Well posedness and approximation for nonhomogeneous fractional differential equations

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Numerical Functional Analysis and Optimization

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Well-Posedness and Approximation for

Nonhomogeneous Fractional Differential

Equations

Ru Liu & Sergey Piskarev

To cite this article: Ru Liu & Sergey Piskarev (2021): Well-Posedness and Approximation for Nonhomogeneous Fractional Differential Equations, Numerical Functional Analysis and Optimization, DOI: 10.1080/01630563.2021.1901117

To link to this article: https://doi.org/10.1080/01630563.2021.1901117

Published online: 22 Mar 2021.

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Well-Posedness and Approximation for

Nonhomogeneous Fractional Differential Equations

Ru Liuaand Sergey Piskarevb

a

College of Computer Science, Chengdu University, Chengdu, Sichuan, P.R China;bScientific Research Computer Center, Lomonosov Moscow State University, Moscow, Russia

ABSTRACT

In this paper, we consider the well-posedness and

approxima-tion for nonhomogeneous fracapproxima-tional differential equaapproxima-tions in

Banach spaces E Firstly, we get the necessary and sufficient

condition for the well-posedness of nonhomogeneous

frac-tional Cauchy problems in the spaces Cb0 đơ0, T; Eỡ: Secondly,

by using implicit difference scheme and explicit difference

scheme, we deal with the full discretization of the solutions of

nonhomogeneous fractional differential equations in time

vari-ables, get the stability of the schemes and the order of

convergence.

ARTICLE HISTORY Received 30 September 2019 Revised 6 March 2021 Accepted 6 March 2021 KEYWORDS

a-times resolvent family; discretization methods; explicit scheme; fractional Cauchy problem; implicit scheme; nonhomogeneous fractional equations MATHEMATICS SUBJECT CLASSIFICATION 45L05; 65M06; 65M12

1 Introduction

A lot of works were devoted to the approximations of C0-semigroups, see[1Ờ5] and the references therein While, other mathematicians consideredthe discrete approximation of integrated semigroups in their papers [6Ờ8]

We all know that Podlubny introduced fractional derivatives, fractional ferential equations, some methods of their solutions and some of theirapplications in his book [9] Ashyralyev and Cakir considered the numer-ical solutions of fractional parabolic partial differential equations [10Ờ15]

dif-In papers [16Ờ19], we dealt with the discrete approximation of the geneous fractional differential equations and semilinear fractional differen-tial equations in Banach spaces Especially in [18,19], we get the stabilityand the order of convergence of implicit difference scheme and explicit dif-ference scheme for homogeneous fractional differential equations In thispaper, we will consider the fulldiscrete approximation of the nonhomoge-neous fractional differential equation in the space Cđơ0, T; Eỡ, which will

Let 0<a<1, we consider the well-posed nonhomogeneous Cauchyproblem:

đDa

tuỡđtỡ Ử Auđtỡ ợ f đtỡ, t 2 đ0, T; uđ0ỡ Ử x, (1.1)where Da

t is the Caputo-Dzhrbashyan derivative

In [20], Ashyralyev and Sobolevskii indicated that in the HẠolder space

Cb0đơ0, T; Eỡ, the analyticity of a C0-semigroup is equivalent to the coercivewell-posedness of nonhomogeneous problem Ashyralyev studied the well-posedness of fractional parabolic partial differential equations [14, 21], andused modified Gauss elimination method to consider their numerical solu-tions [14] In [22], the authors got the maximal regularity as well asapproximation for fractional Cauchy equation in space Cb0đơ0, T; Eỡ: Here

Cb0đơ0, T; Eỡ is the Banach space [20] obtained by completion of the set ofE-valued smooth functions uđỡ on [0,T] in the norm

So, in the second section, we concentrate on the well-posedness of (1.1)

in the HẠolder space Cb0đơ0, T; Eỡ and prove that the analyticity of a-timesresolvent family is the necessary and sufficient condition for the well-pos-edness of(1.1)

and papers [14,22]

 In [14], the initial value of the problem is zero and the correspondingoperator is positive We do not need such assumptions in the pre-sent paper

 The authors in [22] got the maximal regularity for fractional Cauchyequation on space Cb0đơ0, T; Eỡ whenb  a: There is no such restriction

tuỡđtỡ Ử Auđtỡ, t>0; uđ0ỡ Ử x, (1.2)

is well-posed iff A generates an a-times resolvent family Sađ, Aỡ: Weassume from the beginning that resolvent family Sađ, Aỡ satisfiesjjSađt, Aỡjj  Mext, t  0, for some M  1, x  0: In such case, for fka:Rek>xg  qđAỡ, one has

ka1đkaIAỡ1x Ử

đ1 0

ektSađt, Aỡxdt ỬđSađt, Aỡxỡđkỡ, Rek>x, x 2 E,d

where bqðÞ is denoted the Laplace transform of qðÞ: In the paper [16], wehave proved that if the operator A generates an a-times resolvent family

Sað, AÞ which is satisfying jjSaðt, AÞjj  Mext, t  0, then the operator

A is closed and densely defined

Definition 1.1 [23] A family fSaðt,AÞgt0 BðEÞ is called an a-timesresolvent family generated by A if the following conditions are satisfied:

(a) Saðt, AÞ is strongly continuous for t  0 and Sað0, AÞ ¼ I;

(b) Saðt, AÞDðAÞ DðAÞ and ASaðt, AÞx ¼ Saðt, AÞAx for all x 2DðAÞ, t  0;

(c) for x 2 DðAÞ, Saðt, AÞx satisfies the resolvent equation

Saðt, AÞx ¼ x þ

ðt

0

gaðtsÞSaðs, AÞAxds, t  0:

Definition 1.2 An a-times resolvent family Sað, AÞ is called analytic if

Sað, AÞ admits an analytic extension to a sector Rh 0nf0g for some h 0 2ð0, p=2, where Rh 0 :¼ fk 2 C : jargkj<h0g: An analytic solution operator

is said to be of analyticity type ðh0, x0Þ if for each h<h0 and x>x0, there

is M ¼ Mðh, xÞ such that jjSaðz, AÞjj  MexRez, z 2Rh:

Note that Saðt, AÞ for bounded operator A is given by Mittag-Lefflerfunction EaðtaAÞ, i.e Saðt, AÞ ¼ EaðtaAÞ ¼P1

j¼0

ðt a AÞjCðajþ1Þ:

Definition 1.3 [24] A family fPaðt,AÞgt0 of strongly continuous function

Pað, AÞ : ð0, 1Þ ! BðEÞ is called an ða, aÞ-times resolvent family generated

by A if there exists x  0, such that fka: Rek>xg  qðAÞ and

ðkaIAÞ1x ¼

ð1 0

ektPaðt, AÞxdt, Rek>x, x 2 E:

Remark 1.2 [23–25] If A generates an a-times resolvent family Saðt, AÞ forthe case 1<a<2, then A is also the generator of ða, aÞ-times resolvent fam-ily Paðt, AÞ and

Paðt, AÞ ¼ ðga1 aÞðtÞ:

While, when 0<a<1, if A generates an analytic a-times resolvent family

Saðt, AÞ, then it is also the generator of analytic ða, aÞ-times resolventfamily

Paðt, AÞ ¼ 1

2pi

ð

CektRðka; AÞdk,and

đg1a aỡđtỡ Ử Sađt, Aỡ: (1.3)For Pađ, Aỡ, we have the following properties [24, 25]:

Pađt, Aỡx Ử gađtỡx ợ A

đt

0

gađtsỡPađs, Aỡxds, t>0, for any x 2 E,

APađt, Aỡx Ử Pađt, AỡAx, for any x 2 DđAỡ:

When 0<a<1, the following lemma holds:

Lemma 1.1 [25] Let A be the generator of analytic resolvent family Sađt, Aỡ

We have

(1) Pađt, Aỡ 2 BđEỡ and jjPađt, Aỡjj  Mextđ1 ợ ta1ỡ for any t > 0;

(2) For every x 2 E, Pađt, Aỡx 2 DđAỡ and jjAPađt, Aỡjj  Mextđ1 ợ t1ỡ,for any t> 0;

(3) S0ađt, Aỡ Ử APađt, Aỡ for any t > 0, RđPđlỡa đt, Aỡỡ DđAỡ for any integer

l  0 and jjAkPđlỡa đt, Aỡjj  Mextđ1 ợ tl1ađk1ỡỡ for any t > 0, where

Definition 1.5 A function uđỡ 2 Cb0đơ0, T; Eỡ is called a solution to (1)

in Cb0đơ0, T; Eỡ if it is a solution to this problem in Cđơ0, T; Eỡ, tions đDa

func-tuỡđỡ and Auđỡ are belonging to Cb0đơ0, T; Eỡ:

Obviously, if uđỡ is a solution to (1.1) in Cb0đơ0, T; Eỡ, then x 2 DđAỡand f đỡ 2 Cb0đơ0, T; Eỡ: Then we can define the well-posedness of theproblem (1.1) in Cb0đơ0, T; Eỡ as follows

Definition 1.6 The problem(1.1) is well-posed in Cb0đơ0, T; Eỡ, if:

1) For any f đỡ 2 Cb0đơ0, T; Eỡ and x 2 DđAỡ, there exists a unique tion uđtỡ Ử uđt; f đỡ, xỡ to (1.1)in Cb0đơ0, T; Eỡ;

solu-2) The operator uđt; f đỡ, xỡ is continuous as an operator from the space

Cb0đơ0, T; Eỡ DđAỡ to the space Cb0đơ0, T; Eỡ:

Here C0bđơ0, T; Eỡ DđAỡ is equipped with the norm

jjđf đỡ, xỡjjCb

0 đ 0, T ơ ;EỡDđAỡỬ jjf đỡjjCb

0 đ 0, T ơ ;Eỡợ jjxjjE:The semidiscrete approximation on the general discretization scheme ofthe problem (1.1) are the Cauchy problems in Banach spaces En:

đDa

tunỡđtỡ Ử Anunđtỡ ợ fnđtỡ, t 2 đ0, T; unđ0ỡ Ử xn: (1.4)The general approximation scheme, due to [26], can be described in thefollowing way Let En and E be Banach spaces and fpng be a sequence oflinear bounded operators: pn : E ! En, pn 2 BđE, Enỡ, n 2 N, with the prop-erty jjpnxjjEn ! jjxjjE as n ! 1 for any x 2 E:

Definition 1.7 The sequence of elements fxng, xn 2 En, n 2N, is said to

be P-convergent to x 2 E iff jjxnpnxjjEn ! 0 as n ! 1: We write this

as xn!Px:

Definition 1.8 The sequence of bounded linear operators Bn 2 BđEnỡ, n 2

N, is said to be PP-convergent to the bounded linear operator B 2 BđEỡ iffor every x 2 E and for every sequence fxng, xn 2 En, n 2N, such that

xn!P x, one has Bnxn!PBx: We write this as Bn!PPB:

The problem of convergence of solutions of semidiscrete approximation

đDatunỡđtỡ Ử Anunđtỡ, t 2 đ0, T; unđ0ỡ Ử xn,

to solution of problem (1.2) is completely solved by analogy of TheoremABC from Appendix [22, 23, 24, 25] The problem of convergence of solu-tions of (1.4) in Cđơ0, T; Enỡ to the solution of (1.1) in Cđơ0, T; Eỡ can also

be described by ABC TheoremỖs terminology using the conditions (A) and(B) We will address this issue in section 3

2 Necessary and sufficient condition for the well-posedness

in C0bđơ0, T; Eỡ

Obviously, the well-posedness of (1.2) in Cb0đơ0, T; Eỡ imply the ness of it in Cđơ0, T; Eỡ: Then A is the generator of an a-times resolventfamily Sađt, Aỡ, and the solution to (1.2) is Sađt, Aỡx: Furthermore, it fol-lows from the well-posedness of (1.2) in Cb0đơ0, T; Eỡ that

well-posed-jjSađ, AỡxjjCb

0 đ 0, T ơ ;Eỡ MjjxjjE: (2.1)

Lemma 2.1 Let A be a generator of analytic a-times resolvent family Forany 0<t<t ợ s  T and 0  b  1, one has the following inequalities:

jjSaðt þ s, AÞSaðt, AÞjjE!E M2

And, Saðt þ s, AÞSaðt, AÞ ¼Ðtþs

t APaðs, AÞds: From Lemma 1.1, weknow that jjAPaðt, AÞjj  M1ð1 þ t1Þ for any t>0, then

jjSaðt þ s, AÞSaðt, AÞjjE!E M1

We also have

jjSaðt þ s, AÞSaðt, AÞjjE!E 2M1  M2: (2.8)Interpolating (2.7) and (2.8), we obtain (2.2) It follows fromjjAP0

aðs, AÞjj  M1ð1 þ s2Þ for any s>0, that

jjAPaðt þ s, AÞAPaðt, AÞjjE!E ¼ jjÐtþs

jjAPaðt þ s, AÞAPaðt, AÞjjE!E M1

t þ sþM1

t 2M1

t M2

t : (2.10)Interpolating (2.9) and (2.10), we obtain (2.3) It follows fromjjPaðt, AÞjj  M1ð1 þ ta1Þ for any t>0 that

jjga aðt þ s, AÞga aðt, AÞjjE!E

jjPaðt þ s, AÞPaðt, AÞjjE!E¼ jj

where M is independent of b, u0 and f ðÞ:

In fact, it is such a strong condition that the operator A has a boundedinverse A1: When we consider the problem (2.11), it can be replaced bythe condition that the operator ðkIAÞ1 is bounded for some k: The latterone can be easily satisfied While, when it comes to the problem (1.1), suchcondition can not be replaced It means that we do not have the bounded-ness of the operator A1 in this paper Then

where M is independent of b,u0 and f đỡ:

We are going to show that the analyticity of Sađt, Aỡ is a necessary andsufficient condition for the well-posedness of (1) in Cb0đơ0, T; Eỡ:

Theorem 2.1 If the problem (1.1)is well posed in Cb0đơ0, T; Eỡ, then Sađt, Aỡ

is an analytic a-resolvent family

the strong continuity of Sađt, Aỡ, kaIA has a bounded inverse for all plex k with Rek>x: It means that for any u 2 E, kawAw Ử u has aunique solution w Ử đkaIAỡ1u: Clearly, the function uđtỡ Ử Sađt, kaỡw is

com-a solution in Cb0đơ0, T; Eỡ of (1) with f đtỡ Ử Sađt, kaỡuanduđ0ỡ Ử w:Actually, for such uđỡ and f đỡ, the coercivity inequality (2.12) providesthe following inequality

Clearly, jjSađ, kaỡjjCb

0 đơ0, T;Cỡ ! 1, as Rek ! 1: Together with w Ử

đkaIAỡ1u, we have, for sufficiently large x1>x and any kwith Rek>x1,jjka1đkaIAỡ1jjE!E M

jkj M jkxj: Then, Sađt, Aỡ is analytic wTheorem 2.2 Let A be the generator of an analytic a-times resolvent family.Then (1.1) is well posed in Cb0đơ0, T; Eỡ and the coercivity inequality

Firstly, let us consider the estimate of jjAuđỡjjCb

0 đơ0, T;Eỡ: We know thatwđtỡ 2 DđAỡ and jjAwđtỡjjEỬ jjSađt, AỡAxjjE  M1jjAxjjE, 0  t  T:Applying (2.2), we get that, for 0<t<t ợ s  T,

jjAwđt ợ sỡAwđtỡjjEỬ jjSađt ợ s, AỡAxSađt, AỡAxjjE  M2jjAxjjEsb

tb:Then,

bđ1  bỡ,

it follows that for 0  t  T,

Next, we shall estimate the difference Avðt þ sÞAvðtÞ, 0<t<t þ s  T:

We consider the cases t  2s and t>2s, separately When t  2s,

jjAvðt þ sÞAvðtÞjjE jjAvðt þ sÞjjEþ jjAvðtÞjjE 2M3

From (2.2) and the definition of jjf đỡjjCb

0 đơ0, T;Eỡ, we havejjI1jjEⱗ Mjjf jjCb

0 đ 0, T ơ ;Eỡsbtb:jjI2jjE

đ2sỡb

b :Because ts Ử t

2ợ t

2s>t

2, thenjjI2jjE M1jjf jjCb

đt

ts

dsđtsỡ1bỬ M1jjf jjCb

đts

0

sđtsỡ2bsbds Ử

đt 2

0

sđtsỡ2bsbds ợ

đtst 2

sđtsỡ2bsbds

jjAvjjCb

0 đ 0, T ơ ;Eỡⱗbđ1  bỡM jjf jjCb

0 đ 0, T ơ ;Eỡ:Hence,

Secondly, let us consider the estimate of jjujjCb

0 đơ0, T;Eỡ: We can easily getthe estimate jjwjjCb

đơ0, T;Eỡⱗ MjjxjjE: Since

 M1tajjf ðtÞjjEþ M1

ðt

0

dsðtsÞ1absbjjf jjCb

bð1  bÞ jjf jjCb0ð 0, T ½ ;EÞ M4

bð1  bÞjjf jjCb0ð 0, T ½ ;EÞ:Then we consider vðt þ sÞvðtÞ, 0<t<t þ s  T under the case t  2sand t>2s, separately For the case t  2s one has

Applying (2.4), we get that

jjI1jjE ¼ jjga aðt þ s, AÞðf ðt þ sÞf ðtÞÞ

þðga aðt þ s, AÞga aðt, AÞÞf ðtÞjjE

 jjga aðt þ s, AÞjjE!Ejjf ðt þ sÞf ðtÞjjEþjjga aðt þ s, AÞga aðt, AÞjjE!Ejjf ðtÞjjE

ðt

ts

dsðtsÞ1ab

jjI4jjE jjga ađt ợ s, Aỡga ađ2s, AỡjjE!Ejjf đtỡf đt ợ sỡjjE

 đjjga ađt ợ s, AỡjjE!Eợ jjga ađ2s, AỡjjE!Eỡjjf đtỡf đt ợ sỡjjE

 M1đđt ợ sỡaợ đ2sỡaỡjjf jjCb

0 đ 0, T ơ ;Eỡsbtbⱗ Mjjf jjCb

0 đ 0, T ơ ;Eỡsbtb:Putting b Ử 1 in (2.5) and using the method similar to what we used toestimate I5, we obtain

jjvjjCb

0 đ 0, T ơ ;Eỡⱗbđ1  bỡM jjf jjCb

0 đ 0, T ơ ;Eỡ:Hence,

3 The fulldiscrete approximation in Cđơ0, T; Eỡ

Assume that the functions fnđỡ 2 Cđơ0, T; Enỡ converge to the function

f đỡ 2 Cđơ0, T; Eỡ in the sense supt2ơ0,Tjjfnđtỡpnf đtỡjjEn ! 0 as n ! 1:Although Theorems 4.2 and 4.3 hold, when consider the full discretization,

we will impose stronger conditions on operators A, An: First of all, this isdue to the fact that a bounded linear perturbation A ợ B, for bounded B,remove the problem, in general, from the class of well-posed problems.Therefore, we assume that the operators A, An generate C0-semigroups Insuch situation, as was shown in [17], one has Pađt, Anỡxn!PPađt, Aỡx fort>0 as n ! 1, whenever xn!Px for any xn 2 En, x 2 E: Then under condi-tions đAỡ and đB1ỡ from Appendix, one can get the convergence of solu-tions of problems (1.4) to solution of problem (1.1) by major convergenceTheorem So in this section, we investigate approximation of problems

(1.4) and we consider the full discretization of the problem (1.1) by usingthe following implicit difference scheme [18]

Dat Unđỡ Ử AnUnđtkỡ ợ fnđtkỡ, tkỬ ksn, Unđ0ỡ Ử xn, (3.1)