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 . 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  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
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 , 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 non-stationary i.e., the time-dependent 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 space-time fractional evolution equations to the solutions of the corresponding equations associated with the continuous operator. More precisely, we consider the following two systems:
where in (1.2)-(1.3), and are real numbers, denotes the Caputo time-fractional derivative of order and is a nonlinear function satisfying a local Lipschitz continuity condition. Notice that if , the above systems correspond to the semi-linear continuous and discrete heat equations, respectively.
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 Trotter-Kato 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,  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 . 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.
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:
provided that the limit exists for a.e. . The normalization constant is given by
We consider a mesh of fixed side in given by
For a function we define the discrete Laplace operator by
It is well-known that the operator generates a strongly continuous submarkovian semigroup on () given by
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
where we recall that is the submarkovian semigroup given in (1.6) and .
It has been shown in [8, Theorem 1.1] that if , then
where the kernel is given by
Equivalently, we can derive another pointwise explicit formula of as follows: (see e.g. [24, Section 3.1])
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
Next, let be a Banach space. For a fixed real number and a function , we define the Caputo time-fractional derivative as follows:
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 Mittag-Leffler 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 well-known that, for :
namely, for every , the function
is a solution of the scalar valued ordinary differential equation:
Finally, for a real number , we shall denote by the Wright function defined by
The following formula on the moments is well-known (see e.g.[20, 21]):
1.2. Statement of the main results
In this section we state the main results of the paper. Recall that, we consider the following semi-linear space-time fractional order Cauchy problems:
where , are real numbers and denotes the Caputo time-fractional derivative of order given in (1.11).
We introduce our notion of solution.
Let and .
By a local strong solution of the system (1.16), we mean a function , for some , satisfying the following conditions:
the first identity in (1.16) is satisfied for a.e. and all , and the initial condition holds.
If , then we say that or is a global strong solution.
We assume the following condition on :
We mention that, it is easy to see that the assumption (1.17) implies that satisfies the following local-Lipschitz condition:
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  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.
The following theorem is the main result of the paper.
We conclude this section with the following remark.
In this section we give some preliminary results that are needed in the proofs of our main results.
2.1. A version of the Trotter-Kato approximation theorem
A sequence of Banach spaces () together with a sequence of bounded linear maps is said to approximate the Banach space if
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
then for every we have
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 well-known 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.
The operator generates a strongly continuous semigroup of contractions on .
Let , a real number, and consider the following Poisson problem:
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.
The bounded operator (that is, with domain ) generates a uniformly strongly continuous semigroup of contractions on and is given explicitly for by
where () denotes the modified Bessel function of the first kind of order .
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.
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
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
for every and
for every .
It follows from (3) that
where we have used the fact that
which converges to zero as by (3.1).
We notice 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
for every , and