On an initial inverse problem for a diffusion equation with a conformable derivative

R E S E A R C H

Open Access

On an initial inverse problem for a diffusion

equation with a conformable derivative

Tran Thanh Binh

_{, Nguyen Hoang Luc}

_{, Donal O’Regan}

_{and Nguyen H. Can}

*_{Correspondence:} [email protected]

4_{Applied Analysis Research Group,} Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Full list of author information is available at the end of the article

Abstract

In this paper, we consider the initial inverse problem for a diffusion equation with a conformable derivative in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

MSC: 35K05; 35K99; 47J06; 47H10

Keywords: Inverse problem; Diffusion equation; Conformable derivative

1 Introduction

In this paper we consider the following diffusion equation:

D0_γu(t, x) –Bu(t, x) = F(t, x), (t, x)∈(0,To)×Ω, (1)

subject to the boundary conditions

u(t, x) = 0, (t, x)∈(0,To)×∂Ω, (2)

and the initial condition

u(0, x) =u0(x), x∈Ω (3)

where the domainΩis a subset of ad-dimensional spaceRd_{(d= 1, 2, 3 is the dimension} ofΩ), which is a bounded domain with sufficient smooth boundary∂Ω, F is the source term,To> 0 is a fixed value, and D0γ is the conformable derivative [4,14,16].

Fractional differential equations are successful models of real life phenomenon and many authors studied fractional partial differential equations (see e.g. [34–36]). This gives one motivation to study and discuss some of the well known classical differential equa-tions, when some classical derivatives are replaced by fractional derivatives. One of the classical equations is the diffusion equation with the conformable derivative and because of the relationship between the conformable derivative and the classical derivative our equation could be considered as a modified classical diffusion equation.

Khalid et al. [16] introduced the conformable derivative and it was developed in [2–4,6– 9,31,33]. Applications of this derivative were given in [5,14,25,32]. In [22] the existence of solutions to conformable nonlinear differential equations with constant coefficients under mild conditions on the nonlinear term was discussed and in [29] the authors presented the iterative learning control for conformable differential equations. Recently, Machado et al. [1] and Baleanu et al. [15,30] mentioned the critical analysis of the conformable derivative. If the initial datau0 and the source term F are given, Problem (1) satisfying (2) and (3) is called the direct problem. The inverse problem for (1) is less well known. Inverse problems occur when we do not know all the given data. However, by adding some given data, we can discuss inverse problems such as the backward problem (recovering the initial data) or the source identification problem (recovering the source function). Initial inverse problems for fractional Riemann–Liouville or Caputo diffusion equations were discussed in the literature [20,26,27]. However, little is known on the initial inverse problem for the diffusion equation with a conformable derivative.

Motivated by the above, in this paper, we study the initial inverse problem of the diffusion equation with a conformable derivative (1) satisfying (2) and we reconstruct the initial data

g(x) =u(0, x) from the additional data

u(To, x) = h(x), x∈Ω. (4)

Note we cannot observe the data (h, F), so we only get approximate data (hε, Fε) such that

h – hε_L2_(Ω)+F – Fε_L∞_(0,To;L2_(Ω))≤ε, (5)

whereε> 0 is the noise level (in this paper we will also let · denote theL2(Ω) norm). We show that this problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the given data. Indeed, a small error of the given observation can result in that the solution may have a large error. Some regularization method is required for constructing stable approximations for a sought solution.

We use the Landweber method to find a regularized solution and the idea is based on iterative sequences. Using this method, some authors established a fractional method for solving some linear ill-posed models; see, for example, [13,23]. We will consider regu-larized solutions and regularity for the reguregu-larized solution. Also, we present an error estimate of the Landweber regularized solution to the exact solution under an a priori as-sumption using an a priori regularization parameter choice rule, which depends on the noise levelεand the a priori bound condition E of the unknown solution. That means the a priori choice of the regularization parameter depends on the a priori bound of the unknown solution. However, an a priori bound cannot be known exactly in practice, and working with an incorrect value may lead to a bad regularized solution. Therefore, we pro-vide an a posteriori choice of the regularization parameter. We also present a regularized problem and consider the well-posedness of the regularized solution and an error estimate under two parameter choice rules are considered.

2 Preliminaries

2.1 Some basic results

In this section, we introduce some spaces and some basic definitions associated with con-formable derivatives.

Definition 2.1(Conformable Derivative) Let f : [0,∞)→R. Then

D0_γf(s) =lim

ε→0

f(s +εs1–γ) – f(s)

ε , s > 0, (6)

is called the conformable derivative of f of orderγ∈(0, 1].

Some properties of the conformable derivative can be found in [2,4,14] and the refer-ences therein.

Consider the operator –BonL2(Ω) with domainD(–B)⊂H01(Ω)∩H2(Ω). Assume that –Bhas eigenvalues{am}satisfying

0 <a1≤a2≤a3≤ · · · ≤am≤ · · ·

andam→ ∞as m→ ∞with corresponding eigenfunctions em∈H01(Ω)∩H2(Ω). Now

⎧ ⎨ ⎩

Bem(x) = –amem(x), x∈Ω, em(x) = 0, x∈∂Ω,

and we note that there exists a positive constantCsuch thatam≥Cm

2

d _{for m}∈N_and m≥1, wheredis the dimension of the domainΩ; see [10].

For r≥0, consider the Hilbert scale space (see [21])

Hr_{(Ω) =}

v∈L2(Ω) :

∞

k=1

ar_m v, em 2< +∞

, (7)

with the norm

vHr_(Ω)= _∞

k=1

ar_m v, em 2

1 2

.

If r = 0, we haveH0(Ω) =L2(Ω).

For a given real number p≥1, letLp_(0,To;L2_{(Ω)) be the space of all functions such that}

vLp_(0,To;L2_(Ω)):= _To

0

_v(t)p L2_(Ω)dt

1 p

< +∞.

We give two lemmas, which will be needed later.

Lemma 2.1([19,28]) For0 <λ< 1,r> 0and k∈N,we have

Lemma 2.2 Foram> 0,γ> 0,α∈(1₂, 1]and0 <μexp(–2amT

γ

o

γ ) < 1,we get

sup am>0

exp

am Toγ

γ

1 –

1 –μexp

–2am Toγ

γ kα

≤μ1 2_k12_.

Proof First, we obtain

exp

amT γ o

γ

1 –

1 –μexp

–2amT γ o

γ kα

=μ12

μ1 2_exp

–am Toγ

γ –1

1 –

1 –μexp

–2am Toγ

γ kα

. (8)

Let

Ψ(ν) :=ν–21 –1 –ν2k2α,

where μ:=μ12exp(–a_mT

γ

o

γ ). We note that 0 <μ< K12 (see [18]), and this implies that

0 <μexp(–2amT

γ

o

γ ) < 1. Hence, this function is continuous in [0, +∞) whenμ∈(0, 1). Forα∈(1₂, 1) andμ∈(0, 1), from Lemma 3.3 in [18], we obtain

Ψ(ν)≤k. (9)

Combining (8) and (9), we deduce that

sup am>0

exp

amT γ o

γ

1 –

1 –μexp

–2amT γ o

γ kα

≤μ1 2_k12_.

2.2 Solution for the fractional diffusion equation with conformable derivative

Assume that Problem (1) satisfying (2) and (3) (i.e. the direct problem) has a solutionuas follows:

u(t, x) =

∞

m=1

u(t, x), em(x)

em(x).

Note

⎧ ⎨ ⎩

D0_γu(t, x), em(x)–amu(t, x), em(x)=F(t, x), em(x), (t, x)∈(0,To)×Ω, u(0, x), em(x)=u0(x), em(x), x∈Ω,

(10)

whereBu(t, x), em(x)= –amu(t, x), em(x). Using the result in [14] and [22], the solution of the latter problem is

u(t, x) =

∞

m=1

exp

–am tγ

γ

u0(x), em(x)

+

t

0

τγ–1exp

–am tγ–τγ

γ

F(τ, x), em(x)

dτ

Let t =Toand we get (with hm=h, emand Fm(τ) =F(τ, .), em)

hm(x) =exp

–amT γ o

γ

u0(x), em(x)

+

_To

0

τγ–1exp

–am Toγ–τγ

γ

Fm(τ)dτ.

This implies that

u(t, x) =

∞

m=1

exp

–am tγ–Toγ

γ

×

hm–

_To

0

τγ–1exp

–amT γ o –τγ

γ

Fm(τ)dτ

+

t

0

τγ–1exp

–am tγ–τγ

γ

Fm(τ)dτ

em(x). (11)

Let

A1

γ(t,To)v :=

∞

m=1

exp

–am tγ–Toγ

γ

v, emem,

A2 γ(t)v :=

∞

m=1

t

0

τγ–1exp

–am tγ–τγ

γ

v, emdτem,

for v∈L2(Ω), and 0≤τ≤t≤To. Then it follows from (11) that

u(t) =A1_γ(t,To)h –A2_γ(To)F(t)+A2_γ(t)F(t).

Recall for any n > 0, there exists a positive constantP1,n(from elementary calculus note we can takeP1,nto be nnexp(–n)) such that

exp(–z)≤P1,nz–n, z≥0 (12)

and if 0 < s < 1, there exists a positive constant P2,s (we can takeP2,sto be (1 –s)1–s×

exp(s– 1)) such that

exp(–z)≤P2,szs–1, z≥0. (13)

Lemma 2.3 Given0≤τ≤ToandΩ⊂Rdfor any1≤d≤3.

(a) If w∈L2_(Ω),thenA2

γ(t)w ∈L2(Ω)and

_A2

γ(t)wL2_(Ω)≤RwL2_(Ω) whereR= _∞

m=1 1 C2_m4d

1 2

.

(b) If w∈L2_{(Ω)for0 <n= 1,thenA}2

γ(t)w ∈Hn(Ω)and

_A2

γ(t)w2_Hn_(Ω)≤P1,nT γ(1–n) o

γ(1 –n)

t

0

(c) If w∈Hq+1_{(Ω)for0 <q< 1and0≤t}

Proof (a) Note

_A2

d_{, wheredis the dimension of the domainΩ, we know that}

∞

Combining (14), (15) and (16), we deduce that

≤P1,n

where we have noted that

t

and this together with Hölder’s inequality yields

This implies that

We note that

t1

Using Hölder’s inequality, we have

D₂(m, t,γ)2_L2_(Ω)

From the above results, we get

_A2

2.3 Ill-posedness of determining initial data and stability estimate

Recall that the solutionu(t, x) of Problem (4) (i.e. the inverse problem) is given by (11). Let

where

Υm=Υ, em, withΥ= h –A2γ(To)F(τ). (18)

In this paper, our main purpose is to determine the initial valueg(x) from the final data h(x) and the source term F(t, x). We can findg(x) by solving an operator equation as follows:

Kg=Υ,

whereK:L2_(Ω)→L2_{(Ω) is the integral operator defined by}

(Kg)(x) :=

∞

m=1

exp

–am Toγ

γ

g, emem=

Ω

(ξ, x)g(ξ)dξ,

with kernel(·,·) given by

(ξ, x) :=

∞

m=1

exp

–amT γ o

γ

em(ξ)em(x).

Since(ξ, x) =(x,ξ), we know that the operatorKis self-adjoint. Assume thatΥ ∈L2(Ω).

Lemma 2.4 Let h∈L2(Ω)and F∈L∞(0,To;L2(Ω)).ThenΥ as in(18)belongs to L2(Ω) and

ΥL2_(Ω)≤ hL2_(Ω)+RFL∞_(0,To;L2_(Ω)).

Proof From the definition ofΥ, we get

ΥL2_(Ω)≤h –A2_γ(To)F(τ)

L2_(Ω)

≤ hL2_(Ω)+A2_γ(T_o)F(τ)

L2_(Ω).

Using Lemma2.3, we deduce that

_A2

γ(To)F(t)L2_(Ω)≤RF(t)_L2_(Ω)≤RFL∞(0,To;L2(Ω)). (19)

Therefore

ΥL2_(Ω)≤ hL2_(Ω)+RFL∞_(0,To;L2_(Ω)).

This completes the proof.

Therefore,K:L2_(Ω)→L2_{(Ω) is compact operator of infinite rank. HenceKdoes not} have a continuous inverse [24].

To illustrate the ill-posedness of the backward problem, we give an example. Let (h, F) = (0, 0) and (h, F) = (√1

al

el,√1 al

el). It is easy to see that

h – h=√1

al

, and F – F=√1

Hence

lim

l→∞h – h= 0, and llim→∞F – F= 0, (20)

so (h, F) is an approximation of (h, F) when l is large enough. Using (h, F), we get the cor-responding initial datagand the equationΥ as follows:

Υ(x) =

From Parseval’s equality and (15), we get

Υ–Υ2=

On the other hand, we have

g–g2

is,

∞

m=1

exp

2ram Toγ

γ g, em 2

≤E2, for any r≥0, (22)

where E is a positive constant. The a priori bound of the exact solution is necessary for any ill-posed problem, otherwise, the rate of convergence is very slow or the regularization solution is not convergent (see [12]).

3 Landweber regularization method and regularity of the regularized solution In this section, we present a regularized problem by using the Landweber regularization method, and also we consider the well- posedness of the regularized solution. From [17], the operator equationKg=Υ is equivalent to the following equation:

g=I–μK∗Kg+μK∗Υ, (23)

for anyμ> 0. Here,K∗is the adjoint operator ofK, andμ> 0 satisfies 0 <μ< _K12. The

iterative implementation of the Landweber method was constructed in [18]. Denote the Landweber regularization solution by

gk,α(x) =

∞

m=1

exp

am Toγ

γ

1 –

1 –μexp

–2am Toγ

γ kα

Υ, emem, (24)

and the Landweber regularization solution with noisy data by

gε_k,α(x) =

∞

m=1

exp

amT γ o

γ

1 –

1 –μexp

–2amT γ o

γ

kα Υε, em

em, (25)

where α∈(1₂, 1] is called the fractional parameter, and k = 1, 2, 3, . . . is a regularization parameter. Whenα= 1, this is the classical Landweber method.

Hence, we get the Landweber regularization solution of Problem (4) (i.e. the inverse problem):

uk,α(t, x) =

∞

m=1

exp

–am tγ–Toγ

γ

×

1 –

1 –μexp

–2amT γ o

γ kα

Υ, em

+

t

0

τγ–1exp

–am tγ–τγ

γ

Fm(τ)dτ

em(x). (26)

Let us consider the operator

A3

γ(t,To)v :=

∞

m=1

exp

–am tγ–Toγ

γ

×

1 –

1 –μexp

–2am Toγ

γ kα

v, emem,

Lemma 3.1 Givenμ> 0and k∈Nwith k≥1.

Proof (a) First, using Lemma2.2, we obtain

(c) From the definition ofA3

Using Lemma2.2, we get

_A3

From the above results, we deduce that

_A3

The Landweber regularization solution of Problem (4) can be transformed into the form

uk,α(t) =A3γ(t,To)h –A2γ(To)F(t)+A2γ(t)F(t). (27)

and the Landweber regularization solution of Problem (4) with noisy data:

uε_k,α(t) =A3_γ(t,To)hε–A2_γ(To)Fε(t)+A2_γ(t)Fε(t)

=Ξ1(t) +Ξ2(t) +Ξ3(t). (28)

Theorem 3.1

(a) Let hε∈L2_{(Ω)and F∈L}∞_(0,To;L2_{(Ω)).IfΩ⊂R}d_{for any1≤d≤3then}

uε

k,αL∞(0,To;L2_(Ω))≤μ 1 2_k12_hε

L2_(Ω)+

μ1

2_k12 _{+ 1}R_Fε

L∞(0,To;L2_(Ω)).

(b) Given0 <n= 1and1 <p<min(_nγ2,_1–1_γ).Let hε∈L2_(Ω)and

F∈L

2p

p–1_(0,T

o;L2(Ω))∩L∞(0,To;L2(Ω)).Then there existsMdepends onlyTo,p,n,

μ,k and(the regularized solution)uε_k,α∈Lp(0,To,Hn(Ω))∩L∞(0,To;L2(Ω))such that

uε k,αLp_(0,T

o,Hn(Ω))≤Mh ε

L2_(Ω)+Fε_L∞_(0,To;L2_(Ω))+Fε

L

2p

p–1_(0,T

o;L2(Ω))

,

(c) Let hε∈Hs(Ω)for0 <s< 1and F∈L

2p

p–1_(0,T_o;Hq+1_{(Ω))for1 <p<min(}2

γ,1–1γ)and 0 <q< 1.Thenuε_k,α∈C([0,To];L2(Ω)).

Proof (a) Since hε∈L2_{(Ω) so part (a) of Lemma3.1yields}

Ξ1(t)_L2_(Ω)=Aγ3(t,To)hε_L2_(Ω)≤μ 1 2_k12_hε

L2_(Ω).

By a similar method, we obtain

Ξ2(t)_L2_(Ω)=A3γ(t,To)

A2

γ(To)Fε(t)L2_(Ω)≤μ 1

2_k12A2_γ(T_o)Fε_(t)

L2_(Ω).

Assume F∈L∞(0,To;L2(Ω)). Now

Fε(t), em 2

≤ess sup

0≤t≤To

∞

m=1

Fε(t), em 2

=Fε2_L∞_(0,To;L2_(Ω)).

Using Lemma2.3, we deduce that

_A2

γ(To)Fε(t)L2_(Ω)≤RFε(t)_L2_(Ω)≤RFε_L∞_(0,To;L2_(Ω)). (29)

Hence

Ξ2(t)_L2_(Ω)≤μ 1

2_k12R_Fε

L∞(0,To;L2_(Ω)).

On the other hand

Ξ3(t)_L2_(Ω)=A2γ(t)Fε(t)_L2_(Ω)≤RFε_L∞_(0,To;L2_(Ω)).

Combining the above results, we get

uε_k,α(t)_L2_(Ω)=Ξ1(t)(t)_L2_(Ω)+Ξ2(t)_L2_(Ω)+Ξ3(t)_L2_(Ω)

≤μ1 2_k12_hε

L2_(Ω)+

μ1

2_k12 _{+ 1}R_Fε

L∞(0,To;L2(Ω)). (30)

(b) Since hε∈L2_{(Ω) so Lemma3.1with 0 < n yields}

By a similar method, we obtain

Ξ2(t)_Hn_(Ω)=A3γ(t,To)

From (29) we obtain

where

M2(μ, k, p, n) =μ

1 2_k12P

1 2

1,nR

2

γ –n₂

2 2 – pnγ

1 p

T 2–pn2pγ

o .

For n= 1 using Lemma2.3we have

Ξ3(t)_Hn_(Ω)=A2γ(t)Fε(t)_Hn_(Ω)

≤

_P

1,n

γ(1 – n)

1 2

T γ(1–n)2

o

t

0

τγ–1Fε(τ)2_L2_(Ω)dτ 1

2

.

Hölder’s inequality gives

t

0

τγ–1Fε(τ)2_L2_(Ω)dτ= t

0

τp(γ–1)dτ

1 p t

0

Fε(τ)

2p p–1

L2_(Ω)dτ p–1

p

≤ T

γ–1+1p

o

(γp – p + 1)p1 Fε2

L

2p

p–1_(0,To;L2_(Ω)).

From the above results we have

Ξ3(t)_Hn_(Ω)≤ _P

1,n

γ(1 – n)

1 2

T γ(1–n)2

o

T (γ–1)p+12p

o

(γp – p + 1)2p1 Fε

L

2p

p–1_(0,To;L2_(Ω)).

Hence

Ξ3Lp_(0,To,Hn_(Ω))

≤

_P

1,n

γ(1 – n)(γp – p + 1)1p 1

2

Tγ–γn+12 +2p3

o Fε L

2p p–1_(0,T

o;L2_(Ω))

≤M3(μ, k, p, n)Fε L

2p p–1_(0,T

o;L2(Ω))

, (33)

where

M3(μ, k, p, n) =

_P

1,n

γ(1 – n)(γp – p + 1)1p 1

2

Tγ–

γn+1 2 +2p3

o .

Combining (31), (32) and (33) we get

uε

k,αLp_(0,To,Hn_(Ω))≤Mh

ε

L2_(Ω)+Fε_L∞_(0,To;L2_(Ω))+Fε

L

2p p–1_(0,T

o;L2(Ω))

,

where

M=maxMi(μ, k, p, n) :i= 1, 3

(c) We obtain

uε

k,α(t +η) –uεk,α(t)L2_(Ω)≤Ξ1(t +η) –Ξ1(t)_L2_(Ω)

+Ξ2(t +η) –Ξ2(t)_L2_(Ω)+Ξ3(t +η) –Ξ3(t)_L2_(Ω).

Now part (c) of Lemma3.1yields

Ξ1(t +η) –Ξ1(t)_L2_(Ω)=A3γ(t +η,To) –A3γ(t,To)

hε_L2_(Ω)

≤μ1 2_k

1 2P2,s

sγs

(t +η)γ– tγshε_Hs_(Ω).

We use the inequality (a1+a2)ϑ≤aϑ1 +aϑ2 for 0 <ϑ≤1 to get

(t +η)γ– tγ≤ηγ, 0 <γ≤1. (34)

Therefore

Ξ1(t +η) –Ξ1(t)_L2_(Ω)≤μ 1 2_k12P2,s

sγsη γs_hε

Hs_(Ω).

Applying a similar method gives

Ξ2(t +η) –Ξ2(t)_L2_(Ω)=A3γ(t +η,To) –A3γ(t,To)

A2

γ(To)Fε(t)L2_(Ω)

≤μ1 2_k12P2,s

sγsη γs_A2

γ(To)Fε(t)_Hs_(Ω).

Since 0 < s < 1 using Lemma2.3we get

_A2

γ(To)Fε(t)_Hs_(Ω)≤ _P

2,s

γ(1 – s)

1 2

T γ(1–s)2

o

_To

0

τγ–1Fε(τ)2_L2_(Ω)dτ 1

2

≤

_P

2,s

γ(1 – s)(γp – p + 1)1p 1

2

Tγ–γ

s+1 2 +2p1

o Fε 2

L

2p p–1_(0,T

o;L2_(Ω)).

This implies that

Ξ2(t +η) –Ξ2(t)_L2_(Ω)

≤μ1 2_k

1 2P2,s

sγsη γs

_P

2,s

γ(1 – s)(γp – p + 1)1p 1

2

Tγ–γ

s+1 2 +2p1

o Fε 2

L

2p

p–1_(0,To;L2_(Ω)).

Using Lemma2.3and 0 < q < 1 so we have

Ξ3(t +η) –Ξ3(t)_L2_(Ω)

≤

P2,q|(t +

η)γ– tγ|2

γq+1

Tqγ o qγ

t

0

τγ–1Fε(τ)2_Hq–2κ+1_(Ω)dτ 1

2

+

(t +η)γ– tγ

γ

t+η

t

τγ–1Fε(τ)2_L2_(Ω)dτ 1

2

. (35)

Apply Hölder’s inequality and we have

t

0

τγ–1Fε(τ)2_Hq+1_(Ω)dτ ≤ t

0

τp(γ–1)dτ

1 p t

0

Fε(τ)

2p p–1

Hq+1_(Ω)dτ p–1

p

≤ T

γ–1+1_p o

(γp – p + 1)1p Fε2

L

2p p–1_(0,T

o;Hq+1(Ω))

(36)

and

t+η

t

τγ–1Fε(τ)2_L2_(Ω)dτ≤ t+η

t

τp(γ–1)dτ

1 p t+η

t

Fε(τ)

2p p–1

L2_(Ω)dτ p–1

p

≤

_To

0

τp(γ–1)dτ

1 p To

0

Fε(τ)

2p p–1

L2_(Ω)dτ p–1

p

≤ T

γ–1+1p

o

(γp – p + 1)1p Fε2

L

2p

p–1_(0,To;L2_(Ω)). (37)

Combining (34), (35), (36) and (37) and we get

Ξ3(t +η) –Ξ3(t)_L2_(Ω)≤

P2,q η2γ

γq+2

Tqγ+γ–1+1p

o

q(γp – p + 1)1p 1

2

Fε L

2p

p–1_(0,To;Hq+1_(Ω))

+

ηγ

γ

Tγ–1+p1

o

(γp – p + 1)1p 1

2

Fε L

2p p–1_(0,T

o;L2(Ω)) .

Therefore, we conclude thatuε_k,α∈C([0,To];L2_(Ω)).

4 Convergence analysis and error estimate under two parameter choice rules In this section, we choose a regularization parameter k := k(ε) such thatu–u_k,ε_αL2_(Ω)→0

asε→0, and we also consider the convergence analysis between the regularized solution

uε_k,αand the exact solutionu.

4.1 The a priori parameter choice

Theorem 4.1 Let h∈L2(Ω)and F∈L∞(0,To;L2(Ω)).Assume the a priori bound condi-tion(22)holds.If we choose the regularization parameter

k=

E

ε 2

r+1

then we get the following error estimate between the exact solution and its regularization solution with noisy data:

u–uε_k,αL2_(Ω)≤μ–

r

2

r 2

r

2

Er+11 εr+1r ₊μ12_{(1 +}R_)Er+11 εr+1r ₊εR_,

wherekdenotes the largest integer less than or equal to k.

Proof From the triangle inequality, we obtain

u–uε_k,α

L2_(Ω)≤ u–uk,αL2_(Ω)+u_k,α–uε_k,α

L2_(Ω). (38)

Apply part (a) of Theorem3.1and we get

uk,α–uεk,αL2_(Ω)≤uk,α–uεk,αL∞(0,To;L2(Ω))

≤μ1

2_k12_{h – h}ε

L2_(Ω)+

μ1

2_k12 _{+ 1}R_{F – F}ε

L∞(0,To;L2(Ω))

≤μ1 2_k

1

2ε_{(1 +}R) +εR. ₍₃₉₎

On the other hand, we have

u–uk,αL2_(Ω)=A1_γ(t,To) –A3_γ(t,To)Υ

L2_(Ω).

Noteα∈(1₂, 1] and 0 <μ<_K12, so it follows that

u–uk,αL2_(Ω)

=

∞

m=1

exp

–am tγ–Toγ

γ

1 –

1 –

1 –μexp

–2am Toγ

γ

kα

× Υ, emem

L2_(Ω)

≤

∞

m=1

exp

–am tγ–Toγ

γ

1 –μexp

–2am Toγ

γ k

Υ, emem

L2_(Ω)

.

From the definition ofΥ in (17), we get

u–uk,αL2_(Ω)≤

∞

m=1

exp

–am tγ

γ

1 –μexp

–2amT γ o

γ k

g, emem

L2_(Ω)

≤μ–₂r _sup

am>0

1 –μexp

–2am Toγ

γ k

μexp

–2am Toγ

γ r

2

×

∞

m=1

exp

2ramT γ o

Apply Lemma2.1and we obtain

u–uk,αL2_(Ω)≤μ– r 2

r 2

r 2

k–r2_{E. (40)}

Combining (38), (39) and (40) we obtain

u–uε_k,αL2_(Ω)≤μ– r 2

r 2

r 2

k–r2_E+μ21_k12ε_{(1 +}R) +εR.

Choosing the regularization parameter k,

k =

E ε

2 r+1

,

we obtain the error estimate

u–uε_k,α

L2_(Ω)≤μ– r 2

r 2

r 2

Er+11 εr+1r ₊μ12_{(1 +}R)Er+11 εr+1r ₊εR.

4.2 A posteriori parameter choice rule and convergence estimate

In the above result, we obtained an error estimate between the exact solution and its regu-larization solution with noisy data by choosing the a priori parameter k, and this k depends on the noise levelεand the a priori bound condition E. Now, from results in Morozov’s discrepancy principal [11], we choose the regularization parameter k by using an a poste-riori choice rule.

The general a posteriori rule can be formulated as follows:

_Kgε_k,α–Υε_L2_(Ω)≤χε≤Kgε_k–1,α–Υε_L2_(Ω), (41)

whereΥεL2_(Ω)≥χε,χ is a constant independent ofεand k > 0 is the regularization

parameter which makes (41) hold at the first iteration time.

Choosingχ> 1 the following lemma gives a bound for k in terms ofεand E.

Lemma 4.1 Letχ> 1and k satisfies(41).Also assume the a priori bound condition ofg

satisfies(22).Then

k≤r– 1 2μ

1 χ–R– 1

2

r–1_E ε

2

r–1

.

Proof From the definition of k we get

_Kgε_k–1,α–Υε L2_(Ω)

=

∞

m=1

1 –

1 –

1 –μexp

–2am Toγ

γ

k–1α Υε, em

em

≤

Apply Lemma2.1and we obtain

Apply Lemma2.4and we obtain

_Kgε_k–1,α–Υε_L2_(Ω)≤hε– h_L2_(Ω)+RFε– F_L∞_(0,To;L2_(Ω))

This implies that

χε≤(1 +R)ε+μ–r+12

Proof Using the triangle inequality, we obtain

u–uε_k,αL2_(Ω)≤ u–uk,αL2_(Ω)+u_k,α–uε_k,α

L2_(Ω).

Apply the result of Theorem4.1and we obtain

uk,α–uεk,αL2_(Ω)≤μ

Apply Hölder’s inequality and we have

u–uk,αL2_(Ω)

This implies that

From the above we deduce that

u–uε_k,αL2_(Ω)≤μ 1 2_{(1 +}R)

r + 1 2μ

1

2 ₁

χ–R– 1

1 r+1

Er+11 εr+1r

+εR+ (1 +R+χ)r+1r _Er+11 εr+1r _.

Funding

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Author details

1_{Faculty of Natural Sciences, Thu Dau Mot University, Thu Dau Mot City, Vietnam.}2_{Institute of Research and}

Development, Duy Tan University, Da Nang, Vietnam. 3_{School of Mathematics, Statistics and Applied Mathematics,}

National University of Ireland, Galway, Ireland.4_{Applied Analysis Research Group, Faculty of Mathematics and Statistics,}

Ton Duc Thang University, Ho Chi Minh City, Vietnam.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 2 October 2019 Accepted: 6 November 2019

References

1. Abdelhakim, A.A., Machado, J.A.T.: A critical analysis of the conformable derivative. Nonlinear Dyn.95(4), 3063–3073 (2019)

2. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math.279, 57–66 (2015)

3. Alharbia, F.M., Baleanu, D., Abdelhalim, E.: Physical properties of the projectile motion using the conformable derivative. Chin. J. Phys.58, 18–28 (2019)

4. Atangana, A., Baleanu, D., Alsaedi, A.: New properties of conformable derivative. Open Math.13, 889–898 (2015) 5. Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W.: Three-point boundary value problems for conformable fractional

differential equations. J. Funct. Spaces2015, Article ID 706383 (2015)

6. Bouaouid, M., Hilal, K., Melliani, S.: Nonlocal telegraph equation in frame of the conformable time-fractional derivative. Adv. Math. Phys.2019, Article ID 7528937 (2019)

7. Çenesiz, Y., Baleanu, D., Kurt, A., Tasbozan, O.: New exact solutions of Burgers’ type equations with conformable derivative. Waves Random Complex Media27(1), 103–116 (2017)

8. Çenesiz, Y., Kurt, A., Nane, E.: Stochastic solutions of conformable fractional Cauchy problems. Stat. Probab. Lett.124, 126–131 (2017)

9. Chung, W.S.: Fractional Newton mechanics with conformable fractional derivative. J. Comput. Appl. Math.290, 150–158 (2015)

10. Courant, R., Hilbert, D.: Methods of Mathematical Physics: Partial Differential Equations. Wiley, New York (2008) 11. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, vol. 375. Springer, Berlin (1996) 12. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic, Boston (1996) 13. Hochstenbach, M.E., Reichel, L.: Fractional Tikhonov regularization for linear discrete ill-posed problems. BIT Numer.

Math.51(1), 197–215 (2011)

14. Jaiswal, A., Bahuguna, D.: Semilinear conformable fractional differential equations in Banach spaces. Differ. Equ. Dyn. Syst.27(1–3), 313–325 (2019)

15. Jarad, F., Adjabi, Y., Baleanu, D., Abdeljawad, T.: On defining the distributionsδr_{and (δ)}r_{by conformable derivatives.}

Adv. Differ. Equ.2018, 407 (2018)

16. Khalil, A.R., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math.264, 65–70 (2014)

17. Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problem. Springer, Berlin (1996)

18. Klann, E., Ramlau, R.: Regularization by fractional filter methods and data smoothing. Inverse Probl.24(2), 025018 (2008)

19. Louis, A.K.: Inverse Und Schlecht Gestellte Probleme. Springer, Berlin (2013)

20. Luc, N.H., Huynh, L.N., Tuan, N.H.: On a backward problem for inhomogeneous time-fractional diffusion equations. Comput. Math. Appl. To appear

21. McLean, W., McLean, W.C.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University press, Cambridge (2000)

22. Mengmeng, L., Wang, J., O’Regan, D.: Existence and Ulam’s stability for conformable fractional differential equations with constant coefficients. Bull. Malays. Math. Soc. To appear

24. Nair, M.T.: Linear Operator Equation: Approximation and Regularization. World Scientific, Singapore (2009) 25. Tariboon, J., Ntouyas, K.S.: Oscillation of impulsive conformable fractional differential equations. Open Math.14,

497–508 (2016)

26. Tuan, N.H., Hoan, L.V., Tatar, S.: An inverse problem for an inhomogeneous time-fractional diffusion equation: a regularization method and error estimate. Comput. Appl. Math.38(2), 32 (2019)

27. Tuan, N.H., Huynh, L.N., Ngoc, T.B., Zhou, Y.: On a backward problem for nonlinear fractional diffusion equations. Appl. Math. Lett.92, 76–84 (2019)

28. Vainikko, G.M., Veretennikov, A.Y.: Iteration Procedures in Ill-Posed Problems. Nauka, Moscow (1986) (in Russian) 29. Wang, X., Wang, J.R., Shen, D., Zhou, Y.: Convergence analysis for iterative learning control of conformable fractional

differential equations. Math. Methods Appl. Sci.41(17), 8315–8328 (2018)

30. Yusuf, A., Inc, M., Aliyu, A.I., Baleanu, D.: Solitons and conservation laws for the (2 + 1)-dimensional Davey–Stewartson equations with conformable derivative. J. Adv. Phys.7(2), 167–175 (2018)

31. Yusuf, A., Inc, M., Aliyu, A.I., Baleanu, D.: Conservation laws, soliton-like and stability analysis for the time fractional dispersive long-wave equation. Adv. Differ. Equ.2018, 319 (2018)

32. Zhong, W., Wang, L.: Basic theory of initial value problems of conformable fractional differential equations. Adv. Differ. Equ.2018, 321 (2018)

33. Zhou, H.W., Yang, S., Zhang, S.Q.: Conformable derivative approach to anomalous diffusion. Phys. A, Stat. Mech. Appl. 491, 1001–1013 (2018)

34. Zhou, Y.: Attractivity for fractional evolution equations with almost sectorial operators. Fract. Calc. Appl. Anal.21(3), 786–800 (2018)

35. Zhou, Y., He, J.W., Ahmad, B., Tuan, N.H.: Existence and regularity results of a backward problem for fractional diffusion equations. Math. Methods Appl. Sci., 1–16 (2019).https://doi.org/10.1002/mma.5781