# Convergence of solutions of discrete semi-linear space-time fractional evolution equations

Let (-Δ)_c^s be the realization of the fractional Laplace operator on the space of continuous functions C_0(R), and let (-Δ_h)^s denote the discrete fractional Laplacian on C_0(Z_h), where 0<s<1 and Z_h:={hj: j∈Z} is a mesh of fixed size h>0. We show that solutions of fractional order semi-linear Cauchy problems associated with the discrete operator (-Δ_h)^s on C_0(Z_h) converge to solutions of the corresponding Cauchy problems associated with the continuous operator (-Δ)_c^s. In addition, we obtain that the convergence is uniform in t in compact subsets of [0,∞). We also provide numerical simulations that support our theoretical results.

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## 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

 (−Δ)su=f in R and (−Δh)suh=f in Zh, (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 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:

 {Dαtu+(−Δ)su=f(u) in (0,∞)×R,u(0,⋅)=u0 in R, (1.2)

and

 {Dαtuh+(−Δh)suh=f(uh) in (0,∞)×Zh,uh(0,⋅)=u0h in Zh, (1.3)

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.

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

 limh↓0sup0≤t≤T∥Rhu(t)−uh(t)∥C0(Zh)=0, (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 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, [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:

 (−Δ)su(x):= CN,sP.V.∫Ru(x)−u(y)|x−y|N+2sdy = (1.5)

provided that the limit exists for a.e. . The normalization constant is given by

 CN,s:=s22sΓ(N2+s)πN2Γ(1−s).

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

 Zh:={hj:j∈Z}.

For a function we define the discrete Laplace operator by

 −Δhu(jh):=−u((j+1)h)−2u(jh)+u((j−1)h)h2.

It is well-known that the operator generates a strongly continuous submarkovian semigroup on () given by

 etΔhf(jh)=e−2th2∑n∈ZIn−j(2th2)f(nh),t≥0, (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

 (−Δh)su(jh):=1Γ(−s)∫∞0(etΔhu(jh)−u(jh))dtt1+s, (1.7)

where we recall that is the submarkovian semigroup given in (1.6) and .

Let

 ℓs:={u:Zh→R such % that ∑m∈Z|u(mh)|(1+|m|1+2s)<∞}.

It has been shown in [8, Theorem 1.1] that if , then

 (−Δh)su(jh)=∑m∈Z,m≠j(u(jh)−u(mh))Rsh(j−m), (1.8)

where the kernel is given by

 Rsh(m):=⎧⎪ ⎪⎨⎪ ⎪⎩4sΓ(12+s)√π|Γ(−s)|⋅Γ(|m|−s)h2sΓ(|m|+1+s),%ifm∈Z∖{0},=0 if m=0.

Equivalently, we can derive another pointwise explicit formula of as follows: (see e.g. [24, Section 3.1])

 (−Δh)su(jh)=∑n∈ZKsh(j−n)u(nh), (1.9)

where

 Ksh(n):=(−1)nΓ(2s+1)Γ(1+s+n)Γ(1+s−n)h2s.

We notice that for all and each (see e.g. [24, Proposition 1]).

Next, we let the Banach space

 C0(Zh):={f:Zh→R with lim|jh|→∞f(jh)=0},

be endowed with the sup norm

 ∥f∥C0(Zh):=supj∈Z|f(jh)|.

We also define as

 C0(R):={f:f∈C(R)∩L∞(R) and lim|x|→∞f(x)=0}.

and we endow it with the sup norm.

If and , we define the space of Hölder continuous functions

 Cγ,k(R):={U∈C(R):[U(k)]C0,γ(R):=supx,y∈R,x≠y|U(k)(x)−U(k)(y)||x−y|γ<∞}.

For a real number , we define the bounded linear map

 Rh:C0(R)→C0(Zh),u↦Rhu:Zh→R,(Rhu)(jh)=u(jh). (1.10)

Next, let be a Banach space. For a fixed real number and a function , we define the Caputo time-fractional derivative as follows:

 Dαtu(t):=ddt∫t0(t−τ)−αΓ(1−α)(u(τ)−u(0))dτ. (1.11)

We notice that if is smooth, then (1.11) is equivalent to the classical definition of given by

 Dαtu(t):=∫t0(t−τ)−αΓ(1−α)u′(τ)dτ.

If and the function is smooth enough, one can show that .

The Mittag-Leffler function with two parameters is defined as follows:

 Eα,β(z):=∞∑n=0znΓ(αn+β),α>0,β∈C,z∈C.

We have that is an entire function. In the literature, the notation is frequently used. It is well-known that, for :

 DαtEα,1(ztα)=zEα,1(ztα),t>0,z∈C, (1.12)

namely, for every , the function

is a solution of the scalar valued ordinary differential equation:

 Dαtu(t)=zu(t),t>0,0<α<2.

Finally, for a real number , we shall denote by the Wright function defined by

 Φα(z):=∞∑k=0(−z)kΓ(αk+1)k!,z∈C. (1.13)

The following formula on the moments is well-known (see e.g.

[20, 21]):

 ∫∞0Φα(t)tpdt=Γ(p+1)Γ(αp+1),∀p>−1. (1.14)

For more details on time fractional derivatives, the Mittag-Leffler and the Wright functions, we refer the reader to [13, 14, 25, 26, 28] and the references therein.

### 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:

 {Dαtu+(−Δ)su=f(u) in (0,T)×R,u(0,⋅)=u0 in R (1.15)

and

 {Dαtuh+(−Δh)suh=f(uh) in (0,T)×Zh,uh(0,⋅)=u0h in Zh, (1.16)

where , are real numbers and denotes the Caputo time-fractional derivative of order given in (1.11).

We introduce our notion of solution.

###### Definition 1.1.

Let and .

1. By a local strong solution of the system (1.15), we mean a function , for some , satisfying the following conditions:

• ;

• for a.e. ;

• the first identity in (1.15) holds poitwise for a.e. , and the initial condition is satisfied.

2. 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 :

 f∈C1(R) and f(0)=0. (1.17)

We mention that, it is easy to see that the assumption (1.17) implies that satisfies the following local-Lipschitz condition:

 |f(ξ1)−f(ξ2)|≤M|ξ1−ξ2| (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.

Let , and assume that satisfies (1.17). Then the following assertions hold.

1. For every , there exists such that (1.15) has a unique local strong solution .

2. For every , there exists such that (1.16) has a unique local strong solution .

The following theorem is the main result of the paper.

###### Theorem 1.3.

Let , be real numbers and assume that satisfies (1.18). Let and let be the associated unique local strong solution of (1.16) on . Then, there exists such that the corresponding unique local strong solution on of (1.15) satisfies

 limh↓0sup0≤t≤T∥Rhuα(t)−uαh(t)∥C0(Zh)=0, (1.19)

for all , where and is the operator defined in (1.10).

We conclude this section with the following remark.

###### Remark 1.4.

Regarding the convergence results in Theorem 1.3, unfortunately we do not know analytically the rate of convergence. But our simulations results obtained in Section 4 clearly show that we have a rate of convergence which depends on the parameter .

## 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 Trotter-Kato 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

 limn→∞∥Πnf∥Xn=∥f∥X for all f∈X.

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

 limn→∞∥Πnf−fn∥Xn=0 and limn→∞∥ΠnAf−Anfn∥Xn=0, (2.1)

then for every we have

 limn→∞sup0≤t≤S∥ΠnT(t)f−Tn(t)Πnf∥Xn=0, (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

 D((−Δ)s2):={u∈Hs(R):(−Δ)su∈L2(R)},

where the fractional order Sobolev space is defined by

 Hs(R):={u∈L2(R):∫R∫R|u(x)−u(y)|2|x−y|1+2sdxdy<∞}.

Then, it is nowadays well-known that the operator generates a submarkovian semigroup in .

Let be the part of on . That is,

 {D((−Δ)sc):={U∈C0(R)∩D((−Δ)s2):(−Δ)sU∈C0(R)},(−Δ)scU:=(−Δ)s2U.

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:

 λU+(−Δ)sU=F in R. (2.3)

By a weak solution of (2.3), we mean a function such that the identity

 λ∫RUVdx+Cs2∫R∫R(U(x)−U(y))(V(x)−V(y))|x−y|1+2sdxdy=∫RFVdx,

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

 C0(Zh):={f:Zh→R with lim|jh|→∞f(jh)=0}.

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

 Th(t)f(jh)=e−2th2s∑n∈ZIn−j(2th2s)f(nh), (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

 ∥Th(t)f∥C0(Zh)≤∥f∥C0(Zh)∑n∈Ze−2th2sIn(2th2s)=∥f∥C0(Zh),

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.

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

 ∥(−Δh)s(RhU)−Rh((−Δ)sU)∥C0(Zh)≤C[U]C0,γhγ−2s.
2. If and , then there is a constant (independent of and ) such that

 ∥(−Δh)s(RhU)−Rh((−Δ)sU)∥C0(Zh)≤C[U′]C0,γhγ−2s+1.

## 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

 limh↓0∥Rhg−gh∥C0(Zh)=0. (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

 limh↓0sup0≤t≤T∥Rh(T(t)u0)−Th(t)Rhu0∥C0(Zh)=0. (3.2)

It suffices to prove (3.2) for every in a dense subspace of . Indeed, let . It follows from Theorem 2.6 that

 limh↓0∥(−Δh)sRhu0−Rh((−Δ)su0)∥C0(Zh)=0. (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

 u1(t)=T(t)u0+∫t0T(t−τ)f(u1(τ))dτ, (3.4)

for every and

 u1h(t)=Th(t)u0h+∫t0Th(t−τ)f(u1h(τ))dτ, (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 ,

 Rhu1(t)= Rh(T(t)u0)+∫t0Rh(T(t−τ)f(u1(τ)))dτ = (Rh(T(t)u0)−Th(t)Rhu0)+Th(t)Rhu0+∫t0Rh(T(t−τ)f(u1(τ)))dτ = o(h)+Th(t)Rhu0+∫t0(Th(t−τ)Rhf(u1(τ))+o(h))dτ = o(h)+Th(t)Rhu0+∫t0Th(t−τ)f(Rhu1(τ))dτ+to(h). (3.6)

It follows from (3) that

 Rhu1(t)−u1h(t)= (1+t)o(h)+Th(t)Rhu0−Th(t)u0h +∫t0Th(t−τ)(f(Rhu1(τ))−f(u1h(τ)))dτ = (2+t)o(h)+∫t0Th(t−τ)(f(Rhu1(τ))−f(u1h(τ)))dτ, (3.7)

where we have used the fact that

 ∥Th(t)Rhu0−Th(t)u0h∥C0(Zh)=∥Th(t)(Rhu0−u0h)∥C0(Zh)≤∥Rhu0−u0h∥C0(Zh),

which converges to zero as by (3.1).

Now set

 Ψh(t):=∥Rhu1(t)−u1h(t)∥C0(Zh),0≤t≤T.

We notice that

 Ψh(0)=o(h).

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

 Ψh(t)≤Kh+K∫t0Ψh(τ)dτ,0≤t≤T.

Using Gronwall’s inequality, the above estimate implies that

 Ψh(t)≤KheKt for all t∈[0,T].

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

 uα(t)=Sα(t)u0+∫t0Pα(t−τ)f(uα(τ))dτ (3.8)

for every , and

 uαh(t)=Sαh(t)u0h+∫t0Pαh(t−τ)f(uαh(