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Articles

The inverse source problem for time-fractional diffusion equation: stability analysis and regularization

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Pages 969-996 | Received 27 Aug 2013, Accepted 12 Sep 2014, Published online: 16 Oct 2014

Abstract

In the present paper, we consider an inverse source problem for a fractional diffusion equation. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. Based on an a priori assumption, we give the optimal error bound analysis and a conditional stability result. Moreover, we use the Fourier regularization method to deal with this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Meanwhile, a new a posteriori parameter choice rule is also proposed. For the a priori and the a posteriori regularization parameters choice rules, we all obtain the convergence error estimates which are all order optimal. Numerical examples are presented to illustrate the validity and effectiveness of this method.

AMS Subject Classifications:

1 Introduction

In recent years, a lot of attention has been devoted to the study of fractional differential equations. Fractional derivative calculus and fractional differential equations have been used recently to describe a range of problems in physical, chemical, biology, mechanical engineering, signal processing and systems identification, electrical, control theory, finance and fractional dynamics, refer to [Citation1Citation3]. One of the major advantages of the fractional derivatives is that they can be considered as a super set of integer-order derivatives. Thus, fractional derivatives have the potential to accomplish what integer-order derivatives cannot do.[Citation4] A history of the development of fractional differential operators can be found in [Citation5, Citation6]. Time-fractional diffusion equation is deduced by replacing the standard time derivative with a time-fractional derivative and can be used to describe the superdiffusion and subdiffusion phenomena.[Citation7] As pointed out in [Citation8], many anomalous diffusion phenomena have been observed which show different aspects from the classical diffusion. For example, Adams and Gelhar [Citation9] pointed out that field data in the saturated zone of a highly heterogeneous aquifer are not well simulated by the classical advection–diffusion equation which is based on the random walk, and the data indicate slower diffusion than the classical one. The slow diffusion is characterized by the long-tailed profile in the spatial distribution of densities as the time passes; see also Zhou and Selim [Citation10]. Such slow diffusion is called the anomalous diffusion. Since [Citation9] there have been many studies for better models, and from the practical viewpoint, the anomalous diffusion is seriously concerned, e.g. with the quantitative environmental problems such as evaluation of underground contaminants. In particular, Berkowitz et al. [Citation11] and Hatano and Hatano [Citation12] have applied the continuous-time random walk to the underground environmental problem. The direct problems, i.e. initial value problems and initial boundary value problems for time-fractional diffusion equation have been studied extensively in recent years.[Citation13Citation18] However, in some practical situations, part of boundary data, or initial data, or diffusion coefficients, or source term may not be given and we want to find them by additional measured data which will yield to some fractional diffusion inverse problems. Such as in [Citation19], the authors used the optimal regularization method to solve a fractional order backward heat conduction problem in two-dimensional space.

In this paper, we first consider the following inverse source problems of determining the unknown source term f(t), in the following time-fractional diffusion equation:1.1 0+αu(x,t)-uxx(x,t)=f(t),x>0,t>0,u(x,0)=0,x0,u(0,t)=0,t0,u(x,t)|xbounded,t0,u(1,t)=g(t),t0,1.1 where the time-fractional derivative 0+αu(x,t) is the Caputo fractional derivative of order α(0<α1) defined by [Citation1]1.2 0+αu(x,t)=1Γ(1-α)0tu(x,s)sds(t-s)α,0<α<1,1.2 1.3 0+αu(x,t)=u(x,t)t,α=1,1.3 f(t) denotes the source (sink) term. Our purpose is to identify f(t) from the additional data u(1,t)=g(t). Since the data g(t) is based on (physical) observation, there must be measurement errors, and we assume the measured data function gδ(t)L2(R), and satisfies1.4 g-gδδ,1.4 where · denotes L2 -norm, the constant δ>0 represents a noise level.

For α=1, the inverse source problem for standard diffusion equation has been investigated in many articles for different forms of heat source.[Citation20Citation28] To the authors’ knowledge, there were few papers for identifying an unknown source in a fractional diffusion equation by regularization method. In [Citation29], using the analytic continuation and Laplace transform, the authors proved the uniqueness of the identification of the unknown source dependent only on spatial variable for the fractional diffusion equation in a bound domain. In [Citation30], using the coupled method, the authors identified the unknown source for the spatial fractional diffusion equations. In [Citation31], the authors used an optimal perturbation regularization algorithm to identify the unknown source which depends only on the spatial variable for the spatial fractional diffusion equations. In [Citation32], the authors used the boundary element method combined with a generalized Tikhonov regularization method to identify the unknown source which depends only on the time variable for the time-fractional diffusion equation. In [Citation33], the authors used the Fourier method to identify the unknown source which depends only on the spatial variable for the space-fractional diffusion equation, but the regularization parameter is a priori choice rule.

It is well known that the Fourier regularization method is well studied and being widely used as regularization methods in many ill-posed problems. So far, this method has been successfully applied to the inverse heat conduction problems,[Citation34Citation36] the Cauchy problems for the Laplace equation and Helmholtz equation,[Citation37Citation39] the backward heat equation,[Citation40] the numerical differentiation,[Citation41] the numerical analytic continuation [Citation42] and the identification problems for the unknown heat source functions.[Citation43, Citation44]

However, many above works focus on a priori choice of the regularization parameter. Generally speaking, there is a defect in any a priori methods; i.e. the a priori choice of the regularization parameter depends obviously on the a priori bound E of the unknown solution. But the a priori bound E cannot be known exact in practice, and working with a wrong constant E may lead to the bad regularization solution. In this paper, one of our purposes is that we consider the a posteriori choice of the regularization parameter for the Fourier regularization method. The other purpose is that we prove an optimal error bound with Höder type for the identification in a special ‘source condition’. For a priori and a posteriori choices of the regularization parameter, we obtain the Hölder type stability estimates which are order optimal. Moreover, the comparison of numerical effect between a priori and a posteriori methods is provided.

The problem is ill-posed in the sense of Hadamard, i.e. small changes in the measured data can blow up in the solution. The ill-posedness can be seen by solving the problem in the frequency domain. In order to analyse the problem (Equation1.1) in L2(R), we define all functions to be zero for t<0. The notation · denotes L2-norm, and1.5 f^(ξ):=12π-e-iξtf(t)dt1.5 is the Fourier transform of the function f(t).

The problem (Equation1.1) can now be formulated in frequency space as follows:1.6 (iξ)αu^(x,ξ)-u^xx(x,ξ)=f^(ξ),ξR,x>0,u^(0,ξ)=0,ξR,u^(x,ξ)|xbounded,ξR,u^(1,ξ)=g^(ξ),ξR.1.6 By elementary calculations, we get1.7 f^(ξ)=(iξ)α1-e-(iξ)α2g^(ξ),1.7 i.e.1.8 f(t)=12π-eiξt(iξ)α1-e-(iξ)α2g^(ξ)dξ.1.8 From the right side of (Equation1.7) or (Equation1.8), we know1.9 (iξ)α1-e-(iξ)α2=|ξ|α1-2e-|ξ|α2cos(π4α)cos(|ξ|α2sin(π4α))+e-2|ξ|α2cos(π4α),as|ξ|.1.9 Therefore, when we consider our problem in L2(R), the exact data function g^(ξ) must decay. However, the measured data function gδ(t), which is merely in L2(R), does not possess such a decay property in general. Thus, if we try to obtain the unknown source f(t), high-frequency components in the error are magnified and can destroy the solution. It is impossible to solve the problem (Equation1.1) by using the classical method. In the following section, we will use the Fourier regularization method to deal with the ill-posed problem. Before doing that, we impose an a priori bound on the input data, i.e.1.10 f(·)pE,p>0,1.10 where E>0 is a constant, ·p denotes the norm in sobolev space Hp(R) defined by1.11 f(·)p:=-|f^(ξ)|2(1+ξ2)pdξ12.1.11 The outline of the paper is as follows. Section 2 gives the optimal error bound for problem (Equation1.1). In Section 3, we prove the conditional stability estimate for problem (Equation1.1). The order-optimal error estimates are obtained for the a priori and the a posteriori parameter choice rules in Section 4. In Section 5, some numerical examples are proposed to show the effectiveness of this method. In Section 6, we use the Fourier regularization method to identify the unknown source which is separable variable, i.e. f(x,t)=λ(x)f(t). Section 7 puts an end to this paper with a brief conclusion.

2 Preliminary result and optimal error bound for problem (1.1)

2.1 Preliminary result

Consider an arbitrary ill-posed operator equation [Citation45Citation49]:2.1 Ax=y2.1 where AL(X,Y) is a linear bounded between infinite-dimensional Hilbert spaces X and Y with non-closed range R(A) of A. Assume that yδY are available noisy data with y-yδδ. Any operator R:YX can be considered as a special method for approximately solving (Equation2.1), and the approximate solution of (Equation2.1) is given by Ryδ.

Let MX be a bounded set. Let us introduce the worst case error (δ,R) for identifying x from yδ as.[Citation45Citation48]2.2 (δ,R):=sup{Ryδ-x|xM,yδY,Ax-yδδ}.2.2 The best possible error bound (or optimal error bound) is defined as the infimum over all mappings R:YX:2.3 ω(δ):=infR(δ,R).2.3 Now let us review some optimality results if the set M=Mφ,E is given by2.4 Mφ,E={xX|x=[φ(AA)]12v,vE},2.4 where the operator function φ(AA) is well defined via spectral representation [Citation47Citation50]2.5 φ(AA)=0aφ(λ)dEλ,2.5 where AA=0aλdEλ is the spectral decomposition of AA, Eλ denotes the spectral family of the operator AA, and a is a constant such that AAa with a= if AA is unbounded. In the case when A:L2(R)L2(R) is a multiplication operator, Ax(s)=γ(s)x(s), the operator function φ(AA) has the form2.6 φ(AA)x(s)=φ(|γ(s)|2)x(s).2.6 Then a method R0 is called [Citation48]

(i)

optimal on the set Mp,E if (δ,R0)=ω(δ,E) holds;

(ii)

order optimal on the set Mp,E if (δ,R0)Cω(δ,E) with C1 holds.

In order to derive an explicit (best possible) optimal error bound for the worst case error (δ,R) defined in (Equation2.2), we assume that the function φ in (Equation2.5) satisfies the following assumption:

Assumption 2.1.1

([Citation47, Citation48, Citation50]).The function φ(λ):(0,a](0,) in (Equation2.6), where a is a constant such that AAa, is continuous and has the following properties:

(i)

limλ0φ(λ)=0;

(ii)

φ is strictly monotonically increasing on (0,a];

(iii)

ρ(λ)=λφ-1(λ):(0,φ(a)](0,aφ(a)] is convex.

Under Assumption 2.1.1, the next theorem gives us a general formula for the optimal error bound.

Theorem 2.1.2

([Citation47, Citation48, Citation50]).Let Mφ,E be given by (Equation2.4), let Assumption 2.1.1 be satisfied, and let δ2E2σ(AAφ(AA)), where σ(AA) denotes the spectrum of operator AA, then2.7 ω(δ,E)=Eρ-1δ2E2.2.7

2.2 Optimal error bound for problem (1.1)

In this section we consider problem (Equation1.1) and deal with the question concerning the best possible worst case error (Equation2.3) for identifying f(t) from noisy data gδ(t)L2(R) provided (Equation1.2) and fMp,E hold, where Mp,E is given by2.8 fMp,E={fL2(R)|fpE,p>0},2.8 where ·p denotes the norm in Sobolev space Hp:2.9 H0(R)=L2(R),Hp(R)={vL2(R)|vp<},2.9 with2.10 f(·)p:=-|f^(ξ)|2(1+ξ2)pdξ12.2.10 Let us formulate problem (Equation1.1) as an operator equation2.11 Af=g2.11 with linear operator AL(L2(R),L2(R)). Obviously, this equation is equivalent to the operator equation in the frequency space2.12 A^f^=g^,A^=FAF-1,2.12 where F:L2(R)L2(R) is the (unitary) Fourier transformation operator that maps any function v(t)L2(R) into its Fourier transform v^(ξ). From (Equation1.7), we obtain2.13 1-e-(iξ)α2(iξ)αf^(ξ)=g^(ξ).2.13 So2.14 A^=1-e-(iξ)α2(iξ)α,2.14 which shows that A^:L2(R)L2(R) in (Equation2.13) is a linear and bounded multiplication operator, where the inverse A^-1 is unbounded. Since A^=1-e-(iξ)α2¯(iξ)α¯, we obtain2.15 A^A^=A^A^=1-e-(iξ)α22|(iξ)α|2=1-2e-|ξ|α2cos(π4α)cos|ξ|α2sinπ4α+e-2|ξ|α2cosπ4α|ξ|2α.2.15 The smoothness condition (Equation2.8) can also be transformed into an equivalent ‘source condition’ in the frequency domain. From (Equation2.10) we have that condition (Equation2.8) is equivalent to the condition2.16 f^M^p,E={f^L2(R)|f^pE,p>0},2.16 where2.17 f^(·)p:=-|f^(ξ)|2(1+ξ2)pdξ12.2.17 This condition can be reformulated into an equivalent ‘source condition’ with a set of the structure (Equation2.4).

Proposition 2.2.1

Consider the operator Equation (Equation2.11). Then the set M^p,E given in (Equation2.16) is equivalent to the general source set2.18 M^φ,E={f^L2(R)|,[φ(AA)]-12f^E},2.18 where φ=φ(λ) is given (in parameter representation) by2.19 λ(r)=1-2e-rcos(π4α)cos(rsin(π4α))+e-2rcos(π4α)r4,φ(r)=(1+r4α)-p,0r<.2.19

Proof

Comparing (Equation2.16) with (Equation2.18), we obtain2.20 φ(A^A^)=(1+ξ2)-p.2.20 From this representation and (Equation2.15), we obtain that φ is given (in parameter representation) by λ(ξ)=1-2e-|ξ|α2cos(π4α)cos(|ξ|α2sin(π4α))+e-2|ξ|α2cos(π4α)|ξ|2α, φ(ξ)=(1+ξ2)-p,ξR. We substitute |ξ|α2=r and obtain (Equation2.19).

We will discuss properties of the function φ=φ(λ)(λ(0,)) which is given (in parameter representation) by (Equation2.19) in the following.

Proposition 2.2.2

The function φ(λ) defined by (Equation2.19) is continuous and has the following properties:

(i)

limλ0φ(λ)=0;

(ii)

φ is strictly monotonically increasing;

(iii)

ρ(λ)=λφ-1(λ) is strictly monotonically increasing and possesses the following parameter representation:2.21 λ(r)=(1+r4α)-p,ρ(r)=(1+r4α)-p1-2e-rcos(π4α)cos(rsin(π4α))+e-2rcos(π4α)r4,0r<.2.21

(iv)

ρ-1(λ) is strictly monotonically increasing and possesses the following parameter representation:2.22 λ(r)=(1+r4α)-p1-2e-rcos(π4α)cos(rsin(π4α))+e-2rcos(π4α)r4,ρ-1(r)=(1+r4α)-p,0r<.2.22

(v)

For the inverse function ρ-1(λ), there holds2.23 ρ-1(λ)=λpp+α(1+o(1)),forλ0.2.23

Proof

The proof of (i), (ii), (iii) and (iv) is obvious. We only give the proof of (v). Let F(λ)=ρ-1(λ)λ-pp+α, we obtainlimλ0F(λ)=limr1+r4α-p1+r4αp2p+α×r4pp+α(1-2e-rcos(π4α)cos(rsin(π4α))+e-2rcos(π4α))pp+α=limrr-4pα+4p2α(p+α)+4pp+α=limrr0=1.The proof of (Equation2.23) is completed.

Proposition 2.2.3

The function ρ(λ) defined by (Equation2.20) is strictly convex.

Proof

The proof is similar to [Citation43, Citation47, Citation48, Citation50], and we omit it.

Now we will formulate our main result of this section concerning the best possible worst case error ω(δ,E) defined in (Equation2.2) for identifying the solution f(t) of problem (Equation1.1) from noisy data gδ(t)L2(R) under condition (Equation1.4) and fMp,E , where the set Mp,E is given by (Equation2.8). Since the Fourier operator F is unitary (i.e. F-1=F), we introduce the optimal error bound by2.24 ω(δ,E)=ω^(δ,E):=infsup{R^g^δ-f^|f^M^p,E,g^δL2(R),g^-g^δδ},2.24 where R^ is an arbitrary method for approximately solving (Equation2.12), and ‘inf’ means the minimum over all methods R^:L2(R)L2(R).

Theorem 2.2.4

Suppose conditions (Equation1.2) and (Equation2.18) hold. Then the optimal error bound for solving problem (Equation1.1) is:2.25 ω(δ,E)=δpp+αEαp+α(1+o(1)),forδ0.2.25

Proof

Combining (Equation2.7) with (Equation2.23), we obtainω(δ,E)=Eρ-1δ2E2=Eδ2E2pp+α(1+o(1))=δpp+αEαp+α(1+o(1)),forδ0.

3 The conditional stability result

In this section, we establish the stability estimate for the problem (Equation1.1).

Lemma 3.1

If x>1, the following inequality holds:3.1 11-e-x<2.3.1

Theorem 3.2

Suppose problem (Equation1.1) has a solution in the Sobolev space Hp under the a priori assumption. Let f(t) be the solution of problem (Equation1.1) with the exact data g(t). Then the following estimate holds:3.2 f(·)2(cos(π4α))2g(·)+2pp+αEαp+αg(·)pp+α.3.2

Proof

According to (Equation1.7), and by the Parseval identity, we havef(·)2=f^(·)2=(iξ)α1-e-(iξ)α2g^(ξ)2=|ξ|α2cos(π4α)1(iξ)α1-e-(iξ)α2g^(ξ)2dξ+|ξ|α2cos(π4α)1(iξ)α1-e-(iξ)α2g^(ξ)2dξ=:A1+A2.Let |ξ|α2cos(π4α)=s, and |ξ|α=(scos(π4α))2. As |s|1, we obtain3.3 (iξ)α1-e-(iξ)α2|ξ|α1-e-|ξ|α2cos(π4α)=1(cos(π4α))2s21-e-s2(cos(π4))2.3.3 So3.4 A12(cos(π4α))22g^(·)2.3.4 Using the Hölder inequality, we obtainA2=|ξ|α2cos(π4α)1(iξ)α1-e-(iξ)α2g^(ξ)2dξ=|ξ|α2cos(π4α)1|f^(ξ)|2dξ=|ξ|α2cos(π4α)1[(1+ξ2)p|f^(ξ)|2]αp+α[(1+ξ2)-α|f^(ξ)|2]pp+αdξ|ξ|α2cos(π4α)1([(1+ξ2)p|f^(ξ)|2]αp+α)p+ααdξαp+α×|ξ|α2cos(π4α)1([(1+ξ2)-α|f^(ξ)|2]pp+α)p+αpdξpp+α=|ξ|α2cos(π4α)1(1+ξ2)p|f^(ξ)|2dξαp+α×|ξ|α2cos(π4α)1(1+ξ2)-α|f^(ξ)|2dξpp+α=|ξ|α2cos(π4α)1(1+ξ2)p|f^(ξ)|2dξαp+α×|ξ|α2cos(π4α)1(1+ξ2)-α(iξ)α1-e-(iξ)α22|g^(ξ)|2dξpp+αf(·)p2αp+αsup|ξ|α2cos(π4α)1(1+ξ2)-α|ξ|α1-e-|ξ|α2cos(π4α)2pp+αg^(·)2pp+αsup|ξ|α2cos(π4α)1(1+ξ2)-α|ξ|α1-e-|ξ|α2cos(π4α)2pp+αE2αp+αg^(·)2pp+αsup|ξ|α2cos(π4α)111-e-|ξ|α2cos(π4α)2pp+αE2αp+αg^(·)2pp+α22pp+αE2αp+αg^(·)2pp+α.So3.5 f(·)A1+A2A1+A22(cos(π4α))2g^(·)+2pp+αEαp+αg^(·)pp+α.3.5

Remark 3.3

Suppose the functions f1(t) and f2(t) are the solutions of problem (Equation1.1) with the exact data g1(t) and g2(t), respectively. Let f1(·)-f2(·)pE, then there holds the following estimate:3.6 f1(·)-f2(·)2(cos(π4α))2g1(·)-g2(·)+Eαp+α2pp+αg1(·)-g2(·)pp+α.3.6 From (Equation3.5), it is obviously that f1(·)-f2(·)0 when g1(·)-g2(·)0.

4 The Fourier regularization method and the error estimates

4.1 The a priori parameter choice

It is obvious that the ill-posedness of problem (Equation1.1) is caused by disturb of the high frequencies. A natural way to stabilize the problem (Equation1.1) is to eliminate all high frequencies from the solution f(t). This idea has appeared earlier in [Citation51] and the authors considered the IHCP, called this method the Fourier regularization.

We define a regularization approximation solution of problem (Equation1.1) for noisy data gδ(t) as follows:4.1 fδ,ξmax(t):=12π-eiξt(iξ)α1-e-(iξ)α2g^δ(ξ)χmaxdξ,4.1 which is called the Fourier truncation regularized solution of problem (Equation1.1), where χmax is the characteristic function of the interval [-ξmax,ξmax], i.e.4.2 χmax(ξ)=1,|ξ|ξmax,0,|ξ|>ξmax,4.2 and ξmax is a constant which will be selected appropriately as regularization parameter. The main conclusion of this section is

Theorem 4.1.1

Let f(t) given by (Equation1.8) be the exact solution of (Equation1.1) and fδ,ξmax(t) be its regularization approximation given by (Equation4.1). Let assumptions (Equation1.4) and priori condition (Equation1.10) hold. If we select4.3 ξmax=Eδ1p+α,4.3 then there holds the following estimate:4.4 f(·)-fδ,ξmax(·)3δpp+αEαp+α(1+o(1)),asδ0.4.4

Proof

Using the Parseval formula and the triangle inequality, we knowf(·)-fδ,ξmax(·)=f^(·)-f^δ,ξmax(·)=(iξ)α1-e-(iξ)α2g^(ξ)-(iξ)α1-e-(iξ)α2g^δ(ξ)χmax(iξ)α1-e-(iξ)α2g^(ξ)-(iξ)α1-e-(iξ)α2g^(ξ)χmax+(iξ)α1-e-(iξ)α2g^(ξ)χmax-(iξ)α1-e-(iξ)α2g^δ(ξ)χmax=(iξ)α1-e-(iξ)α2g^(ξ)(1-χmax)+(iξ)α1-e-(iξ)α2(g^(ξ)-g^δ(ξ))χmax=|ξ|>ξmax(iξ)α1-e-(iξ)α2g^(ξ)2dξ12+|ξ|ξmax((iξ)α1-e-(iξ)α2(g^(ξ)-g^δ(ξ)))2dξ12|ξ|>ξmax(f^(ξ)(1+ξ2)p2(1+ξ2)-p2)2dξ12+sup|ξ|ξmax(iξ)α1-e-(iξ)α2|ξ|ξmax(g^(ξ)-g^δ(ξ))2dξ12sup|ξ|>ξmax(1+ξ2)-p2|ξ|>ξmax(f^(ξ)(1+ξ2)p2)2dξ12+sup|ξ|ξmax|ξ|α1-e-|ξ|α2cos(π4α)δ.1ξmaxpE+sup|ξ|ξmax|ξ|α1-e-|ξ|α2cos(π4α)δ.Let A(ξ)=|ξ|α1-e-|ξ|α2cos(π4α) and |ξ|α2cos(π4α)=s. So A(ξ) can be rewritten as A(s)=s2(cos(π4α))2(1-es). As 0<s1, we obtain A(s)2(cos(π4α))2. As 1sξmaxα2cos(π4α), we obtain A(s)2ξmaxα. So4.5 sup|ξ|ξmax|ξ|α1-e-|ξ|α2cos(π4α)2(cos(π4α))2+2ξmaxα.4.5 Thusf(·)-fδ,ξmax(·)1ξmaxpE+2ξmaxαδ+2(cos(π4α))2δ=3δpp+αEαp+α+2(cos(π4α))2δ=3δpp+αEαp+α(1+o(1)),asδ0.

4.2 The a posteriori choice rule

In this section, we consider the a posteriori regularization parameter choice rule. Choose the regularization parameter ξmax as the solution of the equation4.6 (1-χmax(ξ))g^δ(ξ)=τδ,τ>1isaconstant,4.6 where χmax(ξ) is defined by (Equation4.2). To establish existence and uniqueness of solution for Equation (Equation4.6), we need the following lemma and remark:

Lemma 4.2.1

Let ρ(ξmax):=(1-χmax(ξ))g^δ(ξ), then for δ>0, there hold

(a)

ρ(ξmax) is a continuous function;

(b)

limξmaxρ(ξmax)=0;

(c)

limξmax0ρ(ξmax)=g^δ;

(d)

ρ(ξmax) is a strictly decreasing function.

The proof is very easy and we omit it here.

Remark 4.2.2

To establish existence and uniqueness of solution for Equation (Equation4.6), we always suppose 0<δ<gδ.

To establish the error estimate for the a posteriori choice rule of the regularization parameter, we need the following lemmas:

Lemma 4.2.3

If ξmax is the solution of Equation (Equation4.6), then the following inequality holds:4.7 |ξmax|2E(τ-1)δ1p+α.4.7

Proof

Due to (Equation1.10), we obtain(1-χmax(ξ))g^(ξ)=|ξ|ξmax|g^(ξ)|2dξ12=|ξ|ξmax|(iξ)α1-e-(iξ)α2|2|g^(ξ)|2×(1+ξ2)p|(iξ)α1-e-(iξ)α2|-2(1+ξ2)-pdξ12sup|ξ|ξmax||1-e-(iξ)α2(iξ)α|(1+ξ2)-p2|Esup|ξ|ξmax2|ξ|α+pE2|ξmax|p+αE.So4.8 (1-χmax(ξ))g^(ξ)2|ξmax|p+αE.4.8 On the other hand, using the triangle inequality, (Equation1.2) and (Equation4.6), we obtain(1-χmax(ξ))g^(ξ)=(1-χmax(ξ))(g^(ξ)-g^δ(ξ)+g^δ(ξ))=(1-χmax(ξ))g^δ(ξ)+(1-χmax(ξ))(g^(ξ)-g^δ(ξ))(1-χmax(ξ))g^δ(ξ)-(1-χmax(ξ))(g^(ξ)-g^δ(ξ))τδ-δ=(τ-1)δ.So4.9 (1-χmax(ξ))g^(ξ)(τ-1)δ.4.9 Combining (Equation4.8) with (Equation4.9), we obtain4.10 (τ-1)δ(1-χmax(ξ))g^(ξ)2|ξmax|p+αE.4.10 So4.11 |ξmax|2E(τ-1)δ1p+α.4.11

Lemma 4.2.4

If ξmax is the solution of Equation (Equation4.6), then the following inequality also holds:4.12 (1-χmax(ξ))g^(ξ)(τ+1)δ.4.12

Proof

Due to the (Equation1.4) and (Equation4.6), we obtain(1-χmax(ξ))g^(ξ)=(1-χmax(ξ))(g^(ξ)-g^δ(ξ)+g^δ(ξ))(1-χmax(ξ))(g^(ξ)-g^δ(ξ)+(1-χmax(ξ))g^δ(ξ)δ+τδ=(τ+1)δ.

Now we give the main result of this section.

Theorem 4.2.5

Assume the conditions (Equation1.2) and (Equation1.10) hold and take the solution ξmax of Equation (Equation4.6) as the regularization parameter, then there holds the following error estimate:4.13 f(·)-fδ,ξmax(·)(2(τ+1))pp+α+22τ-1αp+αEαp+αδpp+α(1+o(1)),asδ0.4.13

Proof

Using the Parseval formula and the triangle inequality, we obtainf(·)-fδ,ξmax(·)=f^(·)-f^δ,ξmax(·)=(iξ)α1-e-(iξ)α2g^(ξ)-(iξ)α1-e-(iξ)α2g^δ(ξ)χmax(iξ)α1-e-(iξ)α2g^(ξ)-(iξ)α1-e-(iξ)α2g^(ξ)χmax+(iξ)α1-e-(iξ)α2g^(ξ)χmax-(iξ)α1-e-(iξ)α2g^δ(ξ)χmax=|ξ|>ξmax|(iξ)α1-e-(iξ)α2|2|g^(ξ)|2dξ12+|ξ|ξmax(iξ)α1-e-(iξ)α2(g^(ξ)-g^δ(ξ))2dξ12=I1+I2.Using the Hölder inequality and (Equation4.12), we obtainI12=|ξ|>ξmax|(iξ)α1-e-(iξ)α2|2|g^(ξ)|2dξ=|ξ|>ξmax|(iξ)α1-e-(iξ)α2|2|g^(ξ)|2αp+α|g^(ξ)|2(1-αp+α)dξ|ξ|>ξmax(|(iξ)α1-e-(iξ)α2|2|g^(ξ)|2αp+α)p+ααdξαp+α×|ξ|>ξmax|g^(ξ)|2(1-αp+α))p+αpdξpp+α=|ξ|>ξmax|(iξ)α1-e-(iξ)α2|2(p+α)α|g^(ξ)|2dξαp+α|ξ|>ξmax|g^(ξ)|2dξpp+α=|ξ|>ξmax|(iξ)α1-e-(iξ)α2|2pα||(iξ)α1-e-(iξ)α2g^(ξ)|2dξαp+α|ξ|>ξmax|g^(ξ)|2dξpp+α=|ξ|>ξmax(1+ξ2)-p|(iξ)α1-e-(iξ)α2|2pα(1+ξ2)p|f^(ξ)|2dξαp+α×|ξ|>ξmax|g^(ξ)|2dξpp+αsup|ξ|>ξmax(1+ξ2)-p||ξ|α1-e-|ξ|α2cos(π4α)|2pααp+αE2αp+α|ξ|>ξmax|g^(ξ)|2dξpp+αsup|ξ|>ξmax11-e-|ξ|α2cos(π4α)2pp+αE2αp+α(1-ξmax)|g^(ξ)|2pp+α22pp+αE2αp+α((τ+1)δ)2pp+α.So4.14 I12pp+αEαp+α((τ+1)δ)pp+α=2pp+α(τ+1)pp+αEαp+αδpp+α.4.14 Combining (Equation1.4) with (Equation4.5), we obtainI22=|ξ|ξmax(iξ)α1-e-(iξ)α2(g^(ξ)-g^δ(ξ))2dξsup|ξ|ξmax|ξ|α1-e-|ξ|α2cos(π4α)2δ22(cos(π4α))2+2ξmaxα2δ2.Using (Equation4.7), we obtainI22(cos(π4α))2+2ξmaxαδ22E(τ-1)δαp+αδ+2(cos(π4α))2δ=22τ-1αp+αEαp+αδpp+α+2(cos(π4α))2δ.So4.15 I222τ-1αp+αEαp+αδpp+α+2(cos(π4α))2δ.4.15 Combining (Equation4.14) with (Equation4.15), we obtainf(·)-fδ,ξmax(·)2pp+α(τ+1)pp+αEαp+αδpp+α+22τ-1αp+αEαp+αδpp+α+2(cos(π4α))2δ.=((2(τ+1))pp+α+22τ-1αp+α)Eαp+αδpp+α(1+o(1)),asδ0.The proof of Theorem 4.2.5 is completed.

5 Several numerical examples

In this section, we present three numerical examples intended to illustrate the usefulness of the proposed methods. The numerical results verify the validity of the theoretical results of these methods. Moreover, we would like to compare the a posteriori parameter choice (Equation4.6) with the a priori parameter choice rule (Equation4.3).

Since it is difficult to find a simple exact solution for Equation (Equation1.1), in this section we first solve the following direct problem for the given exact solution f(t) by a finite difference scheme (FDS) in (0,2)×(0,10):5.1 0+αu(x,t)-uxx(x,t)=f(t),x(0,2),t(0,10),u(x,0)=0,x[0,2],u(0,t)=0,t[0,10],u(2,t)=0,t[0,10],5.1 and use the finite difference solution to obtain the exact data function g(t). Then we added a normally distributed perturbation to each data function and obtained vectors gδ(t). Finally, we obtained the regularization solutions through solving the inverse problem. The bisection method is used to solve the Equation (Equation4.6), where we choose τ=1.1.

Denote the discrete points in the space interval [0, 2] as xj=jh,j=0,1,2,,m with the space step size h=2/m and the grid points in the time interval [0, 10] as tn=nk,n=0,1,2,,s, where k=10/s represents the time step size. Let ujn be the difference approximation to u(xj,tn). Then the difference scheme is given as follows5.2 -1h2uj-11+σα,k+2h2uj1-1h2uj+11=σα,kuj0+f1,5.2 and for n=2,3,,s5.3 -1h2uj-1n+σα,k+2h2ujn-1h2uj+1n=σα,kujn-1-σα,ki=2nϑi(α)(ujn-i+1-ujn-i)+fn,j=1,2,,m-1,5.3 with boundary conditions5.4 u0n=uMn=0;n=1,2,,s5.4 and initial condition5.5 uj0=0,j=1,2,,m-1,5.5 where5.6 σα,k=1Γ(1-α)11-α1kα,ϑi(α)=i1-α-(i-1)1-α.5.6 In the following numerical examples, we set the space step size h=2/100 and time step size k=10/100.

Example 1

Consider a smooth heat source:5.7 f(t)=1t2e-1t2,t>0,0,t0.5.7 Suppose that the sequence {gk}k=0n represents samples from the function g(t) on an equidistant grid, then we add a random uniform perturbation to each data, which forms the vector gδ, i.e.5.8 gδ=g+εrandn(size(g)),5.8 where5.9 g=(g(t1),,g(tn))T,ti=(i-1)Δt,Δt=10n-1,i=1,2,,n.5.9 The function ‘randn(·)’ generates arrays of random numbers whose elements are normally distributed with mean 0, variance σ2=1 and standard deviation σ=1. ‘Randn(size(g))’ returns an array of random entries that is of the same size as g. The total noise level δ can be measured in the sense of Root Mean Square Error(RMSE) according to5.10 δ=gδ-gl2=1ni=1n(gi-giδ)212.5.10 Moreover, we need to make the vector gδ periodical [Citation51] and then we take the discrete Fourier transform for the vector gδ. The approximation of the regularization solution is computed by using FFT algorithm.[Citation51]

Example 2

Consider a piecewise smooth heat source:5.11 f(t)=0,0t2.5,0.4t-1,2.5<t5,-0.4t+3,5<t7.5,0,7.5<t10.5.11

Example 3

[Citation32]Consider the following discontinuous case:5.12 f(t)=0,0t10/3,1,10/3<t20/3,0,20/3<t10.5.12 From Figures , we can find that the smaller ε, the better the computed approximation is, and the smaller the α is, the better the computed approximation is. Moreover, we can also easily find that a posteriori parameter choice also works well. From Figures , it can be seen that the numerical solution is less ideal than that of Example 1. It is not difficult to see that the well-known Gibbs phenomenon and the recovered data near the non-smooth and discontinuities points are not accurate. Taking into consideration of the ill-posedness of the problem, the results presented in Figures are reasonable.

Figure 1. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.1 with Example 1: (a) ε=0.01, (b) ε=0.001.

Figure 1. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.1 with Example 1: (a) ε=0.01, (b) ε=0.001.

Figure 2. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.3 with Example 1: (a) ε=0.01, (b) ε=0.001.

Figure 2. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.3 with Example 1: (a) ε=0.01, (b) ε=0.001.

Figure 3. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.6 with Example 1: (a) ε=0.01, (b) ε=0.001.

Figure 3. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.6 with Example 1: (a) ε=0.01, (b) ε=0.001.

Figure 4. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.1 with Example 2: (a) ε=0.01, (b) ε=0.001.

Figure 4. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.1 with Example 2: (a) ε=0.01, (b) ε=0.001.

Figure 5. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.3 with Example 2: (a) ε=0.01, (b) ε=0.001.

Figure 5. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.3 with Example 2: (a) ε=0.01, (b) ε=0.001.

Figure 6. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.6 with Example 2: (a) ε=0.01, (b) ε=0.001.

Figure 6. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.6 with Example 2: (a) ε=0.01, (b) ε=0.001.

Figure 7. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.1 with Example 3: (a) ε=0.01, (b) ε=0.001.

Figure 7. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.1 with Example 3: (a) ε=0.01, (b) ε=0.001.

Figure 8. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.3 with Example 3: (a) ε=0.01, (b) ε=0.001.

Figure 8. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.3 with Example 3: (a) ε=0.01, (b) ε=0.001.

Figure 9. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.6 with Example 3: (a) ε=0.01, (b) ε=0.001.

Figure 9. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.6 with Example 3: (a) ε=0.01, (b) ε=0.001.

6 Identifying the unknown source of separable variables

In this section, we will use the Fourier regularization method to identify the unknown source of separable variables in the following time-fractional diffusion equation:6.1 0+αu(x,t)-uxx(x,t)=λ(x)f(t),x>0,t>0,u(x,0)=0,x0,u(0,t)=0,t0,u(x,t)|xbounded,t0,u(1,t)=g(t),t0,6.1 where λ(x)f(t) denotes the source (sink) term. We assume the space-dependent source term λ(x) is known and satisfied6.2 λ(x)C[0,+),0<p0λ(x)q0.6.2 The time-dependent source term f(t) is unknown. Our purpose is to identify f(t) from the additional data u(1,t)=g(t).

Using the Fourier transform, we obtain the solution of the problem (Equation6.1) in frequency space as follows:6.3 f^(ξ)=(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^(ξ).6.3 So6.4 f(t)=12π-eiξt(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^(ξ).6.4 Using the Fourier regularization method, we give the Fourier regularization solution of problem (Equation6.1) for noisy data gδ(t) as follows:6.5 fηmaxδ(t):=12π-eiξt(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^δ(ξ)χmaxdξ,6.5 where χmax is the characteristic function of the interval [-ηmax,ηmax], i.e.6.6 χmax(ξ)=1,|ξ|ηmax,0,|ξ|>ηmax,6.6 and ηmax is a constant which will be selected appropriately as regularization parameter.

6.1 A priori choice rule

Theorem 6.1.1

Let f(t) given by (Equation6.4) be the exact solution of (Equation6.1) and fηmaxδ(t) be the Fourier regularization solution given by (Equation6.5). Let assumptions (Equation1.2) and priori condition (Equation1.10) hold. If we select6.7 ηmax=Eδ1p+α,6.7 then we obtain the following estimate:6.8 f(·)-fηmaxδ(·)p0+2p0δpp+αEαp+α(1+o(1)),asδ0.6.8

Proof

Using the Parseval formula and the triangle inequality, we knowf(·)-fηmaxδ(·)=f^(·)-f^ηmaxδ(·)=(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^(ξ)-(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^δ(ξ)χmax(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^(ξ)-(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^(ξ)χmax+(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^(ξ)χmax-(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^δ(ξ)χmax=(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^(ξ)(1-χmax)+(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτ(g^(ξ)-g^δ(ξ))χmax=|ξ|>ηmax(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτg^(ξ)2dξ12+|ξ|ηmax(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτ(g^(ξ)-g^δ(ξ))2dξ12|ξ|>ηmaxf^(ξ)(1+ξ2)p2(1+ξ2)-p22dξ12+sup|ξ|ηmax(iξ)α201λ(τ)sinh((iξ)α2(τ-1))dτ|ξ|ηmax(g^(ξ)-g^δ(ξ))2dξ12sup|ξ|>ηmax(1+ξ2)-p2|ξ|>ηmax(f^(ξ)(1+ξ2)p2)2dξ12+sup|ξ|ηmax(iξ)α2p001sinh((iξ)α2(τ-1))dτδ.1ηmaxpE+sup|ξ|ηmax(iξ)αp0(1-cosh((iξ)α2))δ1ηmaxpE+sup|ξ|ηmax|ξ|αp0(1-e-|ξ|α2cos(π4α))δ1ηmaxpE+2p0ηmaxαδ+2p0(cos(π4α))2δ=p0+2p0δpp+αEαp+α+2p0(cos(π4α))2δ=p0+2p0δpp+αEαp+α(1+o(1)),asδ0.

6.2 The a posteriori choice rule

In this section, we consider the a posteriori regularization parameter choice rule. Choose the regularization parameter ηmax as the solution of the equation6.9 (1-χmax(ξ))g^δ(ξ)=τδ,τ>1isaconstant,6.9 where χmax(ξ) is defined by (Equation6.6).

In order to obtain the error estimate for the a posteriori choice rule of the regularization parameter, we need the following lemmas:

Lemma 6.2.1

If ηmax is the solution of Equation (Equation6.9), then we obtain the following inequality hold:6.10 |ηmax|2q0E(τ-1)δ1p+α.6.10

Proof

The proof of Lemma 6.2.1 is similar to the proof of Lemma 4.2.3, we omit it.

Lemma 6.2.2

If ηmax is the solution of Equation (Equation6.9), then we also get the following inequality:6.11 (1-χmax(ξ))g^(ξ)(τ+1)δ.6.11

Proof

The proof of Lemma 6.2.2 is similar to the proof of Lemma 4.2.4, we omit it

Now we give the main result of this section.

Theorem 6.2.3

Assume the conditions (Equation1.4) and (Equation1.10) hold and take the solution ηmax of Equation (Equation6.9) as the regularization parameter, then there holds the following error estimate:6.12 f(·)-fηmaxδ(·)2p0(τ+1)pp+α+2p0(2q0τ-1)αp+αEαp+αδpp+α(1+o(1)),asδ0.6.12

Proof

The proof of Theorem 6.2.3 is similar to the proof of Theorem 4.2.5, we omit it.

6.3 Numerical example

In this section, we choose an example to show the effectiveness and stability of our proposed method.

Example 4

Take a source function λ(x)=1+sinx and f(t)=e-t.

From Figures , we can also find that the smaller ε, the better the computed approximation is, and the smaller the α is, the better the computed approximation is. Meanwhile, we can easily find that a posteriori parameter choice also works well. So, the Fourier regularization method is very effective for identifying the unknown source which is separable variable.

Figure 10. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.1 with Example 4: (a) ε=0.01, (b) ε=0.001.

Figure 10. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.1 with Example 4: (a) ε=0.01, (b) ε=0.001.

Figure 11. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.3 with Example 4: (a) ε=0.01, (b) ε=0.001.

Figure 11. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.3 with Example 4: (a) ε=0.01, (b) ε=0.001.

Figure 12. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.6 with Example 4: (a) ε=0.01, (b) ε=0.001.

Figure 12. The comparison of the numerical effects between the exact solution and its computed approximations for p=1, α=0.6 with Example 4: (a) ε=0.01, (b) ε=0.001.

7 Conclusions

In this paper, the Fourier regularization method is used to identify the unknown source term which depends only on the time variable or is separable variable for the time-fractional diffusion equation. Under the a priori condition, we gave the optimal error bound for this problem. Moreover, we obtained the stability estimate using the conditional stability. Thirdly, for the a priori and the a posteriori parameter choice rules, we all obtain the Höder type error estimates which are all order-optimal. Finally, the numerical experiments show the proposed methods work effectively.

Notes

The project is supported by the National Natural Science Foundation of China (No.11171136, No.11261032), the Distinguished Young Scholars Fund of Lan Zhou University of Technology (Q201015), the basic scientific research business expenses of Gansu province college and the Natural Science Foundation of Gansu province (1310RJYA021).

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