(t_{n-1})+e^{-s_{\ell}{\Delta
t}}\int_{t_{n-1}}^{t_{n}}e^{-s_{\ell}(t_{n})-s}\Pi_{1}u(s)\,ds,$

which again involves only knowledge of the values $u(t_{n-1})$ and $u(t_{n})$
and can be done explicitly. In the context of FODEs, the values $u(t_{j})$ are
approximated by values $u_{j}$ and the initial value $u_{0}$ is prescribed.
Correspondingly, one sets $H_{\ell}(t_{0})=0$. The computational complexity of
algorithms based on SOE hinges on the number $N_{k}$ of terms in the
approximation to be small. By [206, Thm. 4.2] one has
$N_{k}=O(|\log(\varepsilon{\Delta t})|^{2})$ for a prescribed accuracy
$\varepsilon$ so that for accuracy requirement $\varepsilon=({\Delta
t})^{\beta}$ ($\beta>0$) one arrives at complexity $O(n\log^{2}n)$.

A method somewhat similar to the technique of using sums of exponentials is
the “fast and oblivious” CQ of [222]. It is a CQ that is applicable if the
Laplace transform $K$ of the kernel $k$ is sectorial, i.e., it is holomorphic
on a sector $\\{z\in{\mathbb{C}}\,|\,|\operatorname{arg}z|<\pi/2+\delta\\}$
for some $\delta>0$ and satisfies certain decay conditions as
$|z|\rightarrow\infty$. A typical representative of this setting are operators
of the form $\mathfrak{\tilde{D}}^{\alpha}_{0,t}u+Au$ where $A$ is an elliptic
spatial operator. Fast and oblivious CQ also reduces the cost to
$O(n\log^{\beta}n)$ ($\beta\geq 0$) by introducing a few auxiliary variables
that track the history part in an agglomerated fashion (with controllable
error); see [213, Sec. 8.2] and the related [206, Sec. 3.5], [223], [224] for
details.

####  7.1.4 Spectral Galerkin Methods for FODEs

The techniques described in Sections 7.1.1, 7.1.2 have the form of time-
marching schemes. A different approach is obtained by spectral techniques,
which come in variants such as spectral Galerkin, spectral Petrov-Galerkin,
and spectral collocation methods. Starting points concerning the vast
literature can be found in the contributions [225] and [226] of the handbook
[204],the monograph [227] as well as [206, Sec. 12]. See also [228, 229, 230,
231] and references therein. While we discuss high order spectral Galerkin
methods, Galerkin methods can also be formulated for fixed order methods
similar to the SEM discussed below.

Consider (75) with $\alpha\in(0,1)$ and $\lambda>0$. The classical Spectral
Method seeks to approximate the solution from the space
${\mathcal{P}}_{N}:=\operatorname{span}\\{t^{i}\,|\,i=0,\ldots,N\\}$ of
polynomials of degree $N$. In view of the expected (weak) singularity at the
initial point $t=0$ (cf. Remark Remark) better approximation spaces are
required and this has led to the study of several classes of approximation
spaces:  (i) The first class, associated with the names of polyfractonomials
[231] or Generalized Jacobi functions (GJF) [225, 229] consists (in the
present case of a singularity only at $t=0$) of the space
$t^{\beta}{\mathcal{P}}_{N}$, where the parameter $\beta$ can be chosen to
match the singular behavior.  (ii) The second class of functions are “Müntz
polynomials” $\\{\pi(t^{\beta})\,|\,\pi\in{\mathcal{P}}_{N}\\}$ [232].  (iii)
The third class are the recent “Log orthogonal polynomials”
$\\{t^{\beta}\pi(\log t)\,|\,\pi\in{\mathcal{P}}_{N}\\}$ for chosen
$\beta\in{\mathbb{R}}$, [228], [206, Sec. 12.2].  (iv) The last class is the
very versatile class of spectral element methods (SEM), where approximation is
effected by piecewise polynomials on suitable meshes, in particular the so-
called geometric mesh that is strongly refined towards the singularities, see
[233, Sec. 3.3.6] in the context of PDEs and, in the context of fractional
operators, [230]. Indeed, the geometric mesh is a proven tool to approximate
functions with singularities and requires only knowledge of the location of
the singularities but not of their precise form. The choices (i), (ii) lead to
spectral approximation, i.e., a rate $O(N^{-\gamma})$ for any $\gamma>0$ when
approximating functions of the form $u(t)=t^{\beta}\tilde{u}(t)$ for smooth
$\tilde{u}$ and known $\beta$; the class (ii) can still have good
approximation properties for a larger class of functions with singularities at
$t=0$ when the tuning parameter $\beta$ is chosen suitable.

The classes (iii), (iv) are more versatile and do not even need a priori
knowledge of the precise structure of the singular behavior at $t=0$ for
spectral convergence. It should be noted that Remark Remark informs us of the
expected solution structure and that the classes (i) might only resolve the
leading order singularity so that only fixed algebraic rates of convergence
are achieved.

Conceptually, a simple way to make use of the good approximation properties of
the above spaces are Galerkin methods. The functional framework are the
Sobolev spaces $H^{\beta}(0,T)$, $\beta\in(0,1)$, which are the completions of
$C^{\infty}(0,T)$ under the $H^{\beta}(0,T)$-norm given by the Slobodeckij
norm
$\|u\|^{2}_{H^{\beta}(0,T)}:=\|u\|^{2}_{L^{2}(0,T)}+\int_{t=0}^{T}\int_{\tau=0}^{T}\frac{|u(t)-u(\tau)|^{2}}{|t-\tau|^{1+2\beta}}\,dt\,d\tau$.
Taking, e.g., the Riemann-Liouville derivative in (75), multiplying the
equation by a test function $v$ from $H^{\alpha/2}(0,T)$ and integrating over
$(0,T)$ one arrives at the weak formulation to find $u\in H^{\alpha/2}(0,T)$
such that

$\int_{0}^{T}\mathfrak{D}^{\alpha}_{0,t}u(t)v(t)\,dt=\int_{0}^{T}\mathfrak{D}^{\alpha/2}_{0,t}u\,\mathfrak{D}^{\alpha/2}_{t,T}v\,dt=:b(u,v)=\ell(v):=\int_{0}^{T}f(t)v(t)\,dt$
(82)

for all $v\in H^{\alpha/2}(0,T)$, [234], [206, Sec. 12.1]. One should note
that for $\alpha\in(0,1)$ functions from $H^{\alpha/2}(0,T)$ do not have a
trace at $t=0$ so that the implied initial condition $u(0)=0$ is only
satisfied in a weak sense, [234, Rem. 2.2]. A Galerkin discretization of (82)
is immediately formulated by seeking, for a given a subspace $V_{N}\subset
H^{\alpha/2}(0,T)$ of dimension $N:=\operatorname{dim}V_{N}$, an approximation
$u_{N}\in V_{N}$ such that

$b(u_{N},v)=\ell(v)\qquad v\in V_{N}.$ (83)

Once a basis of $V_{N}$ is selected, the approximation $u_{N}$ is obtained as
the solution of a linear system of equations. The key feature of the weak
formulation (82) is that the bilinear form, although not symmetric (due to the
presence of the different operators $\mathfrak{D}^{\alpha}_{0,t}$ and
$\mathfrak{D}^{\alpha}_{t,T}$), is elliptic on $H^{\alpha/2}(0,T)$ by [235,
234] in the present case $\alpha\in(0,1)$. In fact,

$c_{1}\|u\|^{2}_{H^{\alpha/2}(0,T)}\leq b(u,u)\leq
c_{2}\|u\|^{2}_{H^{\alpha/2}(0,T)},\qquad|b(u,v)|\leq
c_{2}\|u\|_{H^{\alpha/2}(0,T)}\|v\|_{H^{\alpha/2}(0,T)}\qquad\forall u,v\in
H^{\alpha/2}(0,T),$

for some $c_{1}$, $c_{2}$ depending only on $\alpha$ and $T$. This implies
unique solvability of both the continuous problem (82) and the discrete
problem (83) for any $f\in(H^{\alpha/2}(0,T))^{\prime}$. Additionally, the
Galerkin method is quasi-optimal, i.e,.

$\|u-u_{N}\|_{H^{\alpha/2}(0,T)}\leq\frac{c_{2}}{c_{1}}\inf_{v\in
V_{N}}\|u-v\|_{H^{\alpha/2}(0,T)}.$ (84)

We conclude that with good choices of approximation spaces $V_{N}$, high
accuracy of Galerkin methods can be achieved. The Galerkin method leads to
fully populated, non-symmetric system matrices. Its entries can be computed
with suitable Gauss-Jacobi quadrature rules: For case (iv) of the SEM based on
standard piecewise polynomials, these calculations are given in [230]. For
case (i), i.e., polyfractonomials or GJF, it is observed in [231] and [225,
Lemma 6] or [229] that fractional derivatives of GJFs can be expressed in
terms of Jacobi polynomials. This motivates the choice of basis of the space
GJF space $t^{\beta}{\mathcal{P}}_{N}$. Indeed, by [225, Lemma 6], [229] one
has (in certain parameter ranges) that
$\mathfrak{D}^{\alpha}_{-1,t}(1+t)^{\alpha+\alpha_{2}}P^{(\alpha_{1}-\alpha,\alpha_{2}+\alpha)}_{n}(t)=\frac{\Gamma(n+\alpha_{2}+\alpha+1)}{\Gamma(n+\alpha_{2}+1)}P^{(\alpha_{1},\alpha_{2})}_{n}(t)$
with $P^{(\alpha_{1},\alpha_{2})}_{n}$ denoting a Jacobi polynomial defined,
as usual, on the interval $(-1,1)$. That is, fractional derivatives of
functions of the form $(1+t)^{\alpha}P^{(\alpha,\beta)}_{n}$ can be expressed
through Jacobi polynomials, albeit corresponding to different parameters. This
allows one to evaluate the “stiffness matrix” arising from
$\int_{0}^{T}\mathfrak{D}^{\alpha/2}_{0,t}u\mathfrak{D}^{\alpha/2}_{t,T}v\,dt$
explicitly by a special basis of the GJF space. For (75) with $\lambda\neq 0$
or (76), also the “mass matrix” arising from $\lambda\int_{0}^{T}uv$ has to be
computed; it involves the computation of weighted $L^{2}$-inner products of
polynomials and is thus accessible through Gauss-Jacobi quadratures. The
computation of the matrix entries for case (ii) of the Müntz polynomials is
done in [232]. For case (iii), i.e., the “log orthogonal” case (with
$\beta=0$), basis functions of ${\mathcal{P}}_{N}$ related to the Laguerre are
chosen and lead to efficient computations of the system matrix, [206, Sec.
12.5].

Often, Spectral Methods are not applied as a Galerkin method but as a more
general Petrov-Galerkin (PG) method. This generalization is based on two
spaces, a trial space $U_{N}$ and a test space $V_{N}$ and takes the form:
find $u_{N}\in U_{N}$ such

$b(u_{N},v)=\ell(v)\qquad\forall v\in V_{N}.$ (85)

The choice $V_{N}=U_{N}$ reverts to the Galerkin method. Good choices of the
variational formulation (i.e., the form $b$) and $V_{N}$ can have a
significant impact on the computational complexity. In simple cases, the
resulting system matrix turns out to be sparse or even diagonal, [231] and
[225]. The flexibility in the choice of the test space $V_{N}$ often leads to
simplified ways of setting up the system matrix and can allow one to give it
some structure. We refer to [236], where symmetric system matrices result from
a suitable choice of basis of $V_{N}$; one reason for preferring symmetric
matrices over general non-symmetric ones is the availability of fast solver
technology.

Especially when considering problems such as (76) with a discretization of the
spatial operator $A$, a (Petrov-) Galerkin discretization with its fully
populated stiffness and mass matrix leads to very large problems. A common
strategy is to try to “decouple” as described, e.g., in [206, Sec. 12.5]. To
that end, one makes a change of basis of the temporal space $V_{N}$ so that
the (temporal) stiffness matrix ${\mathbf{A}}$ and the (temporal) mass matrix
${\mathbf{M}}$ are simultaneously diagonalized, which leads to $N$ fully
decoupled spatial problems. If a simultaneous diagonalization is not possible
(or not advisable for stability reasons), e.g., because ${\mathbf{A}}$ and
${\mathbf{M}}$ are not symmetric, then bringing them simultaneous to upper
triangular form [237, Thm. 7.7.2] is an option and allows one to sequentially
solve $N$ spatial problems.

###  7.2 Numerical methods for Fractional Operators in Multi-d

Although the Riemann-Liouville and the Caputo derivative are 1D constructions,
they can, in principle, be employed to define various fractional operators in
multi-d. Here, we focus instead on the fractional Laplacian
$(-\Delta)^{\gamma/2}$ and restrict our discussion only to two different
approaches to defining $(-\Delta)^{\gamma/2}$ for $\gamma\in(0,2)$ that we
already introduced in Section 2.2, namely, the spectral fractional Laplacian
(SFL) and the integral fractional Laplacian (IFL). Recall that these operator
coincide on ${\mathbb{R}}^{d}$, but they differ when restricted to bounded
domains $\Omega$; for example, the boundary singularities of the solutions of
(86) and (93) differ, [238]. Good surveys of these and related operators are
[239, 238, 44, 227].

####  7.2.1 Numerical Methods for the Spectral Fractional Laplacian

Let $\Omega\subset{\mathbb{R}}^{d}$ be a bounded Lipschitz domain. Recall the
definition of the SFL from Section 2.2. The standard model problem involving
the SFL is then: given $f\in H^{-\gamma/2}(\Omega)$ find
$u\in\widetilde{H}^{\gamma/2}(\Omega)$ such that

$(-\Delta)^{\gamma/2}u=f\mbox{ in $\Omega$},\qquad u=0\quad\mbox{ on
$\partial\Omega$.}$ (86)

#### Rational Approximations

Very good discussions of various methods based on rational approximations of
matrix functions are given in [240] and [241], which we follow here. A rather
intuitive approach to treat (86) numerically is to discretize the (positive)
Laplacian $(-\Delta)$ and work with powers of the resulting system matrix. For
example, as described in [239, Sec. 3.2.1], one may compute discrete
eigenfunctions and eigenvalues and mimick the definition of
$(-\Delta)^{\gamma/2}$ by replacing the continuous eigenpairs
$(\phi_{i},\lambda_{i})$ by the discrete ones. This approach, called the
discrete eigenfunction method (DEM), is the basic version of methods that
first discretize the Laplacian and then take matrix powers. To be specific,
consider a FEM discretization based on a space $V_{N}\subset
H^{1}_{0}(\Omega)$ with basis $\\{\varphi_{i}\,|\,i=1,\ldots,N\\}$ with the
symmetric stiffness matrix ${\mathbf{A}}\in{\mathbb{R}}^{N\times N}$ with
entries
${\mathbf{A}}_{ij}=(\nabla\varphi_{j},\nabla\varphi_{i})_{L^{2}(\Omega)}$ and
the symmetric mass matrix ${\mathbf{M}}\in{\mathbb{R}}^{N\times N}$ with
entries ${\mathbf{M}}_{ij}=(\varphi_{j},\varphi_{i})_{L^{2}(\Omega)}$. With
the load vector ${\mathbf{f}}\in{\mathbb{R}}^{N}$ with entries
$\mathbf{f}_{i}=(f,\varphi_{i})_{L^{2}(\Omega)}$, the DEM amounts to
approximating the solution $u$ of (86) by
$u^{N}=\sum_{i}{\mathbf{u}}^{N}_{i}\varphi_{i}$ with the vector
${\mathbf{u}}^{N}\in{\mathbb{R}}^{N}$ given by (see [240, Sec. 2] for details)

${\mathbf{u}}^{N}=\left({\mathbf{M}}^{-1}{\mathbf{A}}\right)^{-\gamma/2}{\mathbf{f}}.$
(87)

(87) indicates that, in contrast to the DEM, we may avoid computing all
eigenpairs of the matrix pencil $({\mathbf{A}},{\mathbf{M}})$ if we can find
good approximations of
$\left({\mathbf{M}}^{-1}{\mathbf{A}}\right)^{-\gamma/2}$. For example, we may
use rational approximations $r(x)\approx x^{-\gamma/2}$ and approximate
${\mathbf{u}}^{N}\approx{\mathbf{u}}^{N,rat}:=r({\mathbf{M}}^{-1}{\mathbf{A}}){\mathbf{f}}$.
Assuming that $r(x)=O(1/x)$ for $x\rightarrow\infty$, we can write write $r$
in terms of its poles $d_{j}\in{\mathbb{C}}$ as
$r(x)=\sum_{j}c_{j}(x-d_{j})^{-1}$ for suitable $c_{j}\in{\mathbb{C}}$; thus
the evaluation of $r({\mathbf{M}}^{-1}{\mathbf{A}}){\mathbf{f}}$ reduces to
solving (possibly in parallel) linear systems of equations of the form
$({\mathbf{A}}-d_{j}{\mathbf{M}})\widetilde{\mathbf{u}}=c_{j}^{-1}{\mathbf{M}}{\mathbf{f}}$.
For large scale problems, such a system might require preconditioning;
however, for $d_{j}$ with $\operatorname{Re}d_{j}>0$, these problems
correspond to discretizations of reaction-diffusion equations, for which
preconditioners are available. By [240, Thm. 1], the additional error
$u^{N}-u^{N,rat}:=\sum_{i}{\mathbf{u}}^{N}_{i}\varphi_{i}-\sum_{i}{\mathbf{u}}^{N,rat}_{i}\varphi_{i}$
incurred by the rational approximation is controlled by
$\|u^{N}-u^{N,rat}\|_{L^{2}(\Omega)}\leq\|(\cdot)^{-\gamma/2}-r\|_{L^{\infty}([\lambda_{min},\lambda_{max}])}\|f\|_{L^{2}(\Omega)}$,
where $\lambda_{min}$, $\lambda_{max}>0$ are the extremal eigenvalues of the
pencil $({\mathbf{A}},{\mathbf{M}})$. Algorithms for determining rational
approximations $r$ proposed in the literature include proposed BURA, AAA (see
[240]) and BRASIL [242]; see also [243]. Instead of trying to find best
rational approximation, one may explicitly construct approximations. For
example, starting from the formula

$x^{-s}=2\frac{\sin(\pi
s)}{\pi}\int_{-\infty}^{\infty}e^{2sy}(1+e^{2y}x)^{-1}\,dy,$ (88)

we may approximate the integral by a truncated trapezoidal rule (see [240] for
details) to get a rational approximation. This method is closely related to
the “Balakrishnan formula” (92) below. Rational approximations that bring in
reduced basis techniques can be found in [244]. An established technique to
compute the matrix $g({\mathbf{K}})$ for a holomorphic $g$ and a matrix
${\mathbf{K}}\in{\mathbb{R}}^{N\times N}$ is the contour integral method,
described, e.g., in [241] and is applicable here with $g(x)=x^{-\gamma/2}$.
Discretizing the contour integral by a quadrature results in having to solve
several linear systems of equations and can, in fact, be understood as a
specific way of constructing a rational approximation of the function $g$.

#### Extension Methods

Surveys for these methods include [238] and the literature reviews in [245,
246]. The above rational approximation methods start from a discretization of
the underlying operator $-\Delta$ and then compute fractional powers on the
discrete level. Alternative approaches start from representations of
$(-\Delta)^{\gamma/2}$ or its inverse that can then be discretized. One way of
achieving this is to use the so-called Caffarelli-Silvestre extension [177],
which for the present bounded domain case is due to [247, 248]. The extension
$U=U(x,y)$ (with $x\in\Omega$, $y\in(0,\infty)$) of the function $u$ solves
the following problem on the semi-infinite cylinder
${\mathcal{C}}:=\Omega\times(0,\infty)$:

$\displaystyle\nabla_{(x,y)}\cdot\left(y^{1-\gamma}\nabla_{(x,y)}U\right)$
$\displaystyle=0\quad\mbox{ in ${\mathcal{C}}$},$ (89) $\displaystyle\quad U$
$\displaystyle=0\quad\mbox{on $\partial\Omega\times(0,\infty)$},$ (90)
$\displaystyle\lim_{y\rightarrow 0+}-y^{1-\gamma}\partial_{y}U(\cdot,y)$
$\displaystyle=2^{\gamma-1}\frac{\Gamma(\gamma/2)}{\Gamma(1-\gamma/2)}f\quad\mbox{
on $y=0$.}$ (91)

The sought solution $u$ of (86) turns out to be $U|_{y=0}$, [177, 247, 248] so
that $U$ is indeed an extension of $u$ to ${\mathcal{C}}$. Hence, we may
approximate $u$ by discretizing the (degenerate) elliptic problem (89). This
can be achieved by established methods, e.g., a FEM, [249, 245, 246]. Since
$U$ decays exponentially as $y\rightarrow 0$, one can replace the infinite
cylinder ${\mathcal{C}}$ by a truncated one. The main difficulty with (89) is
that the coefficient $y^{1-\gamma}$ degenerates as $y\rightarrow 0$, which
requires mesh refinement near $y=0$, see [249, 245, 246] for details. An
analysis of the solution $U$ of (89) also reveals that $U$ typically has a
singularity at $\partial\Omega\times\\{0\\}$ so that also its trace $u$ has a
singularity at $\partial\Omega$, [238]. In the interest of efficiency,
especially for higher order methods, this mandates (anisotropic) mesh
refinement near $\partial\Omega$, see [245, 246] where exponential convergence
of high order methods on suitably refined meshes is proved. Discretizations of
the fractional Laplacian can be combined with time discretizations in time-
dependent problems, see, e.g., [250, 251], where again exponential convergence
is achieved.

#### Balakrishnan Formula

Somewhat surprisingly, using the Dunford-Taylor functional calculus, a formula
for the solution $u$ of (86) can be derived, the so-called Balakrishnan
formula [252]:

$u=\frac{\sin(\pi\gamma/2)}{\pi}\int_{-\infty}^{\infty}e^{-(\gamma/2)y}\left(\operatorname{I}+e^{-y}(-\Delta)\right)^{-1}\,dy.$
(92)

(Note that this can formally be obtained from (88) by setting $x=(-\Delta)$
and changing the integration variable.)

As is worked out in more detail in Section 8.1, this integral can be
discretized with a quadrature rule as proposed in [252, 246], e.g., the
truncated trapezoidal rule leading to a sum of elliptic boundary value
problems. The error incurred by this semidiscretization is root exponential in
the number of quadrature points, [253, Thm. 3.1]. The resulting elliptic
problems can be approximated by established discretization techniques, e.g.,
FEM. Algebraic [252] convergence for low order methods and exponential
convergence of high order methods on suitable meshes [246] is achieved. The
Balakrishnan formula has natural extensions to the time(-fractional) problems,
see, e.g., [254, 251].

#### Discretization based on Parabolic Tools

In addition to representations in terms of the extension or the Balakrishnan
formula, the inverse of the SFL can be expressed in terms of the heat
semigroup. Thus, numerical methods can be based on parabolic solvers, [255].
Further methods based on parabolic solvers include methods due to
Vabishchevich described in [240, Sec. 8].

####  7.2.2 The Integral Fractional Laplacian

The integral fractional Laplacian (IFL) is an operator defined on
${\mathbb{R}}^{d}$. Several equivalent definitions are available, [48]. One
may define it in terms of the Fourier transformation ${\mathcal{F}}$ by
$(-\Delta)^{\gamma/2}u:={\mathcal{F}}^{-1}(|\xi|^{\gamma}{\mathcal{F}}u)$. In
principle, this suggests already a numerical realization via FFT, if the IFL
is considered on a (sufficiently large) periodic box. Another class of methods
suitable for the IFL in full space are specially designed spectral methods,
[256, 257, 258, 259, 260, 261], and, at least in principle, methods based on
approximating by so-called sinc-functions, [262, 263], which have also been
used on bounded domains. We consider the model problem

$(-\Delta)^{\gamma/2}u=f\quad\mbox{ in $\Omega$},\qquad u=0\quad\mbox{ on
${\mathbb{R}}^{d}\setminus\overline{\Omega}$.}$ (93)

A weak formulation of (93), which is derived from the representation (7), is
to find $u\in\widetilde{H}^{\gamma/2}(\Omega)$ such that

$a(u,v):=\frac{c_{d,\gamma}}{2}\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{d+\gamma}}\,dx\,dy=\ell(v):=\int_{\Omega}f(x)v(x)\,dx$
(94)

for all $v\in\widetilde{H}^{\gamma/2}(\Omega)$. (The functions $u$, $v$
defined on $\Omega$ are extended by $0$ outside $\Omega$.) The bilinear form
$a$ is continuous and coercive on $\widetilde{H}^{\gamma/2}(\Omega)$, [249].
Galerkin methods are then obtained by selecting a closed subspace
$V_{N}\subset\widetilde{H}^{\gamma/2}(\Omega)$ and seeking $u_{N}\in V_{N}$
such that $a(u_{N},v)=\ell(v)$ for all $v\in V_{N}$. Céa’s lemma immediately
gives the quasi-optimality result
$\|u-u_{N}\|_{\widetilde{H}^{\gamma/2}(\Omega)}\leq C\inf_{v\in
V_{N}}\|u-v\|_{\widetilde{H}^{\gamma/2}(\Omega)}$. The spaces $V_{N}$ may be
chosen as spaces of piecewise polynomials on a mesh. Algebraic convergence for
low order methods is shown in [249, 245, 264, 265]. As in the case of the SFL,
the solution $u$ of (93) has strong singularities at $\partial\Omega$ [266] so
that (anisotropic) refinement is advantageous; exponential convergence of high
order methods on suitably anisotropically refined meshes is obtained in [267].

The realization of the Galerkin method faces many challenges that are
structurally similar to issues encountered and overcome in the boundary
element method (BEM) [268]: Setting up the system matrix requires the
evaluation of (hyper-)singular integrals for which we refer to [268, Sec. 5]
for the simpler case of the BEM, to [269] and the realizations [270, 271] in
the context of the IFL. Furthermore, the Galerkin matrix is fully populated.
However, the matrix compression techniques that were already described in
Section 7.1.3 such as ${\mathcal{H}}$-matrices [220] can be brought to bear
and lead to data-sparse representations of the matrix with logarithmic-linear
complexity for storage and matrix-vector multiplication; see also [272]. A
rather comprehensive discussion of various numerical aspects of Galerkin
methods for the IFL can be found in [271]. Optimal preconditioning of the
resulting stiffness matrices based on multilevel techniques is addressed in
[273, 274] and in [275] based on operator preconditioning. Adaptivity of FEM
is analyzed in [276].

The IFL admits a Caffarelli-Sivestre extension very similar to (89) where the
semi-infinite cylinder is replaced with a half-space
${\mathbb{R}}^{d}\times{\mathbb{R}}^{+}$ (see also the discussion in Section
6.1.2). Numerical methods based on this extension have recently been proposed
in [277]. The Dunford-Taylor calculus is also applicable and leads to
representations of the IFL similar to the Balakrishnan formula (92). Numerical
methods based on this have been proposed in [278]. We finally point to
numerical methods that exploit the connection of the IFL with $\alpha$-stable
Lévy processes, [279, 280].

##  8 Numerical Continuation: Fractional PDEs

In addition to theoretical investigations, it is also useful to look at the
bifurcation structure of steady states to understand how solutions behave far
from the homogeneous state or critical points when a parameter is varied.
Moreover, by varying further parameters, the bifurcation structure itself also
changes, which could reveal the effect of crucial quantities on the onset of
non-trivial steady states. To compute the bifurcation structure of stationary
steady states, a spatial discretization is usually combined with numerical
continuation techniques [21, 22, 23], i.e., one wants to combine spatial
discretization methods from Section 7 with analytical bifurcation formulas
[15, 16]. For classical reaction–diffusion systems on bounded domains (and
$d=1,2,3$), the global structure can be computed numerically with, for
instance, the advanced continuation/bifurcation software pde2path [23].
Recently, it has also been used beyond its standard setting, namely to treat
cross-diffusion systems [281, 282], optimal control problems [283], network
dynamics [284], and differential geometric PDEs in parametric form [285].

It is also quite natural to extend the class of problems treated by pde2path
to fractional diffusion PDEs to exploit the software continuation capabilities
and routines. In fact, it allows for branch point and Hopf point continuation,
continuation of relative equilibria (e.g., traveling waves and rotating
waves), branch switching from periodic orbits (Hopf pitchfork/transcritical
bifurcation, and period doubling). Therefore, in [148], new capabilities for
non-standard diffusion problems involving the fractional Laplacian in the
spectral definition on 1D domains have been developed and tested for three
important benchmark problems in the non-local setting, aiming at better
understanding the effect of fractional diffusion on the steady state
bifurcation structure of generic fractional reaction–diffusion systems.

In the following, we show how to implement a discretization of the spectral
fractional Laplacian and its embedding within the numerical continuation
package pde2path, and we demonstrate results for the Swift–Hohenberg equation
in the non-local setting (48) obtained thanks to the new continuation
capabilities [148].

###  8.1 Fractional PDEs in pde2path

In this section, the FEM discretization of the fractional Laplacian and its
implementation in pde2path are presented. We do not aim to provide a complete
guide to the software for beginner users here; for the basic setup and for the
notation used in the following we refer to [286, 287, 23, 288].

The continuation software pde2path in its standard setting is based on the FEM
discretization of the stationary elliptic problem exploiting the package OOPDE
[289] for the FEM discretization. In order to adapt it to handle (spectral)
fractional reaction–diffusion equations (42) endowed with homogeneous
Dirichlet boundary conditions (we refer to [148] for the case of homogeneous
Neumann boundary conditions), we need to discretize the spectral fractional
Laplacian (43) to converted the problem (42) into the algebraic system

$G(\texttt{u},p)=K_{tot}\mathtt{u}-F_{tot}(\mathtt{u})=0,$ (95)

where $\texttt{u}\in\mathbb{R}^{n_{p}N}$ contains the nodal values of $u$ with
$n_{p}$ mesh points, the matrix $K_{tot}\in\mathbb{R}^{n_{p}N\times n_{p}N}$
and the vector $F_{tot}\in\mathbb{R}^{n_{p}N}$ correspond to the finite
element discretization of the diffusion and reaction terms (and of the
boundary conditions) from equations (42). To this end, in [148] the
Balakrishnan formula representation [290, 48] in the version appearing in
[291, formula (4), p.260] has been discretized via a sinc quadrature for the
integral as well as a finite element discretization in space; see also Section
7.2 in particular (92) and the discussion following it. This method has been
proposed in [292, 238] for the inverse formula (namely for the inverse
fractional Laplacian representation, $\Delta^{-\gamma/2}u$), while in [148]
this method has been adapted to the direct formula (96), that is, for
$\gamma\in(0,2)$ and $u\in\widetilde{H}^{\gamma/2}(\Omega)$

$\Delta^{\gamma/2}u(x)=\frac{\sin(\gamma\pi/2)}{\pi}\int_{0}^{\infty}\Delta(\nu
I-\Delta)^{-1}u(x)\nu^{\gamma/2-1}\textnormal{d}\nu.$ (96)

Here are the steps leading to the discretization. Thanks to the change of
variable $\nu={\textnormal{e}}^{\eta}$
($\textnormal{d}\nu={\textnormal{e}}^{\eta}\textnormal{d}\eta$) in equation
(96) and then approximating the integral using sinc quadrature [293] one
obtains

$\displaystyle\Delta^{\gamma/2}u(x)$
$\displaystyle=\frac{\sin(\gamma\pi/2)}{\pi}\int_{-\infty}^{\infty}({\textnormal{e}}^{\eta}I-\Delta)^{-1}\Delta
u(x){\textnormal{e}}^{\eta\gamma/2}~{}\textnormal{d}\eta$
$\displaystyle\approx\frac{\sin(\gamma\pi/2)}{\pi}\kappa\sum_{l=-\infty}^{\infty}{\textnormal{e}}^{\kappa
l\gamma/2}({\textnormal{e}}^{\kappa l}I-\Delta)^{-1}\Delta u(x),$

where $\kappa$ is the stepsize parameter coming from the sinc quadrature.
Choosing suitable values of $n_{+}$ and $n_{-}$ for the truncation, the sum is
approximated by

$\Delta^{\gamma/2}u(x)\approx\frac{\sin(\gamma\pi/2)}{\pi}\kappa\sum_{l=-n_{-}}^{n_{+}}{\textnormal{e}}^{\kappa
l\gamma/2}v_{l}(x),$ (97)

where $v_{l}\in\mathbb{U}(\mathcal{\tau})$ is the solution of the Galerkin
variational problem

${\textnormal{e}}^{\kappa l}\int_{\Omega}v_{l}w+\int_{\Omega}\nabla
v_{l}\cdot\nabla w=\int_{\Omega}\nabla u\cdot\nabla w\qquad\forall
w\in\mathbb{U}(\mathcal{\tau})$

and $\mathbb{U}(\mathcal{\tau})$ is the finite element space satisfying
homogeneous Dirichlet boundary conditions. The variational problem is then
equivalent to the matrix formulation

$({\textnormal{e}}^{\kappa l}M+K)\mathtt{v_{l}}=K\mathtt{u},$ (98)

where $M$ and $K$ are the classical FEM mass and stiffness matrices
respectively, and $\mathtt{v_{l}}$ denotes the node vector for $v_{l}$ and can
be obtained as the solution of the linear system (98). Therefore, the
discretization requires solving $n_{-}+n_{+}+1$ independent linear systems to
obtain an approximation of $\Delta^{\gamma/2}u$.

However, to interface the fractional Laplacian via a discretization of the
Balakrishnan formula with the pde2path setup, a matrix approximation of the
operator $\Delta^{\gamma/2}$ itself is needed. This matrix, denoted by
$K_{\gamma}$ to emphasize its dependence on the fractional order $\gamma$, can
be built column by column by _testing_ the proposed approximation method on
the standard basis of Lagrangian $\mathbb{P}_{1}$ elements. Finally, once the
matrix $K_{\gamma}$ has been built, the standard stiffness matrix $K$ has to
be replaced by $-MK_{\gamma}$ in the expression of $K_{tot}$ in the Matlab
routines relevant to the algebraic system of the form (95). We refer to [148]
for the detailed description of the implementation in pde2path, the Matlab
code and the choice of the quantities $\kappa,\,n_{+},\,n_{-}$. In addition,
the Matlab code is also freely available in the GitHub folder [294].

###  8.2 The Fractional Swift–Hohenberg Equation

Thanks to the new capabilities of the continuation software, it is possible to
investigate the effect of the fractional order $\gamma$ on the steady state
bifurcation structure of benchmark problems. As in Section 5, we consider as a
case study the fractional Swift–Hohenberg equation with competing
cubic–quintic nonlinearities (48) and homogeneous Dirichlet boundary
conditions. For the classical Swift–Hohenberg equation (obtained with
$\gamma=2$), it is known that bifurcation diagrams exhibit the so-called
homoclinic snaking [295, 296, 297]. A non-trivial branch of stationary
solutions bifurcates from a branch point and, for a suitable range of the main
bifurcation parameter, undergoes several fold bifurcations leading to multi-
stability of the solutions along the branch that are often important localized
patterns. Looking at the bifurcation diagram, this branch presents the
characteristic snaking-type structure. Recently, in [148], also the
bifurcation structure of the Swift–Hohenberg equation with nonlocal reaction
terms (but also other well-known PDEs) has been investigated with a focus on
understanding how the snaking scenario changes under the nonlocal influence
(namely decreasing the fractional power $\gamma$) as already carried out for
nonlocal reaction terms in [155]. As a first step, the Swift–Hohenberg
equation can be converted into a system of second-order PDEs by substituting
$(u_{1},u_{2})=(u,\Delta^{\gamma/2}u)$ in equation (48). The resulting system
is

$\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}\partial_{t}\begin{pmatrix}u_{1}\\\
u_{2}\end{pmatrix}=\begin{pmatrix}-\Delta^{\gamma/2}u_{2}-2u_{2}-(1-p)u_{1}+qu_{1}^{3}-u_{1}^{5}\\\
\Delta^{\gamma/2}u_{1}-u_{2}\end{pmatrix},$

which fits the pde2path extended framework. As in [148], we consider equation
(48) defined on a bounded interval $\Omega=(-5\pi,5\pi)\subset\mathbb{R}$ of
length $L=10\pi$ together with homogeneous Dirichlet boundary conditions
$u(-5\pi)=u(5\pi)=0$ and
$\Delta^{\gamma/2}u(-5\pi)=\Delta^{\gamma/2}u(5\pi)=0$. In the system two
parameters are involved: we fix the parameter $q=2$, while parameter $p$ is
taken as the bifurcation parameter. For the space discretization, we choose
the mesh size $h=0.04$, corresponding to $n_{p}=786$ mesh points. The
quantities $\kappa$, $n_{+}$ and $n_{-}$ in the quadrature approximation (97)
of the fractional Laplacian have been chosen in [148] following [292,
Corollary 2] where a similar discretization method was used. In detail, the
value of $\kappa$ is related to the mesh size $h$ and the values $n_{+}$ and
$n_{-}$ are chosen proportional to $1/\kappa^{2}$ as follows

$\kappa=\frac{1}{|\log(h)|},\qquad
n_{+}\coloneqq\left\lceil\frac{\pi^{2}}{4(1-\gamma/2)\kappa^{2}}\right\rceil,\qquad
n_{-}\coloneqq\left\lceil\frac{\pi^{2}}{2\gamma\kappa^{2}}\right\rceil.$

It is then possible to investigate some of the effects of super-diffusion on
the steady state bifurcation structure and solution profiles. We highlight
here the main effects that were observed for the bifurcation points on the
homogeneous branch, the snaking branch in the bifurcation structure, and the
solution profile.

Bifurcation points on the homogeneous branch. As already mentioned in Section
5 from equation (49), the bifurcation points on the homogeneous branch
accumulate at the value of the bifurcation parameter at which a change of
stability of the homogeneous solution occurs, namely at $p_{c}=0$, as the
fractional order $\gamma$ decreases.

\begin{overpic}[width=433.62pt]{SH-s09-bifdiag}
\put(-8.0,45.0){\rotatebox{90.0}{$\|u\|_{L^{2}}$}} \put(99.0,5.0){$p$}
\end{overpic} (a) $\gamma=1.8$

\begin{overpic}[width=433.62pt]{SH-s07-bifdiag}
\put(-8.0,45.0){\rotatebox{90.0}{$\|u\|_{L^{2}}$}} \put(99.0,5.0){$p$}
\end{overpic} (b) $\gamma=1.4$

\begin{overpic}[width=433.62pt]{SH-s05-bifdiag}
\put(-8.0,45.0){\rotatebox{90.0}{$\|u\|_{L^{2}}$}} \put(99.0,5.0){$p$}
\end{overpic} (c) $\gamma=1$

\begin{overpic}[width=433.62pt]{SH-s03-bifdiag}
\put(-8.0,45.0){\rotatebox{90.0}{$\|u\|_{L^{2}}$}} \put(99.0,5.0){$p$}
\end{overpic} (d) $\gamma=0.6$

Figure 3: The snaking branch of the fractional Swift–Hohenberg equation (48)
gets stretched (“bloated”) as the fractional order $\gamma$ decreases. The
bifurcation diagrams with respect to the bifurcation parameter $p$ have been
computed thanks to the new capabilities of the continuation software pde2path.
The equation is considered on a bounded domain $\Omega=(-5\pi,5\pi)$ with
homogeneous Dirichlet boundary conditions, and parameter $q=2$. Thick and thin
lines denote stable and unstable solutions, while circles and crosses indicate
branch and fold points respectively. This figure was published in [148],
Copyright Elsevier (2021). \begin{overpic}[width=173.44534pt]{SH-solutions-s}
\put(-5.0,35.0){\rotatebox{90.0}{$u(x)$}} \put(100.0,5.0){$x$} \end{overpic}
Figure 4: Change in the solution profile to the fractional Swift–Hohenberg
equation (48) as the fractional order $\gamma$ decreases. The equation is
considered on a bounded domain $\Omega=(-5\pi,5\pi)$ with homogeneous
Dirichlet boundary conditions, and parameter $q=2$. In the bifurcation
diagrams, the solutions correspond to $p=-0.4$ and are located on the first
(blue) branch bifurcating from the homogenous branch. Different colors denote
different fractional orders: blue for $\gamma=1.8$, red for $\gamma=1$ and
black for $\gamma=0.6$. This figure was published in [148], Copyright Elsevier
(2021).

Bifurcation structure and snaking branch. The fractional order also affects
the bifurcation structure of problem (48). In order to get a general
impression of the effect, Figure 3 compares the bifurcation diagram for
decreasing fractional orders (namely $\gamma=1.8$, $\gamma=1.4$, $\gamma=1$
and $\gamma=0.6$). The homogeneous branch is shown in black, the first branch
bifurcating at $p=0$ from the homogeneous branch in blue and the first
(homoclinic) snaking branch in red. For $\gamma=1.8$ the bifurcation structure
is very similar to the one of the standard Swift–Hohenberg equation but it
deforms as $\gamma$ varies. The most evident changes regard the snaking
branch. In fact, while the width of the snaking branch significantly increases
(it gets stretched or “bloated”) as the fractional order decreases, the back-
and-forth oscillations forming the snaking tend to tilt on the left for
smaller $\gamma$. Further, also the number of turns on the snaking branches
varies: we pass from $8$ pair of turns for $\gamma=1.8$ to $9$ pairs for
$\gamma=1.4$ and $\gamma=1$, decreasing further to $6$ pairs for $\gamma=0.6$.
Since the first snaking branch corresponds to front solutions as shown in
Figure 5, at each snaking turn one oscillation is added to the solution
profile. This means that the number of turns determines the final number of
oscillations in the solution and consequently that the snaking branch has to
reconnect to other periodic branches originating from subsequent bifurcations
on the homogeneous instead of the first periodic branch. In particular, for
$\gamma=1$ and $\gamma=0.6$ it reconnects to the branches originating from the
third and ninth bifurcation points, respectively, on the homogeneous black
branch (also with different wave numbers). Note that the deformation of the
snaking branch is known to occur for Swift–Hohenberg equations with non-local
reaction [155] and that the re-connection of the snaking branch not to the
first branch also occurs for standard (Laplacian) Swift–Hohenberg equations
varying the domain length [298]. A less evident deformation is that the non-
homogeneous steady states have slightly larger $L^{2}$-norm values as the
fractional order $\gamma$ decreases and has an effect on the solution
profiles.

\begin{overpic}[width=390.25534pt]{SH-s03-sn-labeled}
\put(-10.0,45.0){\rotatebox{90.0}{$\|u\|_{L^{2}}$}} \put(100.0,5.0){$p$}
\put(10.0,58.0){(a)} \put(40.0,67.0){(b)} \put(65.0,75.0){(c)} \end{overpic}

\begin{overpic}[width=433.62pt]{SH-s03-sn-sol-pt5300}
\put(-7.0,45.0){\rotatebox{90.0}{$u(x)$}} \put(95.0,5.0){$x$} \end{overpic}
(a)

\begin{overpic}[width=433.62pt]{SH-s03-sn-sol-pt5600} \put(95.0,5.0){$x$}
\end{overpic} (b)

\begin{overpic}[width=433.62pt]{SH-s03-sn-sol-pt5900} \put(95.0,5.0){$x$}
\end{overpic} (c)

Figure 5: Change in the solution profile to the fractional Swift–Hohenberg
equation (48) along the upper part of the snaking branch. The equation is
considered on a bounded domain $\Omega=(-5\pi,5\pi)$ with homogeneous
Dirichlet boundary conditions, and parameter $q=2$, while the fractional order
is $\gamma=0.6$. This figure was published in [148], Copyright Elsevier
(2021).

Solution profile. Finally, it has been observed that, for a fixed value of the
bifurcation parameter, the non-homogeneous stationary solutions tend to have
flatter peaks or valleys as the fractional order of the Laplacian decreases.
This effect has been named as “sharpening the teeth” of the spatial solution
profiles. In Figure 4, solution profiles located on the first (blue) branch
bifurcating from the homogenous branch at $p=4$ for different values of the
fractional order $\gamma$ (see Figure 3) are shown. Whereas the solution
profiles for $\gamma=1.8$ and $\gamma=1.4$ resemble the basic periodic pattern
specific to the branch (i.e., to the eigenfunction associated to the
corresonding mode), we clearly see that at $\gamma=1$ the profile
significantly “flattens” towards the maximum of oscillation. The same happens
to the solution profile along the upper part of the snaking branch for
$\gamma=0.6$, as illustrated in Figure 5. Moving up the snaking branch, namely
increasing the parameter $p$, the oscillations in the solution profile
significantly enlarge while the peaks and valleys are squared off.

Moreover, in [148] it has been shown that all these effects are also common to
the fractional Allen–Cahn equation and the fractional Schnakenberg system,
contributing to a better understanding of the effects of the fractional
Laplacian on the steady solutions of generic reaction–diffusion systems on
bounded domains. Overall, there are currently still no additional detailed
numerical continuation case studies available for various fractional
dissipative PDEs but we anticipate this research direction to naturally grow
in the coming years.

##  9 Conclusion & Outlook

In this chapter, we have summarized several important research directions for
fractional dissipative PDEs. As mentioned in the introduction, it is
impossible at this point to provide a completely comprehensive overview of the
area as it has grown so rapidly in recent years. Yet, even in a relatively
condensed format, several conclusions for fractional dissipative PDEs have
become apparent. First, the general dynamics approach known for classical
dissipative PDEs can be transferred in many cases. One starts to analyze the
invariant sets, studies their local stability via spectral theory, gets a more
global analytical view via gradient/Lyapunov-function structures, repeats the
strategy for moving patterns, and finally transitions to numerical
discretization and numerical continuation to fill out the dynamical picture.
Second, the technical approach often differs significantly. Many methods from
classical derivative calculus are not available for fractional derivatives.
This often necessitates very significant technical extensions, e.g., for
transform methods, spectral theory, energy estimates, bifurcation theory,
sub/super-solutions, new integral discretizations, and updated numerical
continuation packages. For some problems, entirely new methods have been
employed such as converting by extension back to local derivatives. Third,
once methods are available, one notices often significant quantitative changes
of the dynamics. This includes, e.g., changes for the spectrum, space-time
asymptotics, bifurcation points, numerical cost/accuracy or spatial structure
of patterns. Yet, there are also numerous phenomena from classical dissipative
PDEs that are qualitatively robust. This is re-assuring as one wants physical
dissipative systems to behave robustly under small structural perturbations of
the model.

Currently, the field of fractional dissipative PDEs still poses immense
challenges as only the dynamics of some basic phenomena have been understood,
so we can only outline a few possible future directions here: (I) We have
mostly focused on purely time-fractional or purely space-fractional PDEs with
Riemann-Liouville, Caputo and Riesz–Feller derivatives. Although there is a
good motivation for this viewpoint regarding diffusive processes, it is
certainly plausible for applications to directly think in terms of space-time
modelling leading to nonlocal operators in space and time. For example, for
subdiffusive processes on complex networks, one expects the mean-field limit
PDEs to have nonlocal operators in space and time, potentially even new
coupled operators beyond current standard examples. (II) Since we have mostly
focused on low-dimensional, or even scalar, problems, the analysis of the
existence and stability theory for steady states was still somewhat tractable.
Once spatial multi-component systems are considered, spectral problems are
likely going to become much more complicated, which is an effect already well-
known for classical dissipative PDEs. (III) The last issue also connects to a
topic we have not covered here concerning invariant/inertial manifolds for
fractional dissipative PDEs. We conjecture that existing techniques will
transfer once suitable spectral gap conditions are satisfied but verifying
these conditions could be much more challenging as already indicated in our
discussion of spectral theory for time-fractional PDEs. (IV) On a global
level, the theory of nonlocal gradient flows is extremely active currently in
multiple directions, so we anticipate that fractional PDEs will be embedded
within these efforts as key benchmark examples. (V) One the level of pattern
formation, we only touched upon traveling waves, which have a relatively
simple (translation) symmetry group and upon Turing patterns generated at
local bifurcation points. Yet, for classical reaction-diffusion PDEs, there
are many other phenomena of key interest such as chaos, chimeras, modulated
patterns, and spirals, just to name a few. It is likely that fractional
derivatives can fundamentally alter the mechanisms for these in analogy to
pattern formation results observed when one transitions from lattice or all-
to-all network models to intermediate-range nonlocal coupling. (VI) From a
numerical perspective, some excellent discretization schemes do exist but we
are still very far from having practical software packages available that
would allow for a seamless use of these schemes to study fractional nonlinear
dynamics. Hence, a major software development effort in this direction would
be highly welcome. (VII) Lastly, the PDEs we focused on had relatively simple
polynomial nonlinearities and mostly semilinear structures providing a
connection to several classical parabolic PDEs. Yet, many complex systems
applications are bound to lead us far beyond these mathematical base cases.

Acknowledgments: FA was supported by the Austrian Science Fund (FWF) via the
FWF-funded SFB # F65 and FWF # P33151. GA was supported by JSPS KAKENHI Grant
Numbers JP21KK0044, JP21K18581, JP20H01812 and JP20H00117 as well as by the
Research Institute for Mathematical Sciences, an International Joint
Usage/Research Center located in Kyoto University. CK would like to thank the
VolkswagenStiftung for support via a Lichtenberg Professorship. JMM was
supported by Austrian Science Fund (FWF) via the SFB “Taming complexity in
partial differential systems” (grant SFB F65). CS is a member of the INdAM-
GNFM Group and was partially supported by the OeAD project FR 03/2022.

## References

  * [1] N.F. Britton.  Reaction-Diffusion Equations and Their Applications to Biology.  Academic Press, 1986.
  * [2] P. Grindrod.  Patterns and Waves: The Theory and Applications of Reaction-Diffusion Equations.  Clarendon Press, 1991.
  * [3] J.D. Murray.  Mathematical Biology II: Spatial Models and Biomedical Applications.  Springer, 3rd edition, 2003.
  * [4] H. Amann.  Linear and Quasilinear Parabolic Problems: Volume I: Abstract Linear Theory.  Springer, 1995.
  * [5] L.C. Evans.  Partial Differential Equations.  AMS, 2002.
  * [6] A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems.  Springer, 1995.
  * [7] F. Rothe.  Global Solutions of Reaction-Diffusion Systems.  Springer, 1984.
  * [8] J. Smoller.  Shock Waves and Reaction-Diffusion Equations.  Springer, 1994.
  * [9] J.C. Robinson.  Infinite-Dimensional Dynamical Systems.  CUP, 2001.
  * [10] R. Temam.  Infinite-Dimensional Dynamical Systems in Mechanics and Physics.  Springer, 1997.
  * [11] D. Henry.  Geometric Theory of Semilinear Parabolic Equations.  Springer, Berlin Heidelberg, Germany, 1981.
  * [12] C. Kuehn.  PDE Dynamics: An Introduction.  SIAM, 2019.
  * [13] G. Schneider and H. Uecker.  Nonlinear PDEs: A Dynamical Systems Approach.  AMS, 2017.
  * [14] S.-N. Chow and J.K. Hale.  Methods of Bifurcation Theory.  Springer, 1982.
  * [15] H. Kielhoefer.  Bifurcation Theory: An Introduction with Applications to PDEs.  Springer, 2004.
  * [16] Yu.A. Kuznetsov.  Elements of Applied Bifurcation Theory.  Springer, New York, NY, 3rd edition, 2004.
  * [17] E. Zeidler.  Nonlinear Functional Analysis and its Applications III: Variational Methods and Optimization.  Springer, 2013.
  * [18] T. Kato.  Perturbation Theory for Linear Operators.  Springer, 1980.
  * [19] T. Kapitula and K. Promislow.  Spectral and Dynamical Stability of Nonlinear Waves.  Springer, 2013.
  * [20] B. Sandstede.  Stability of travelling waves.  In B. Fiedler, editor, Handbook of Dynamical Systems, volume 2, pages 983–1055. Elsevier, 2001.
  * [21] H. Dankowicz and F. Schilder.  Recipes for Continuation.  SIAM, 2013.
  * [22] E.J. Doedel, A. Champneys, F. Dercole, T. Fairgrieve, Y. Kuznetsov, B. Oldeman, R. Paffenroth, B. Sandstede, X. Wang, and C. Zhang.  Auto 2007p: Continuation and bifurcation software for ordinary differential equations (with homcont).  http://cmvl.cs.concordia.ca/auto, 2007.
  * [23] H. Uecker, D. Wetzel, and J.D.M. Rademacher.  pde2path \- a Matlab package for continuation and bifurcation in 2D elliptic systems.  Numerical Mathematics: Theory, Methods and Applications, 7(1):58–106, 2014.
  * [24] S.M. Allen and J.W. Cahn.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening.  Acta Metallurgica, 27(6):1085–1905, 1979.
  * [25] N. Chafee and E.F. Infante.  A bifurcation problem for a nonlinear partial differential equation of parabolic type.  Applic. Anal., 4(1):17–37, 1974.
  * [26] J. Nagumo, S. Arimoto, and S. Yoshizawa.  An active pulse transmission line simulating nerve axon.  Proc. IRE, 50:2061–2070, 1962.
  * [27] M.C. Cross and P.C. Hohenberg.  Pattern formation outside of equilibrium.  Rev. Mod. Phys., 65(3):851–1112, 1993.
  * [28] F. Schlögl.  Chemical reaction models for non-equilibrium phase transitions.  Zeitschrift für Physik, 253(2):147–161, 1972.
  * [29] M. Hairer.  Regularity structures and the dynamical $\Phi^{4}_{3}$ model.  arXiv:1508.05261, pages 1–46, 2015.
  * [30] P. Kirrmann, G. Schneider, and A. Mielke.  The validity of modulation equations for extended systems with cubic nonlinearities.  Proc. R. Soc. Edinburgh A, 122(1):85–91, 1992.
  * [31] B. Fiedler and C. Rocha.  Heteroclinic orbits of semilinear parabolic equations.  J. Differen. Equat., 125(1):239–281, 1996.
  * [32] Bernold Fiedler and Arnd Scheel.  Spatio-temporal dynamics of reaction-diffusion patterns.  In Trends in nonlinear analysis, pages 23–152. Springer, Berlin, 2003.
  * [33] B. Sandstede.  Stability of $N$-fronts bifurcating from a twisted heteroclinic loop and an application to the FitzHugh-Nagumo equation.  SIAM J. Math. Anal., 29(1):183–207, 1998.
  * [34] W.J. Beyn and V. Thümmler.  Freezing solutions of equivariant evolution equations.  SIAM J. Appl. Dyn. Syst., 3(2):85–116, 2004.
  * [35] R. Metzler and J. Klafter.  The random walk’s guide to anomalous diffusion: a fractional dynamics approach.  Phys. Rep., 339(1):1–77, 2000.
  * [36] Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev.  Fractional integrals and derivatives.  Gordon and Breach Science Publishers, Yverdon, 1993.  Theory and applications, Edited and with a foreword by S. M. Nikol’skiUı, Translated from the 1987 Russian original, Revised by the authors.
  * [37] David A. Benson, Rina Schumer, Mark M. Meerschaert, and Stephen W. Wheatcraft.  Fractional dispersion, Lévy motion, and the MADE tracer tests.  Transp. Porous Media, 42(1-2):211–240, 2001.
  * [38] D. Del Castillo-Negrete.  Asymmetric transport and non-Gaussian statistics of passive scalars in vortices in shear.  Physics of Fluids, 10(3):576, 1998.
  * [39] D. del Castillo-Negrete, B. A. Carreras, and V. E. Lynch.  Fractional diffusion in plasma turbulence.  Physics of Plasmas, 11(8):3854–3864, 07 2004.
  * [40] E. Hanert, E. Schumacher, and E. Deleersnijder.  Front dynamics in fractional-order epidemic models.  Journal of theoretical biology, 279(1):9–16, June 2011.
  * [41] Ralf Metzler, Jae-Hyung Jeon, Andrey G. Cherstvy, and Eli Barkai.  Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.  Phys. Chem. Chem. Phys., 16:24128–24164, 2014.
  * [42] Rainer Klages, G. (Günter) Radons, and Igor M. Sokolov.  Anomalous transport : foundations and applications.  WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2008.
  * [43] Rina Schumer, Mark M. Meerschaert, and Boris Baeumer.  Fractional advection-dispersion equations for modeling transport at the earth surface.  Journal of Geophysical Research: Earth Surface, 114(F4), 2009.
  * [44] Marta D’Elia, Qiang Du, Christian Glusa, Max Gunzburger, Xiaochuan Tian, and Zhi Zhou.  Numerical methods for nonlocal and fractional models.  Acta Numer., 29:1–124, 2020.
  * [45] V. Méndez, S. Fedotov, and W. Horsthemke.  Reaction-transport systems.  Springer Series in Synergetics. Springer, Heidelberg, 2010.  Mesoscopic foundations, fronts, and spatial instabilities.
  * [46] K. Sato.  Lévy processes and infinitely divisible distributions, volume 68 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1999.  Translated from the 1990 Japanese original, Revised by the author.
  * [47] Serena Dipierro, Xavier Ros-Oton, Joaquim Serra, and Enrico Valdinoci.  Non-symmetric stable operators: regularity theory and integration by parts.  Adv. Math., 401:Paper No. 108321, 100, 2022.
  * [48] Mateusz Kwaśnicki.  Ten equivalent definitions of the fractional Laplace operator.  Fract. Calc. Appl. Anal., 20(1):7–51, 2017.
  * [49] William McLean.  Strongly elliptic systems and boundary integral equations.  Cambridge University Press, Cambridge, 2000.
  * [50] Kai Diethelm.  The analysis of fractional differential equations, volume 2004 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2010.  An application-oriented exposition using differential operators of Caputo type.
  * [51] Vicenç Méndez, Daniel Campos, and Frederic Bartumeus.  Stochastic Foundations in Movement Ecology: Anomalous Diffusion, Front Propagation and Random Searches.  Springer Series in Synergetics. Springer-Verlag Berlin Heidelberg, 1st edition, 2014.
  * [52] Bangti Jin.  Fractional differential equations—an approach via fractional derivatives, volume 206 of Applied Mathematical Sciences.  Springer, Cham, 2021.
  * [53] I. M. Sokolov and J. Klafter.  From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 15(2):026103, 06 2005.
  * [54] Rico Zacher.  Time fractional diffusion equations: solution concepts, regularity, and long-time behavior.  In Anatoly Kochubei and Yuri Luchko, editors, Volume 2 Fractional Differential Equations, pages 159–180. De Gruyter, Berlin, Boston, 2019.
  * [55] W. R. Schneider and W. Wyss.  Fractional diffusion and wave equations.  Journal of Mathematical Physics, 30(1):134–144, 01 1989.
  * [56] Francesco Mainardi, Antonio Mura, Gianni Pagnini, and Rudolf Gorenflo.  Sub-diffusion equations of fractional order and their fundamental solutions.  In K. Taş, J. A. Tenreiro Machado, and D. Baleanu, editors, Mathematical Methods in Engineering, pages 23–55. Springer Netherlands, Dordrecht, 2007.
  * [57] Ralf Metzler and Joseph Klafter.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics.  Journal of Physics A: Mathematical and General, 37(31):R161–R208, 2004.
  * [58] Denis Matignon.  Stability results for fractional differential equations with applications to control processing.  In Computational engineering in systems applications, volume 2, pages 963–968. 1996.
  * [59] Oana Brandibur, Roberto Garrappa, and Eva Kaslik.  Stability of systems of fractional-order differential equations with Caputo derivatives.  Mathematics, 9(8), 2021.
  * [60] Francesco Mainardi, Yuri Luchko, and Gianni Pagnini.  The fundamental solution of the space-time fractional diffusion equation.  Fractional Calculus and Applied Analysis, 4(2):153–192, 2001.
  * [61] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo.  Theory and Applications of Fractional Differential Equations, volume 204 of North-Holland Mathematics Studies.  Elsevier Science, Amsterdam, 1st edition, 2006.
  * [62] A. N. Kochubei.  Diffusion of fractional order.  Differential Equations, 26(4):485–492, 1990.
  * [63] Samuil D. Eidelman and Anatoly N. Kochubei.  Cauchy problem for fractional diffusion equations.  Journal of Differential Equations, 199(2):211–255, 2004.
  * [64] Jichen Yang and Jens D. M. Rademacher.  Reaction-subdiffusion systems and memory: spectra, Turing instability and decay estimates.  IMA J. Appl. Math., 86(2):247–293, 2021.
  * [65] William McLean, Kassem Mustapha, Raed Ali, and Omar Knio.  Well-posedness of time-fractional advection-diffusion-reaction equations.  Fract. Calc. Appl. Anal., 22(4):918–944, 2019.
  * [66] Rudolf Gorenflo, Yuri Luchko, and Francesco Mainardi.  Wright functions as scale-invariant solutions of the diffusion-wave equation.  Journal of Computational and Applied Mathematics, 118(1):175–191, 2000.
  * [67] Francesco Mainardi, Antonio Mura, and Gianni Pagnini.  The M-Wright function in time-fractional diffusion processes: a tutorial survey.  International Journal of Differential Equations, 2010:1–29, 2010\.
  * [68] Vicente Vergara and Rico Zacher.  Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods.  SIAM Journal on Mathematical Analysis, 47(1):210–239, 2015.
  * [69] Jukka Kemppainen, Juhana Siljander, Vicente Vergara, and Rico Zacher.  Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\mathbb{R}^{d}$.  Math. Ann., 366(3-4):941–979, 2016.
  * [70] B. I. Henry, T. A. M. Langlands, and S. L. Wearne.  Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations.  Physical Review E, 74(3):031116, 2006.
  * [71] Kazuhiko Seki, Mariusz Wojcik, and M Tachiya.  Fractional reaction-diffusion equation.  The Journal of chemical physics, 119(4):2165–2170, 2003.
  * [72] Kazuhiko Seki, Mariusz Wojcik, and M Tachiya.  Recombination kinetics in subdiffusive media.  The Journal of chemical physics, 119(14):7525–7533, 2003.
  * [73] S. B. Yuste, L. Acedo, and Katja Lindenberg.  Reaction front in an A + B $\to$ C reaction-subdiffusion process.  Physical Review E, 69(3):036126, 2004.
  * [74] T. A. M. Langlands, B. I. Henry, and S. L. Wearne.  Turing pattern formation with fractional diffusion and fractional reactions.  Journal of Physics: Condensed Matter, 19(6):065115, 2007.
  * [75] Y. Nec and A. A. Nepomnyashchy.  Turing instability of anomalous reaction–anomalous diffusion systems.  European Journal of Applied Mathematics, 19(3):329–349, 2008.
  * [76] V. Vergara and R. Zacher.  Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations.  J. Evol. Equ., 17:599–626, 2017.
  * [77] Tran Dinh Ke, Nguyen Nhu Thang, and Lam Tran Phuong Thuy.  Regularity and stability analysis for a class of semilinear nonlocal differential equations in Hilbert spaces.  Journal of Mathematical Analysis and Applications, 483(2):123655, 2020.
  * [78] Nguyen Van Dac, Tran Dinh Ke, and Lam Tran Phuong Thuy.  On stability and regularity for semilinear anomalous diffusion equations perturbed by weak-valued nonlinearities.  Discrete and Continuous Dynamical Systems - S, pages 1–19, 2023\.
  * [79] Nhu Thang Nguyen, Dinh Ke Tran, and Van Dac Nguyen.  Stability analysis for nonlocal evolution equations involving infinite delays.  J. Fixed Point Theory Appl., 25(1):Paper No. 22, 33, 2023.
  * [80] Uttam Kumar and Subramaniam Pushpavanam.  The effect of subdiffusion on the stability of autocatalytic systems.  Chemical Engineering Science, 265:118230, 2023.
  * [81] Bruce I. Henry and Susan L. Wearne.  Fractional reaction–diffusion.  Physica A: Statistical Mechanics and its Applications, 276(3):448–455, 2000.
  * [82] Sergei Fedotov.  Non-markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts.  Physical Review E, 81(1):011117, 2010.
  * [83] Y. Nec and A. A. Nepomnyashchy.  Linear stability of fractional reaction-diffusion systems.  Mathematical Modelling of Natural Phenomena, 2(2):77–105, 2007\.
  * [84] A. A. Nepomnyashchy.  Mathematical modelling of subdiffusion-reaction systems.  Mathematical Modelling of Natural Phenomena, 11(1):26–36, 2016\.
  * [85] B. I. Henry and S. L. Wearne.  Existence of Turing instabilities in a two-species fractional reaction-diffusion system.  SIAM Journal on Applied Mathematics, 62(3):870–887, 2002.
  * [86] T. A. M. Langlands, B. I. Henry, and S. L. Wearne.  Anomalous subdiffusion with multispecies linear reaction dynamics.  Physical Review E, 77(2):021111, 2008.
  * [87] Amanda M. Alexander and Sean D. Lawley.  Reaction-subdiffusion equations with species-dependent movement.  SIAM J. Appl. Math., 81(6):2457–2479, 2021.
  * [88] Björn Sandstede.  Chapter 18 - stability of travelling waves.  In Bernold Fiedler, editor, Handbook of Dynamical Systems, volume 2 of Handbook of Dynamical Systems, pages 983 – 1055. Elsevier Science, 2002.
  * [89] B. I. Henry, T. A. M. Langlands, and S. L. Wearne.  Turing pattern formation in fractional activator-inhibitor systems.  Physical Review E, 72(2):026101, 2005.
  * [90] Y. Kōmura.  Nonlinear semi-groups in Hilbert space.  J. Math. Soc. Japan, 19:493–507, 1967.
  * [91] M.G. Crandall and T.M. Liggett.  Generation of semi-groups of nonlinear transformations on general Banach spaces.  Amer. J. Math., 93:265–298, 1971.
  * [92] H. Brézis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, volume No. 5 of North-Holland Mathematics Studies.  North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Notas de Matemática, No. 50. [Mathematical Notes].
  * [93] V. Barbu.  Nonlinear Volterra equations in a Hilbert space.  SIAM J. Math. Anal., 6:728–741, 1975.
  * [94] V. Barbu.  On a nonlinear Volterra integral equation on a Hilbert space.  SIAM J. Math. Anal., 8:346–355, 1977.
  * [95] V. Barbu.  Existence for nonlinear Volterra equations in Hilbert spaces.  SIAM J. Math. Anal., pages 552–569, 1979.
  * [96] M.G. Crandall and J.A. Nohel.  An abstract functional differential equation and a related nonlinear Volterra equation.  Israel J. Math., 29:313–328, 1978.
  * [97] G. Gripenberg.  An existence result for a nonlinear Volterra integral equation in a Hilbert space.  SIAM J. Math. Anal., 9:793–805, 1978.
  * [98] S.-O. Londen and O.J. Staffans.  A note on Volterra equations in a Hilbert space.  Proc. Amer. Math. Soc., 70:57–62, 1978.
  * [99] T. Kiffe and M. Stecher.  An abstract Volterra integral equation in a reflexive Banach space.  J. Differential Equations, 34:303–325, 1979.
  * [100] Ph. Clément.  On abstract Volterra equations with kernels having a positive resolvent.  Israel J. Math., 36:193–200, 1980.
  * [101] S. Aizicovici.  An abstract doubly nonlinear Volterra integral equation.  Funkcialaj Ekvacioj, 36:479–497, 1993.
  * [102] Ph. Clément and J.A. Nohel.  Asymptotic behavior of solutions of nonlinear volterra equations with completely positive kernels.  SIAM J. Math. Anal., 12:514–535, 1981.
  * [103] Ph. Clément and J.A. Nohel.  Abstract linear and nonlinear Volterra equations preserving positivity.  SIAM J. Math. Anal., 10:365–388, 1979.
  * [104] G. Gripenberg.  An abstract nonlinear volterra equation.  Israel J. Math., 34:198–212, 1979.
  * [105] T. Kiffe.  A perturbation of an abstract Volterra equation.  SIAM J. Math. Anal., 11:1036–1046, 1980.
  * [106] Ph. Clément.  On abstract Volterra equations in Banach spaces with completely positive kernels, volume 1076 of Lecture Notes in Math., pages 32–40.  Springer, Berlin, 1984.
  * [107] G. Gripenberg.  Volterra integro-differential equations with accretive nonlinearity.  J. Differential Equations, 60:57–79, 1985.
  * [108] Ph. Clément and J. Prüss.  Completely positive measures and Feller semigroups.  Math. Ann., 287:73–105, 1990.
  * [109] R. Zacher.  Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces.  Funkcial. Ekvac., 52:1–18, 2009.
  * [110] V. Vergara and R. Zacher.  Lyapunov functions and convergence to steady state for differential equations of fractional order.  Math. Z., 259:287–309, 2008.
  * [111] R. Zacher.  Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients.  J. Math. Anal. Appl., 348:137–149, 2008.
  * [112] R. Zacher.  A De Giorgi–Nash type theorem for time fractional diffusion equations.  Math. Ann., 356:99–146, 2013.
  * [113] V. Vergara and R. Zacher.  A priori bounds for degenerate and singular evolutionary partial integro-differential equations.  Nonlinear Anal., 73:3572–3585, 2010.
  * [114] R. Zacher.  A weak Harnack inequality for fractional evolution equations with discontinuous coefficients.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12:903–940, 2013.
  * [115] E. Affili and E. Valdinoci.  Decay estimates for evolution equations with classical and fractional time-derivatives.  J. Differential Equations, 266:4027–4060, 2019.
  * [116] R. Zacher.  Time fractional diffusion equations: solution concepts, regularity, and long-time behavior, volume 2, pages 159–179.  De Gruyter, Berlin, 2019.
  * [117] R. Zacher.  Maximal regularity of type $l_{p}$ for abstract parabolic Volterra equations.  J. Evol. Equ., 5:79–103, 2005.
  * [118] R. Zacher.  Quasilinear parabolic integro-differential equations with nonlinear boundary conditions.  Differential Integral Equations, 19:1129–1156, 2006.
  * [119] C.G. Gal and M. Warma.  Fractional-in-time semilinear parabolic equations and applications, volume 84 of Mathematics & Applications.  Springer, Cham, 2020.
  * [120] A. Kubica, K. Ryszewska, and M. Yamamoto.  Time-fractional differential equations—a theoretical introduction.  Springer Briefs Math. Springer, Singapore, 2020.
  * [121] G. Akagi.  Fractional flows driven by subdifferentials in Hilbert spaces.  Israel J. Math., 234:809–862, 2019.
  * [122] C.O. Alves and T. Boudjeriou.  Existence and uniqueness of solution for some time fractional parabolic equations involving the 1-Laplace operator.  Partial Differential Equations and Applications, 4, 2023.
  * [123] P. Wittbold, P. Wolejko, and R. Zacher.  Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations.  J. Math. Anal. Appl., 499:20 pp, 2021.
  * [124] A. Capella, J. Dávila, L. Dupaigne, and Y. Sire.  Regularity of radial extremal solutions for some non-local semilinear equations.  Communications in Partial Differential Equations, 36(8):1353–1384, 2011.
  * [125] Nicola Abatangelo and Enrico Valdinoci.  Getting acquainted with the fractional Laplacian.  Contemporary research in elliptic PDEs and related topics, pages 1–105, 2019.
  * [126] R. Servadei and E. Valdinoci.  On the spectrum of two different fractional operators.  Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 144(4):831–855, 2014.
  * [127] Claudia Bucur, Enrico Valdinoci, et al.  Nonlocal diffusion and applications, volume 20.  Springer, 2016.
  * [128] Siwei Duo, Hong Wang, and Yanzhi Zhang.  A comparative study on nonlocal diffusion operators related to the fractional Laplacian.  Discrete & Continuous Dynamical Systems-Series B, 24(1), 2019.
  * [129] Roberta Musina and Alexander I Nazarov.  On fractional laplacians.  Communications in partial differential equations, 39(9):1780–1790, 2014.
  * [130] Kolade M Owolabi and Abdon Atangana.  Modelling and formation of spatiotemporal patterns of fractional predation system in subdiffusion and superdiffusion scenarios.  The European physical Journal Plus, 133:1–13, 2018.
  * [131] Canrong Tian.  Turing pattern formation in a semiarid vegetation model with fractional-in-space diffusion.  Bulletin of mathematical biology, 77:2072–2085, 2015.
  * [132] L.W. Somathilake and K. Burrage.  A space-fractional-reaction-diffusion model for pattern formation in coral reefs.  Cogent Mathematics & Statistics, 5(1):1426524, 2018.
  * [133] Alexander A Golovin, Bernard J Matkowsky, and Vladimir A Volpert.  Turing pattern formation in the Brusselator model with superdiffusion.  SIAM Journal on Applied Mathematics, 69(1):251–272, 2008.
  * [134] Lai Zhang and Canrong Tian.  Turing pattern dynamics in an activator-inhibitor system with superdiffusion.  Physical Review E, 90(6):062915, 2014.
  * [135] J. Schnakenberg.  Simple chemical reaction systems with limit cycle behaviour.  Journal of Theoretical Biology, 81(3):389–400, 1979.
  * [136] M. Alfaro, D. Hilhorst, and H. Matano.  The singular limit of the Allen–Cahn equation and the FitzHugh–Nagumo system.  Journal of Differential Equations, 245(2):505–565, 2008.
  * [137] S. Alama, L. Bronsard, and C. Gui.  Stationary layered solutions in $\mathbb{R}^{2}$ for an Allen–Cahn system with multiple well potential.  Calculus of Variations and Partial Differential Equations, 5(4):359–390, 1997.
  * [138] P.H. Rabinowitz and E.W. Stredulinsky.  Mixed states for an Allen–Cahn type equation.  Communications on Pure and Applied Mathematics, 56(8):1078–1134, 2003.
  * [139] M.J. Ward.  Metastable bubble solutions for the Allen–Cahn equation with mass conservation.  SIAM Journal on Applied Mathematics, 56(5):1247–1279, 1996.
  * [140] I. S Aranson and L. Kramer.  The world of the complex Ginzburg–Landau equation.  Reviews of Modern Physics, 74(1):99–143, 2002.
  * [141] A. Mielke.  The Ginzburg–Landau equation in its role as a modulation equation.  In Handbook of Dynamical Systems, volume 2, pages 759–834. Elsevier, 2002.
  * [142] G. Schneider.  The validity of generalized Ginzburg–Landau equations.  Mathematical Methods in the Applied Sciences, 19(9):717–736, 1996\.
  * [143] T. Kapitula and B. Sandstede.  Instability mechanism for bright solitary-wave solutions to the cubic–quintic Ginzburg–Landau equation.  Journal of the Optical Society of America B, 15(11):2757–2762, 1998\.
  * [144] C. Kuehn.  Numerical continuation and SPDE stability for the 2D cubic–quintic Allen–Cahn equation.  SIAM/ASA Journal on Uncertainty Quantification, 3(1):762–789, 2015\.
  * [145] Franz Achleitner, Christian Kuehn, Jens M. Melenk, and Alexander Rieder.  Metastable speeds in the fractional Allen-Cahn equation.  Appl. Math. Comput., 408:Paper No. 126329, 18, 2021.
  * [146] G. Akagi, G. Schimperna, and A. Segatti.  Fractional Cahn–Hilliard, Allen–Cahn and porous medium equations.  Journal of Differential Equations, 261(6):2935–2985, 2016.
  * [147] A. Bueno-Orovio, D. Kay, and K. Burrage.  Fourier spectral methods for fractional-in-space reaction-diffusion equations.  BIT Numerical mathematics, 54(4):937–954, 2014.
  * [148] N. Ehstand, C. Kuehn, and C. Soresina.  Numerical continuation for fractional pdes: sharp teeth and bloated snakes.  Communications in Nonlinear Science and Numerical Simulation, 98:105762, 2021.
  * [149] J. Swift and P.C. Hohenberg.  Hydrodynamic fluctuations at the convective instability.  Physical Review A, 15(1):319, 1977.
  * [150] G. Schneider.  Diffusive stability of spatial periodic solutions of the Swift–Hohenberg equation.  Communications in Mathematical Physics, 178(3):679–702, 1996.
  * [151] U. Thiele, A.J. Archer, M.J. Robbins, H. Gomez, and E. Knobloch.  Localized states in the conserved Swift–Hohenberg equation with cubic nonlinearity.  Physical Review E, 87(4):042915, 2013.
  * [152] P. Collet and J.P. Eckmann.  The time dependent amplitude equation for the Swift–Hohenberg problem.  Communications in Mathematical Physics, 132(1):139–153, 1990.
  * [153] A. van Harten.  On the validity of the Ginzburg–Landau equation.  Journal of Nonlinear Science, 1(4):397–422, 1991.
  * [154] C. Kuehn and S. Throm.  Validity of amplitude equations for non-local non-linearities.  Journal of Mathematical Physics, 59:071510, 2018.
  * [155] D. Morgan and J.H.P. Dawes.  The Swift–Hohenberg equation with a nonlocal nonlinearity.  Physica D, 270:60–80, 2014.
  * [156] H. Uecker and D. Wetzel.  Numerical results for snaking of patterns over patterns in some 2D Selkov–Schnakenberg reaction-diffusion systems.  SIAM Journal on Applied Dynamical Systems, 13(1):94–128, 2014\.
  * [157] V. A. Volpert, Y. Nec, and A. A. Nepomnyashchy.  Fronts in anomalous diffusion-reaction systems.  Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371(1982):20120179, 18, 2013.
  * [158] Yana Nec, Vladimir A. Volpert, and Alexander A. Nepomnyashchy.  Front propagation problems with sub-diffusion.  Discrete Contin. Dyn. Syst., 27(2):827–846, 2010.
  * [159] Hiroshi Ishii.  Propagating front solutions in a time-fractional fisher-kpp equation, 2023\.
  * [160] F. Achleitner, C. M. Cuesta, and S. Hittmeir.  Travelling waves for a non-local Korteweg–de Vries–Burgers equation.  J. Differential Equations, 257(3):720–758, 2014.
  * [161] C. M. Cuesta and F. Achleitner.  Addendum to “Travelling waves for a non-local Korteweg–de Vries-Burgers equation” [J. Differential Equations 257 (3) (2014) 720–758] [MR3208089].  J. Differential Equations, 262(2):1155–1160, 2017.
  * [162] Franz Achleitner and Christian Kuehn.  Traveling waves for a bistable equation with nonlocal diffusion.  Adv. Differential Equations, 20(9-10):887–936, 2015.
  * [163] D.G. Aronson and H.F. Weinberger.  Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation.  In Partial Differential Equations and Related Topics, volume 446 of Lecture Notes in Mathematics, pages 5–49. Springer, 1974.
  * [164] Paul C. Fife and J. B. McLeod.  The approach of solutions of nonlinear diffusion equations to travelling front solutions.  Arch. Ration. Mech. Anal., 65(4):335–361, 1977.
  * [165] Xavier Cabré and Yannick Sire.  Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions.  Trans. Amer. Math. Soc., 367(2):911–941, 2015.
  * [166] A. Chmaj.  Existence of traveling waves in the fractional bistable equation.  Archiv der Mathematik, 100(5):473–480, May 2013.
  * [167] Changfeng Gui and Mingfeng Zhao.  Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian.  Ann. Inst. H. Poincaré C Anal. Non Linéaire, 32(4):785–812, 2015.
  * [168] Giampiero Palatucci, Ovidiu Savin, and Enrico Valdinoci.  Local and global minimizers for a variational energy involving a fractional norm.  Ann. Mat. Pura Appl. (4), 192(4):673–718, 2013.
  * [169] Y. Nec, A.A. Nepomnyashchy, and A.A. Golovin.  Front-type solutions of fractional Allen-Cahn equation.  Phys. D, 237(24):3237–3251, 2008.
  * [170] V. A. Volpert, Y. Nec, and A. A. Nepomnyashchy.  Exact solutions in front propagation problems with superdiffusion.  Phys. D, 239(3-4):134–144, 2010.
  * [171] D.H. Zanette.  Wave fronts in bistable reactions with anomalous Lévy-flight diffusion.  Physical Review E, 55(1):1181–1184, 1997.
  * [172] H.P. McKean.  Nagumo’s equation.  Advances in mathematics, 4:209–223, 1970.
  * [173] Luyi Ma, Hong-Tao Niu, and Zhi-Cheng Wang.  Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian.  Commun. Pure Appl. Anal., 18(5):2457–2472, 2019.
  * [174] Hongmei Cheng and Rong Yuan.  The stability of traveling waves for Allen-Cahn equations with fractional Laplacian.  Appl. Anal., 101(1):263–273, 2022.
  * [175] Peter W. Bates, Paul C. Fife, Xiaofeng Ren, and Xuefeng Wang.  Traveling waves in a convolution model for phase transitions.  Arch. Rational Mech. Anal., 138(2):105–136, 1997.
  * [176] Xinfu Chen.  Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations.  Adv. Differential Equations, 2(1):125–160, 1997.
  * [177] Luis Caffarelli and Luis Silvestre.  An extension problem related to the fractional Laplacian.  Comm. Partial Differential Equations, 32(7-9):1245–1260, 2007.
  * [178] Mateusz Kwaśnicki.  Harmonic extension technique for non-symmetric operators with completely monotone kernels.  Calc. Var. Partial Differential Equations, 61(6):Paper No. 202, 40, 2022.
  * [179] Maria G. Westdickenberg.  On the metastability of the 1-$d$ Allen-Cahn equation.  J. Dynam. Differential Equations, 33(4):1853–1879, 2021.
  * [180] Franz Achleitner and Christian Kuehn.  Analysis and numerics of traveling waves for asymmetric fractional reaction-diffusion equations.  Commun. Appl. Ind. Math., 6(2):e–532, 25, 2015.
  * [181] Jack Xin.  Front propagation in heterogeneous media.  SIAM Rev., 42(2):161–230, 2000.
  * [182] François Hamel.  Bistable transition fronts in $\mathbb{R}^{N}$.  Adv. Math., 289:279–344, 2016.
  * [183] Masaharu Taniguchi.  Traveling front solutions in reaction-diffusion equations, volume 39 of MSJ Memoirs.  Mathematical Society of Japan, Tokyo, 2021.
  * [184] J. X. Xin.  Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I.  Comm. Partial Differential Equations, 17(11-12):1889–1899, 1992\.
  * [185] C. D. Levermore and J. X. Xin.  Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. II.  Comm. Partial Differential Equations, 17(11-12):1901–1924, 1992\.
  * [186] Christopher K. R. T. Jones.  Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions.  Rocky Mountain J. Math., 13(2):355–364, 1983.
  * [187] Todd Kapitula.  Multidimensional stability of planar travelling waves.  Trans. Amer. Math. Soc., 349(1):257–269, 1997.
  * [188] A.I. Volpert, V.A. Volpert, and V.A. Volpert.  Traveling wave solutions of parabolic systems, volume 140 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 1994.  Translated from the Russian manuscript by James F. Heyda.
  * [189] Wah Wah and Masaharu Taniguchi.  Traveling front solutions for perturbed reaction-diffusion equations.  Math. J. Okayama Univ., 65:125–143, 2023.
  * [190] Xinfu Chen, Jong-Shenq Guo, François Hamel, Hirokazu Ninomiya, and Jean-Michel Roquejoffre.  Traveling waves with paraboloid like interfaces for balanced bistable dynamics.  Ann. Inst. H. Poincaré C Anal. Non Linéaire, 24(3):369–393, 2007.
  * [191] Henri Berestycki and François Hamel.  Generalized transition waves and their properties.  Comm. Pure Appl. Math., 65(5):592–648, 2012.
  * [192] Alberto Farina and Enrico Valdinoci.  Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications.  Indiana Univ. Math. J., 60(1):121–141, 2011.
  * [193] Serena Dipierro and Enrico Valdinoci.  Long-range phase coexistence models: recent progress on the fractional Allen-Cahn equation.  In Topics in applied analysis and optimisation—partial differential equations, stochastic and numerical analysis, CIM Ser. Math. Sci., pages 121–138. Springer, Cham, [2019] ©2019.
  * [194] Alberto Farina and Enrico Valdinoci.  The state of the art for a conjecture of De Giorgi and related problems.  In Recent progress on reaction-diffusion systems and viscosity solutions, pages 74–96. World Sci. Publ., Hackensack, NJ, 2009.
  * [195] O. Savin.  Phase transitions, minimal surfaces and a conjecture of De Giorgi.  In Current developments in mathematics, 2009, pages 59–113. Int. Press, Somerville, MA, 2010.
  * [196] Hardy Chan and Juncheng Wei.  On De Giorgi’s conjecture: recent progress and open problems.  Sci. China Math., 61(11):1925–1946, 2018.
  * [197] I. S. Gradshteyn and I. M. Ryzhik.  Table of integrals, series, and products.  Elsevier/Academic Press, Amsterdam, eighth edition, 2015.  Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll.
  * [198] Hardy Chan and Juncheng Wei.  Traveling wave solutions for bistable fractional Allen-Cahn equations with a pyramidal front.  J. Differential Equations, 262(9):4567–4609, 2017.
  * [199] Luyi Ma and Zhi-Cheng Wang.  On the existence of cylindrically symmetric traveling fronts of fractional Allen-Cahn equation in $\mathbb{R}^{3}$.  Differential Integral Equations, 34(9-10):467–490, 2021.
  * [200] Luyi Ma, Hong-Tao Niu, and Zhi-Cheng Wang.  V-shaped traveling fronts of fractional Allen-Cahn equations.  J. Math. Phys., 63(2):Paper No. 021507, 29, 2022.
  * [201] Hongmei Cheng and Rong Yuan.  The stability of the equilibria of the Allen-Cahn equation with fractional diffusion.  Appl. Anal., 98(3):600–610, 2019.
  * [202] Serena Dipierro, Joaquim Serra, and Enrico Valdinoci.  Improvement of flatness for nonlocal phase transitions.  Amer. J. Math., 142(4):1083–1160, 2020.
  * [203] Serena Dipierro and Enrico Valdinoci.  Some perspectives on (non)local phase transitions and minimal surfaces.  Bull. Math. Sci., 13(1):Paper No. 2330001, 77, 2023.
  * [204] George Em Karniadakis, editor.  Handbook of fractional calculus with applications. Vol. 3.  De Gruyter, Berlin, 2019.  Numerical methods.
  * [205] Changpin Li and An Chen.  Numerical methods for fractional partial differential equations.  Int. J. Comput. Math., 95(6-7):1048–1099, 2018.
  * [206] Bangti Jin and Zhi Zhou.  Numerical treatment and analysis of time-fractional evolution equations, volume 214 of Applied Mathematical Sciences.  Springer, Cham, [2023] ©2023.
  * [207] Martin Stynes.  Singularities.  In Handbook of fractional calculus with applications. Vol. 3, pages 287–305. De Gruyter, Berlin, 2019.
  * [208] Martin Stynes, Eugene O’Riordan, and José Luis Gracia.  Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation.  SIAM J. Numer. Anal., 55(2):1057–1079, 2017.
  * [209] William McLean and Kassem Mustapha.  Time-stepping error bounds for fractional diffusion problems with non-smooth initial data.  J. Comput. Phys., 293:201–217, 2015.
  * [210] Hermann Brunner.  Collocation methods for Volterra integral and related functional differential equations, volume 15 of Cambridge Monographs on Applied and Computational Mathematics.  Cambridge University Press, Cambridge, 2004.
  * [211] C. Lubich.  Convolution quadrature and discretized operational calculus. I.  Numer. Math., 52(2):129–145, 1988.
  * [212] C. Lubich.  Convolution quadrature and discretized operational calculus. II.  Numer. Math., 52(4):413–425, 1988.
  * [213] L. Banjai and F.-J. Sayas.  Integral equation methods for evolutionary PDE, volume 53 of Springer Series in Computational Mathematics.  Springer Verlag, 2022.
  * [214] Ch. Lubich and A. Ostermann.  Runge-Kutta methods for parabolic equations and convolution quadrature.  Math. Comp., 60(201):105–131, 1993.
  * [215] Christian Lubich.  Convolution quadrature revisited.  BIT, 44(3):503–514, 2004.
  * [216] Jing Guo and Maria Lopez-Fernandez.  Generalized convolution quadrature for the fractional integral and fractional diffusion equations, 2023.
  * [217] Lehel Banjai and Charalambos G. Makridakis.  A posteriori error analysis for approximations of time-fractional subdiffusion problems.  Math. Comp., 91(336):1711–1737, 2022.
  * [218] Leslie Greengard and Vladimir Rokhlin.  A new version of the fast multipole method for the Laplace equation in three dimensions.  In Acta numerica, 1997, volume 6 of Acta Numer., pages 229–269. Cambridge Univ. Press, Cambridge, 1997.
  * [219] Johannes Tausch and Jacob White.  Multiscale bases for the sparse representation of boundary integral operators on complex geometry.  SIAM J. Sci. Comput., 24(5):1610–1629, 2003.
  * [220] Wolfgang Hackbusch.  Hierarchical matrices: algorithms and analysis, volume 49 of Springer Series in Computational Mathematics.  Springer, Heidelberg, 2015.
  * [221] William McLean.  Exponential sum approximations for $t^{-\beta}$.  In Contemporary computational mathematics—a celebration of the 80th birthday of Ian Sloan. Vol. 1, 2, pages 911–930. Springer, Cham, 2018\.
  * [222] Achim Schädle, María López-Fernández, and Christian Lubich.  Fast and oblivious convolution quadrature.  SIAM J. Sci. Comput., 28(2):421–438, 2006.
  * [223] J. Dölz, H. Egger, and V. Shashkov.  A fast and oblivious matrix compression algorithm for Volterra integral operators.  Adv. Comput. Math., 47(6):Paper No. 81, 24, 2021.
  * [224] Ustim Khristenko and Barbara Wohlmuth.  Solving time-fractional differential equations via rational approximation.  IMA J. Numer. Anal., 43(3):1263–1290, 2023.
  * [225] Jie Shen and Changtao Sheng.  Spectral methods for fractional differential equations using generalized Jacobi functions.  In Handbook of fractional calculus with applications. Vol. 3, pages 127–155. De Gruyter, Berlin, 2019.
  * [226] Anna Lischke, Mohsen Zayernouri, and Zhongqiang Zhang.  Spectral and spectral element methods for fractional advection-diffusion-reaction equations.  In Handbook of fractional calculus with applications. Vol. 3, pages 157–183. De Gruyter, Berlin, 2019.
  * [227] Mohsen Zayernouri, Li-Liang Wang, Jie Shen, and George Em Karniadakis.  Spectral and Spectral-Element Methods for Fractional Ordinary and Partial Diﬀerential Equations.  Cambridge University Press, 2024.
  * [228] Sheng Chen and Jie Shen.  Log orthogonal functions: approximation properties and applications.  IMA J. Numer. Anal., 42(1):712–743, 2022.
  * [229] Sheng Chen, Jie Shen, and Li-Lian Wang.  Generalized Jacobi functions and their applications to fractional differential equations.  Math. Comp., 85(300):1603–1638, 2016.
  * [230] Zhiping Mao and Jie Shen.  Spectral element method with geometric mesh for two-sided fractional differential equations.  Adv. Comput. Math., 44(3):745–771, 2018.
  * [231] Mohsen Zayernouri and George Em Karniadakis.  Fractional Sturm-Liouville eigen-problems: theory and numerical approximation.  J. Comput. Phys., 252:495–517, 2013.
  * [232] Dianming Hou, Mohammad Tanzil Hasan, and Chuanju Xu.  Müntz spectral methods for the time-fractional diffusion equation.  Comput. Methods Appl. Math., 18(1):43–62, 2018.
  * [233] Ch. Schwab.  $p$\- and $hp$-finite element methods.  Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University Press, New York, 1998.  Theory and applications in solid and fluid mechanics.
  * [234] Xianjuan Li and Chuanju Xu.  A space-time spectral method for the time fractional diffusion equation.  SIAM J. Numer. Anal., 47(3):2108–2131, 2009.
  * [235] Vincent J. Ervin and John Paul Roop.  Variational formulation for the stationary fractional advection dispersion equation.  Numer. Methods Partial Differential Equations, 22(3):558–576, 2006\.
  * [236] Mohsen Zayernouri, Mark Ainsworth, and George Em Karniadakis.  A unified Petrov-Galerkin spectral method for fractional PDEs.  Comput. Methods Appl. Mech. Engrg., 283:1545–1569, 2015.
  * [237] Gene H. Golub and Charles F. Van Loan.  Matrix computations.  Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, fourth edition, 2013.
  * [238] Andrea Bonito, Juan Pablo Borthagaray, Ricardo H. Nochetto, Enrique Otárola, and Abner J. Salgado.  Numerical methods for fractional diffusion.  Comput. Vis. Sci., 19(5-6):19–46, 2018.
  * [239] Anna Lischke, Guofei Pang, Mamikon Gulian, and et al.  What is the fractional Laplacian? A comparative review with new results.  J. Comput. Phys., 404:109009, 62, 2020.
  * [240] Clemens Hofreither.  A unified view of some numerical methods for fractional diffusion.  Comput. Math. Appl., 80(2):332–350, 2020.
  * [241] Kevin Burrage, Nicholas Hale, and David Kay.  An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations.  SIAM J. Sci. Comput., 34(4):A2145–A2172, 2012.
  * [242] Clemens Hofreither.  An algorithm for best rational approximation based on barycentric rational interpolation.  Numer. Algorithms, 88(1):365–388, 2021.
  * [243] Wolfgang Hackbusch.  Computation of best $L^{\infty}$ exponential sums for $1/x$ by Remez’ algorithm.  Comput. Vis. Sci., 20(1-2):1–11, 2019.
  * [244] Tobias Danczul and Joachim Schöberl.  A reduced basis method for fractional diffusion operators I.  Numer. Math., 151(2):369–404, 2022.
  * [245] Lehel Banjai, Jens M. Melenk, Ricardo H. Nochetto, Enrique Otárola, Abner J. Salgado, and Christoph Schwab.  Tensor FEM for spectral fractional diffusion.  Found. Comput. Math., 19(4):901–962, 2019.
  * [246] Lehel Banjai, Jens M. Melenk, and Christoph Schwab.  Exponential convergence of $hp$ FEM for spectral fractional diffusion in polygons.  Numer. Math., 153(1):1–47, 2023.
  * [247] Xavier Cabré and Jinggang Tan.  Positive solutions of nonlinear problems involving the square root of the Laplacian.  Adv. Math., 224(5):2052–2093, 2010.
  * [248] Pablo Raúl Stinga and José Luis Torrea.  Extension problem and Harnack’s inequality for some fractional operators.  Comm. Partial Differential Equations, 35(11):2092–2122, 2010.
  * [249] Ricardo H. Nochetto, Enrique Otárola, and Abner J. Salgado.  A PDE approach to fractional diffusion in general domains: a priori error analysis.  Found. Comput. Math., 15(3):733–791, 2015.
  * [250] Jens Markus Melenk and Alexander Rieder.  $hp$-FEM for the fractional heat equation.  IMA J. Numer. Anal., 41(1):412–454, 2021.
  * [251] Jens Markus Melenk and Alexander Rieder.  An exponentially convergent discretization for space–time fractional parabolic equations using $hp$-FEM.  IMA J. Numer. Anal., 43(4):2352–2376, 2023.
  * [252] Andrea Bonito and Joseph E. Pasciak.  Numerical approximation of fractional powers of elliptic operators.  Math. Comp., 84(295):2083–2110, 2015.
  * [253] Andrea Bonito, Wenyu Lei, and Joseph E. Pasciak.  On sinc quadrature approximations of fractional powers of regularly accretive operators.  J. Numer. Math., 27(2):57–68, 2019.
  * [254] Andrea Bonito, Wenyu Lei, and Joseph E. Pasciak.  Numerical approximation of space-time fractional parabolic equations.  Comput. Methods Appl. Math., 17(4):679–705, 2017.
  * [255] Nicole Cusimano, Félix del Teso, and Luca Gerardo-Giorda.  Numerical approximations for fractional elliptic equations $VIA$ the method of semigroups.  ESAIM Math. Model. Numer. Anal., 54(3):751–774, 2020.
  * [256] Zhiping Mao and Jie Shen.  Hermite spectral methods for fractional PDEs in unbounded domains.  SIAM J. Sci. Comput., 39(5):A1928–A1950, 2017.
  * [257] Tao Tang, Huifang Yuan, and Tao Zhou.  Hermite spectral collocation methods for fractional PDEs in unbounded domains.  Commun. Comput. Phys., 24(4):1143–1168, 2018.
  * [258] Tao Tang, Li-Lian Wang, Huifang Yuan, and Tao Zhou.  Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains.  SIAM J. Sci. Comput., 42(2):A585–A611, 2020.
  * [259] Changtao Sheng, Jie Shen, Tao Tang, Li-Lian Wang, and Huifang Yuan.  Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains.  SIAM J. Numer. Anal., 58(5):2435–2464, 2020.
  * [260] Changtao Sheng, Suna Ma, Huiyuan Li, Li-Lian Wang, and Lueling Jia.  Nontensorial generalised Hermite spectral methods for PDEs with fractional Laplacian and Schrödinger operators.  ESAIM Math. Model. Numer. Anal., 55(5):2141–2168, 2021.
  * [261] Ioannis P. A. Papadopoulos, Timon S. Gutleb, José A. Carrillo, and Sheehan Olver.  A frame approach for equations involving the fractional Laplacian, 2023\.  arXiv:2311.12451.
  * [262] Harbir Antil, Patrick Dondl, and Ludwig Striet.  Approximation of integral fractional Laplacian and fractional PDEs via sinc-basis.  SIAM J. Sci. Comput., 43(4):A2897–A2922, 2021.
  * [263] Harbir Antil, Patrick Dondl, and Ludwig Striet.  Analysis of a sinc-Galerkin method for the fractional Laplacian, 2023\.  arXiv:2212.11581.
  * [264] Juan Pablo Borthagaray, Wenbo Li, and Ricardo H. Nochetto.  Linear and nonlinear fractional elliptic problems.  In 75 years of mathematics of computation, volume 754 of Contemp. Math., pages 69–92. Amer. Math. Soc., [Providence], RI, [2020] ©2020.
  * [265] Juan Pablo Borthagaray, Wenbo Li, and Ricardo H. Nochetto.  Fractional elliptic problems on Lipschitz domains: Regularity and approximation, 2022.
  * [266] Markus Faustmann, Carlo Marcati, Jens Markus Melenk, and Christoph Schwab.  Weighted analytic regularity for the integral fractional Laplacian in polygons.  SIAM J. Math. Anal., 54(6):6323–6357, 2022.
  * [267] M. Faustmann, C. Marcati, J.M. Melenk, and Ch. Schwab.  Exponential convergence of $hp$-FEM for the integral fractional Laplacian in polygons.  arXiv:2209.11468, 2022.
  * [268] Stefan A. Sauter and Christoph Schwab.  Boundary element methods, volume 39 of Springer Series in Computational Mathematics.  Springer-Verlag, Berlin, 2011.  Translated and expanded from the 2004 German original.
  * [269] Alexey Chernov, Tobias von Petersdorff, and Christoph Schwab.  Quadrature algorithms for high dimensional singular integrands on simplices.  Numer. Algorithms, 70(4):847–874, 2015.
  * [270] G. Acosta, F.M. Bersetche, and J.P. Borthagaray.  A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian.  Comput. Math. Appl., 74(4):784–816, 2017.
  * [271] Mark Ainsworth and Christian Glusa.  Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver.  Comput. Methods Appl. Mech. Engrg., 327:4–35, 2017.
  * [272] Michael Karkulik and Jens Markus Melenk.  $\mathcal{H}$-matrix approximability of inverses of discretizations of the fractional Laplacian.  Adv. Comput. Math., 45(5-6):2893–2919, 2019.
  * [273] Markus Faustmann, Jens Markus Melenk, and Maryam Parvizi.  On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion.  ESAIM Math. Model. Numer. Anal., 55(2):595–625, 2021.
  * [274] Juan Borthagaray, Ricardo Nochetto, Shuonan Wu, and Jinchao Xu.  Robust BPX preconditioner for fractional Laplacians on bounded Lipschitz domains.  Math. Comp., 92(344):2439–2473, 2023.
  * [275] Heiko Gimperlein, Jakub Stocek, and Carolina Urzúa-Torres.  Optimal operator preconditioning for pseudodifferential boundary problems.  Numer. Math., 148(1):1–41, 2021.
  * [276] Markus Faustmann, Jens Markus Melenk, and Dirk Praetorius.  Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian.  Math. Comp., 90(330):1557–1587, 2021.
  * [277] Markus Faustmann and Alexander Rieder.  FEM-BEM coupling in fractional diffusion, 2023.
  * [278] Andrea Bonito, Wenyu Lei, and Joseph E. Pasciak.  Numerical approximation of the integral fractional Laplacian.  Numer. Math., 142(2):235–278, 2019.
  * [279] Andreas E. Kyprianou, Ana Osojnik, and Tony Shardlow.  Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian.  IMA J. Numer. Anal., 38(3):1550–1578, 2018.
  * [280] Changtao Sheng, Bihao Su, and Chenglong Xu.  Efficient Monte Carlo Method for Integral Fractional Laplacian in Multiple Dimensions.  SIAM J. Numer. Anal., 61(5):2035–2061, 2023.
  * [281] C. Kuehn and C. Soresina.  Numerical continuation for a fast reaction system and its cross-diffusion limit.  SN Partial Differential Equations and Applications, 1:7, 2020.
  * [282] M. Breden, C. Kuehn, and C. Soresina.  On the influence of cross-diffusion in pattern formation.  Journal of Computational Dynamics, 8(2), 2021.
  * [283] H. Uecker and H. de Witt.  Infinite time horizon spatially distributed optimal control problems with pde2path algorithms and tutorial examples.  arXiv preprint arXiv:1912.11135, 2019.
  * [284] H. Uecker.  pde2path without finite elements, 2021.
  * [285] A. Meiners and H. Uecker.  Differential geometric bifurcation problems in pde2path–algorithms and tutorial examples.  arXiv preprint arXiv:2309.03646, 2023.
  * [286] T. Dohnal, J.D.M. Rademacher, H. Uecker, and D. Wetzel.  pde2path-version 2.0: faster FEM, multi-parameter continuation, nonlinear boundary conditions, and periodic domains — a short manual, 2014.
  * [287] J.D.M. Rademacher and H. Uecker.  The OOPDE setting of pde2path — a tutorial via some Allen–Cahn models.  available at http://www.staff.uni-oldenburg.de/hannes.uecker/pde2path/tuts/actut.pdf, 2018\.
  * [288] Hannes Uecker.  Numerical continuation and bifurcation in Nonlinear PDEs.  SIAM, 2021.
  * [289] U. Prüfert.  OOPDE \- an object oriented approach to finite elements in MATLAB.  Quickstart Guide, available at http://www.mathe.tu-freiberg.de/nmo/mitarbeiter/uwe-pruefert/software, 2014\.
  * [290] A.V. Balakrishnan.  Fractional powers of closed operators and the semigroups generated by them.  Pacific Journal of Mathematics, 10(2):419–437, 1960.
  * [291] K. Yosida.  Functional Analysis.  Springer-Verlag Berlin Heidelberg, 6 edition, 1980.
  * [292] S. Dohr, C. Kahle, S. Rogovs, and P. Swierczynski.  A FEM for an optimal control problem subject to the fractional Laplace equation.  Calcolo, 56(4):37, 2019.
  * [293] J. Lund and K.L. Bowers.  Sinc Methods for Quadrature and Differential Equations, volume 32.  SIAM, 1992.
  * [294] N. Ehstand, C. Kuehn, and C. Soresina.  Supplementary material.  Figures and videos at https://www-m8.ma.tum.de/bin/view/Allgemeines/CinziaSoresinaPublic, matlab scripts at https://github.com/soresina/fractional_pde2path, 2020\.  Accessed December 20, 2020.
  * [295] J. Burke and E. Knobloch.  Snakes and ladders: Localized states in the Swift–Hohenberg equation.  Physics Letters A, 360(6):681–688, 2007.
  * [296] D. Avitabile, D.J.B. Lloyd, J. Burke, E. Knobloch, and B. Sandstede.  To snake or not to snake in the planar Swift–Hohenberg equation.  SIAM Journal on Applied Dynamical Systems, 9(3):704–733, 2010.
  * [297] S.M. Houghton and E. Knobloch.  Swift–Hohenberg equation with broken cubic–quintic nonlinearity.  Physical Review E, 84(1):016204, 2011.
  * [298] A. Bergeon, J. Burke, E. Knobloch, and I. Mercader.  Eckhaus instability and homoclinic snaking.  Physical Review E, 78(4):046201, 2008.