1. Introduction, notations and main results
Fractional order operators have recently emerged as a modeling alternative in various branches of science and technology. In fact, in many situations, the fractional models reflect better the behavior of the system both in the deterministic and stochastic contexts. A number of stochastic models for explaining anomalous diffusion have been introduced in the literature; among them we mention the fractional Brownian motion; the continuous time random walk; the Lévy flights; the Schneider grey Brownian motion; and more generally, random walk models based on evolution equations of single and distributed fractional order in space (see e.g. [9, 15, 27, 32]
). In general, a fractional diffusion operator corresponds to a diverging jump length variance in the random walk. Further applications include, imaging
[2, 4][3], and geophysics [34]. We refer to [10, 11, 12] and the references therein for a complete analysis, the derivation and more applications of fractional order operators.Numerical methods for fractional PDEs have recently received a great deal of attention, see [19] and references therein. The study of convergence of discrete solutions to continuous ones for the associated stationary linear problem, that is, the time independent problem
(1.1) 
where is a real number, denotes the fractional Laplace operator defined on the real line and, for a fixed real number , the operator is the discrete fractional Laplace operator and is a mesh of fixed size , and does not depends on (resp. ), has been first and completely analyzed by Ciaurri et al [8], where the authors have proved the convergence of solutions in spaces. They have also obtained explicit convergence rates.
However, the study of the convergence of discrete to continuous solutions for the nonstationary i.e., the timedependent problem, is completely open.
The main concern of the present paper is to solve this open problem. We show the convergence of solutions of a class of discrete spacetime fractional evolution equations to the solutions of the corresponding equations associated with the continuous operator. More precisely, we consider the following two systems:
(1.2) 
and
(1.3) 
where in (1.2)(1.3), and are real numbers, denotes the Caputo timefractional derivative of order and is a nonlinear function satisfying a local Lipschitz continuity condition. Notice that if , the above systems correspond to the semilinear continuous and discrete heat equations, respectively.
Our main result (Theorem 1.3) states that if , , with , then for every , there exist and , such that the corresponding solutions and of the systems (1.2) and (1.3), respectively, satisfy
(1.4) 
where is a suitable mapping from to (see (1.10) below). In particular, we obtain that for , then taking , we have that the corresponding solutions also satisfy the convergence (1.4). The main tools we shall use are the convergence results for the stationary problem (1.1) (see Theorem 2.6) and a suitable version of the TrotterKato approximation theorem.
We observe that the existing theoretical error estimates are largely limited to either energy type norms or to the
norm, see for instance, [1] where error estimates for finite element discretization has been carried out. There are no existing pointwise error estimates for parabolic fractional PDEs, even though from a practical point of view, it is much easier to observe the pointwise values. In fact, the techniques for pointwise error estimates are significantly different than in the other norms [6]. It is unclear how to extend such arguments to nonlocal/fractional PDEs since solutions of such PDEs do not enjoy enough regularity as the local PDEs, and therefore makes the numerical analysis of such equations very challenging. There are many papers where the authors have assumed that such PDEs have very smooth solutions in order to do the numerical analysis. We refrain from citing those works, but we point out that solutions of fractional PDEs involving do not have enough regularity as in the classical case . We refer to [29, 30] for more details on this topic.The rest of the paper is structured as follows. In Section 1.1 we give some notations and introduce the function spaces needed to investigate our problem. We state the main results of the paper in Section 1.2 and give some preliminary results in Section 2 as they are needed throughout the paper. Section 3 contains the proofs of our main results. Finally in Section 4 we present some numerical simulations that confirm our theoretical findings.
1.1. Notations
In this section we fix some notations and introduce the function spaces needed to investigate our problem. Let be a real number and . The continuous fractional Laplace operator in is defined by the following singular integral:
(1.5) 
provided that the limit exists for a.e. . The normalization constant is given by
If , then we shall denote . We refer to [7, 16, 17, 29, 33] and the references therein for the class of functions for which the limit in (1.1) exists.
We consider a mesh of fixed side in given by
For a function we define the discrete Laplace operator by
It is wellknown that the operator generates a strongly continuous submarkovian semigroup on () given by
(1.6) 
where denotes the modified Bessel function of the first kind and of order (see e.g. [24, Theorem 1.1 and Remark 1] and the references therein).
For , we define the fractional powers of on by
(1.7) 
where we recall that is the submarkovian semigroup given in (1.6) and .
Let
It has been shown in [8, Theorem 1.1] that if , then
(1.8) 
where the kernel is given by
Equivalently, we can derive another pointwise explicit formula of as follows: (see e.g. [24, Section 3.1])
(1.9) 
where
We notice that for all and each (see e.g. [24, Proposition 1]).
Next, we let the Banach space
be endowed with the sup norm
We also define as
and we endow it with the sup norm.
If and , we define the space of Hölder continuous functions
For a real number , we define the bounded linear map
(1.10) 
Next, let be a Banach space. For a fixed real number and a function , we define the Caputo timefractional derivative as follows:
(1.11) 
We notice that if is smooth, then (1.11) is equivalent to the classical definition of given by
If and the function is smooth enough, one can show that .
The MittagLeffler function with two parameters is defined as follows:
We have that is an entire function. In the literature, the notation is frequently used. It is wellknown that, for :
(1.12) 
namely, for every , the function
is a solution of the scalar valued ordinary differential equation:
1.2. Statement of the main results
In this section we state the main results of the paper. Recall that, we consider the following semilinear spacetime fractional order Cauchy problems:
(1.15) 
and
(1.16) 
where , are real numbers and denotes the Caputo timefractional derivative of order given in (1.11).
We introduce our notion of solution.
Definition 1.1.
We assume the following condition on :
(1.17) 
We mention that, it is easy to see that the assumption (1.17) implies that satisfies the following localLipschitz condition:
(1.18) 
for all where is a fixed constant.
Before stating our main results, for the sake of completeness, we include the result on the existence of solutions to the systems (1.15) and (1.16). We notice that this existence result can be found in [18] for the case and in [11, Theorem 4.2.2] for the general case and . Since this is not the main concern of the present article, we will not go into details.
Theorem 1.2.
The following theorem is the main result of the paper.
Theorem 1.3.
We conclude this section with the following remark.
2. Preliminaries
In this section we give some preliminary results that are needed in the proofs of our main results.
2.1. A version of the TrotterKato approximation theorem
Definition 2.1.
A sequence of Banach spaces () together with a sequence of bounded linear maps is said to approximate the Banach space if
We have the following approximation result whose proof can be found in [31, Theorem 2.6] (see also [23]).
Theorem 2.2.
Suppose approximates the Banach space and , are strongly continuous contraction semigroups with generators and , respectively. If for every (or a core of ) there exists a sequence with the properties
(2.1) 
then for every we have
(2.2) 
for every .
2.2. The continuous and discrete fractional Laplace operators
Let and let be the fractional Laplace operator defined in (1.1).
We also let be the selfadjoint operator on with domain
where the fractional order Sobolev space is defined by
Then, it is nowadays wellknown that the operator generates a submarkovian semigroup in .
Let be the part of on . That is,
We have the following result. We include the proof for the sake of completeness.
Proposition 2.3.
The operator generates a strongly continuous semigroup of contractions on .
Proof.
Let , a real number, and consider the following Poisson problem:
(2.3) 
By a weak solution of (2.3), we mean a function such that the identity
holds for every . Firstly, we notice that for every , the problem (2.3) has a unique weak solution. In addition, if , then (see e.g. [16, 29] and their references). This shows that the resolvent leaves the space invariant. Using semigroups theory, the above property implies that the operator also leaves the space invariant for every . Thus, generates a semigroup of contractions on . The strong continuity of the semigroup follows from the fact that is dense in and the proof is finished. ∎
Next, recall that we have defined
Let be the operator defined in (1.7). Then, we have following result.
Proposition 2.4.
The bounded operator (that is, with domain ) generates a uniformly strongly continuous semigroup of contractions on and is given explicitly for by
(2.4) 
where () denotes the modified Bessel function of the first kind of order .
Remark 2.5.
We notice that is an uniformly continuous semigroup of contractions on follows from the positivity of the Bessel functions and [24, Formula (51)] which implies that the estimate
holds for all
The following convergence result taken from [8, Theorem 1.7] will be crucial in the proof of our main results.
Theorem 2.6.
Let and . Then the following assertions holds.

If and , then there is a constant (independent of and ) such that

If and , then there is a constant (independent of and ) such that
3. Proof of the main result
In this section we give the proof of our main result, namely, Theorem 1.3.
Proof of Theorem 1.3.
We prove the result in several steps. Throughout the proof if , are such that , then we shall sometimes write .
Step 1: Let be the mapping defined in (1.10). We claim that and approximate the Banach space in the sense of Definition 2.1. Indeed, firstly it is clear that for every . Secondly, let be a fixed real number. Then, for every , there exists such that . This implies that as . This fact, together with the previous observation imply that for the every and the claim is proved.
We also notice that, since and approximate , it follows that for every , there exists a function such that
(3.1) 
Step 2: Let and be the strongly continuous and contractive semigroups given in Propositions 2.3 and 2.4, respectively. We claim that for every and , we have
(3.2) 
It suffices to prove (3.2) for every in a dense subspace of . Indeed, let . It follows from Theorem 2.6 that
(3.3) 
Using Step 1, the convergence in (3.3) and applying Theorem 2.2, we can deduce that (3.2) holds.
Step 3: We prove (1.19) for the case . Recall that by Theorem 1.2, under the assumption (1.18) on the nonlinearity , there exist local strong solutions and , for some and . In addition, using semigroups theory, we have that
(3.4) 
for every and
(3.5) 
for every .
Next, let be an arbitrary real number. Let be fixed. Choose a function satisfying (3.1). Then, using (3.2) and the representations (3.4)(3.5), we get that for every ,
(3.6) 
It follows from (3) that
(3.7) 
where we have used the fact that
which converges to zero as by (3.1).
Now set
We notice that
Using the fact that for every and the local Lipschitz continuity assumption (1.18) on , we can deduce from (3) that there is a constant (depending only on and the Lipschitz constant ) such that
Using Gronwall’s inequality, the above estimate implies that
We have shown (1.19) for .
Step 4: Next, we prove (1.19) for the case . Firstly, we recall that under the assumption (1.18) on , Theorem 1.2 implies the existence of local strong solutions and , for some and . In addition, using the theory of fractional order Cauchy problems (see e.g. [5, 20, 21, 22]), we have that
(3.8) 
for every , and
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