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Identifying the space source term problem for a generalization of the fractional diffusion equation with hyper-Bessel operator
Advances in Difference Equations volume 2020, Article number: 261 (2020)
Abstract
In this paper, we consider an inverse problem of identifying the source term for a generalization of the time-fractional diffusion equation, where regularized hyper-Bessel operator is used instead of the time derivative. First, we investigate the existence of our source term; the conditional stability for the inverse source problem is also investigated. Then, we show that the backward problem is ill-posed; the fractional Landweber method and the fractional Tikhonov method are used to deal with this inverse problem, and the regularized solution is also obtained. We present convergence rates for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Finally, we present a numerical example to illustrate the proposed method.
1 Introduction
Fractional calculus has a long history in the mathematical theory and has attracted much attention in various fields of the applied science [3, 4, 13, 23]. Fractional differential equations have an important position in the mathematical modeling of different physical systems [1, 10, 21], in engineering [6, 18], [7], and finance [24], in physics, chemistry, medicine, and they describe anomalous diffusion [12, 16, 20].
In this paper, we restore the space source term problem for a generalization of the time-fractional diffusion equation with variable coefficients. The time-fractional diffusion is discussed in this paper as follows:
where Ω is bounded with sufficient smooth boundary ∂Ω in \(\mathbb{R}^{d}\) (\(d \in\mathbb {N}\)), \({T} > 0\) is a fixed value, \(0 < \upbeta<1 \), and \({}^{C} ( t ^{1 - \upbeta} \frac{\partial}{\partial t } )^{\upalpha }\) stands for a regularized Caputo-like counterpart hyper-Bessel operator of order \(0 < \upalpha< 1\). From [10], we have the following formula:
where \(\varGamma(x)\) is a gamma function, and the hyper-Bessel operator\(( t^{1- \upbeta} \frac{\partial }{\partial t} )^{\upalpha} \) was introduced by Dimovski in [8]. Since (2), we see that the study of (1) comes from the definition of hyper-Bessel operator. Some papers [10, 11] used the hyper-Bessel operator to describe heat diffusion for fractional Brownian motion. Some more details on them can be found in [2, 25, 30].
The results for equation (1) were investigated by some recent works [2, 29]. The authors [2] considered two direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart hyper-Bessel operator. They established the existence and uniqueness of solutions to the problem and gave the explicit eigenfunction expansions. In [29], the author investigated the exact solution of the inhomogeneous linear equation and the semilinear equation using fixed point theorems. In practice, some initial data, boundary data, diffusion coefficients, or source terms may not be given. By adding some given data, we can recover them, this is the inverse problem (or backward problem) of the time-fractional diffusion. To the best of our knowledge, the source identification problem for the fractional diffusion equation with hyper-Bessel operator has also been studied very little.
Our purpose in this paper is to find an inversion source problem for (1). Assume that the source term \(\mathcal {F}(\textrm{x} ,t )\) of problem (1) is a forward problem, which can be split into \(\mathscr{F} (\textrm{x} ) \mathcal {Q}(t)\), where \(\mathcal{Q}(t)\) is known in advance. Hence, we want to identify the space source term \(\mathscr{F} (\textrm{x} )\) by using the value of the final time T as follows:
In fact, the measurements are noised, the observation data \(\mathscr {H} \) are obtained by inexact data using some measurements; and so, they are approximated data by \(\mathscr{H} ^{\upepsilon}\) and
where \(\upepsilon > 0\) is a bound on the measurement error. A small error of the given observation \(\mathscr{H}\) can result in that the solution may have a large error. Hence, we have to propose some regularization method in order to recover stable approximations for the unknown space source function.
In this paper, we apply the fractional Landweber method and fractional Tikhonov method to restore the unknown space source function \(\mathscr {F} \). Both methods were studied by Klann and Ramlau [15] when they considered a linear ill-posed problem. Since the a priori bound of the exact solution cannot be known exactly in practice, we need to give a posteriori choice of the regularization parameter. Study for choosing the regularization parameter by the a priori rule is easier than that by a posteriori rule.
The paper is organized as follows. In Sect. 2, we recall some preliminary results. The exact solution, the ill-posedness of the inverse problem, and the conditional stability are also discussed in Sect. 2. In Sects. 3 and 4, we present the fractional Landweber regularization method and the fractional Tikhonov regularization method. The convergence estimate under an a priori assumption for the exact solution and the a posteriori regularization parameter choice rule are considered in there. In the last section, we present a numerical example to illustrate the proposed method.
2 Identifying the space source term problem
2.1 Preliminary results
In this section, we recall some useful results.
Let us consider the operator \(\mathscr{B} \) on the domain \(\mathcal {D}(-\mathscr{B} ):= \mathcal {H}_{0}^{1}(\varOmega)\cap\mathcal {H}^{2}(\varOmega)\), and assume that \(-\mathscr{B} \) has eigenvalues \({a}_{p} \) with corresponding eigenfunction \(w_{p} \in\mathcal {D}(-\mathscr{B} ) \).
Note
and \({a}_{p} \to\infty\) as \(p \to\infty\). The most popular example of \(\mathscr{B} \) is the negative Laplacian operator −Δ on \(\mathcal{L}^{2}({\varOmega})\), we have
From [5], it easy to see that \({a}_{p} \ge C p ^{\frac{2}{d}}\) for C is a constant, \(p \in\mathbb{N}\), and d is the dimensional number of the spatial variable.
Now, let us define fractional powers of \(\mathscr{B} \) and the Hilbert scale spaces. For all \(k \geq0\), we denote by \((-\mathscr{B}) ^{k}\) the following operator:
and
The space \(\mathcal{D}((-\mathscr{B} )^{k} ) \) is a Banach space with the following norm:
It is easy to see that \(\| v \|_{\mathcal{D}((-\mathscr{B} )^{k} ) } = \| (-\mathscr{B} )^{k} v \|_{\mathcal{L}^{2}({\varOmega}) }\). Its domain \(\mathcal{D}((-\mathscr{B} )^{-k})\) is a Hilbert space endowed with the dual inner product \(\langle\cdot,\cdot\rangle_{-k,k}\) taking between \(\mathcal{D}((-\mathscr{B} )^{-k})\) and \(\mathcal {D}((-\mathscr{B} )^{k})\). This generates the norm
Definition 2.1
([21])
The generalized Mittag-Leffler function is defined as follows:
Note \(E_{\upalpha ,\upbeta }(z)\) is an entire function in \(z \in\mathbb{C}\). For convenience, let us set \(E_{\upalpha}(z):= E_{\upalpha,1}(z)\) and \(\boldsymbol {E}(z) := E_{\upalpha,\upalpha}(z)\).
Lemma 2.1
(see [21])
Let\(0 < \upalpha < 2\), and\(\upbeta\in\mathbb{R}\)be arbitrary. Let us suppose thatμis such that\(\frac{\pi \upalpha }{2} < \mu< \min \{ \pi, \pi \upalpha \} \). Then there exists a constant\(\mathbb{M} =\mathbb{M}( \upalpha , \upbeta, \mu) >0 \)such that
Lemma 2.2
Let\(\upalpha\in(0,1)\), then\(E_{\upalpha} (-z) >0\)for any\(z>0\). Moreover, there exist three positive constants\(\mathbb{M}_{\upalpha ,\upbeta}^{-}\), \(\mathbb{M}_{\upalpha,\upbeta}^{+}\), \(\mathbb {M}_{\upalpha,\upbeta}\)such that
If \(\upalpha\in[\upalpha_{0},\upalpha_{1}]\) for any \(0 < \upalpha_{0} < \upalpha_{1} < 1 \), the constants can be chosen, which depends only \(\upalpha_{0}\), \(\upalpha_{1}\).
Lemma 2.3
(see [21])
Let\(\mathtt{c} >0\)and\(0 < \upalpha< 1 \). Then
- (a)
\(\frac{d}{d t} E_{\upalpha} (- \mathtt{c} t ^{\upalpha}) = - \mathtt{c} t^{\upalpha-1} \boldsymbol {E} ( - \mathtt{c} t ^{\upalpha} ) \), \(t>0\);
- (b)
\(\frac{d}{d t} ( t^{\upalpha-1} \boldsymbol {E} (- \mathtt{c} t ^{\upalpha}) )= t^{\upalpha-2} E_{\upalpha,\upalpha-1 } ( - \mathtt{c} t ^{\upalpha} ) \), \(t>0\);
- (c)
\(\partial_{t}^{\upalpha} E_{\upalpha} (- \mathtt{c} t ^{\upalpha}) = - \mathtt{c} E_{\upalpha} ( - \mathtt{c} t ^{\upalpha} ) \), \(t>0\);
- (d)
\(\partial_{t}^{\upalpha} ( {t}^{\upalpha-1} \boldsymbol {E} (- \mathtt{c} t ^{\upalpha}) ) = - \mathtt{c} {t}^{\upalpha-1} \boldsymbol {E} (- \mathtt{c} t ^{\upalpha}) \), \(t>0\).
Lemma 2.4
For\(0 < \mathtt{k} < 1\), \(\mathtt{q} > 0 \), and\(\mathtt{m} \in \mathbb{N}\), we obtain
Lemma 2.5
(see [28])
For some positive constantsr, μ, c, d, we obtain
and
where\(M_{1} = M_{1}(r,d) >0\), \(M_{2} = M_{2}(r,d ) >0\), \(M_{3} = M_{3}(r,d) >0\), \(M_{4} = M_{4}(r,d ) >0\)are independent ofc.
2.2 Solution for a fractional diffusion equation with regularized Caputo-like counterpart of a hyper-Bessel differential operator
Using the Fourier series expansion and the properties of Mittag-Leffler, the exact solution of problem (1) is given by the following form (see [2, 29]):
where \(\textrm{g}_{p} = \langle\textrm{g} , w_{p} \rangle\), \(F_{p}(\uptau) = \mathcal{Q} (\uptau) \langle \mathscr{F} , w_{p} \rangle\) stands for its Fourier coefficient.
Let \(t= {T}\) in, and we obtain
where \(\mathscr{H}_{p}= \langle\mathscr{H},w_{p}\rangle \) with \(p\in \mathbb{N}\), \(p \ge1\).
Lemma 2.6
Let\(\mathcal{Q}: [0,{T} ] \to\mathbb{R}\)be a positive continuous function such that\(\inf_{t \in[0,{T} ]} | \mathcal{Q} (t) |= \mathcal{Q}_{0}\). Assume that\(\| \mathcal{Q} \| _{\infty} = \sup_{t \in[0,{T} ]}| \mathcal{Q} (t)|\), then we get
for all\(p \in\mathbb{N}\).
Proof
First, by Lemma (2.2), we obtain
where \(\overline{\mathbb{M}_{\upalpha,\upbeta}^{+} } := M_{2} ( \frac {2}{\upalpha }, \frac{\beta^{\upalpha }}{a_{p}} )\) by applying Lemma 2.5 for \(r = \frac{2}{\upalpha }\) and \(d = \frac{\beta^{\upalpha }}{a_{p}}\).
Otherwise, we also get
where \(\mathcal{M}= ( 1 - E_{\upalpha} (- \frac{{a}_{p} }{\upbeta^{\upalpha}} {T}^{\upalpha\upbeta}) ) \). □
2.3 Ill-posedness and stability estimates
For any \(\mathcal {H} \in\mathcal{L}^{2}({\varOmega}) \), let \(\mathcal{K} : \mathcal{L}^{2}({\varOmega}) \rightarrow \mathcal{L}^{2}({\varOmega}) \) be the following operator:
where the kernel \(\upmu(\cdot,\cdot)\) is
Since (8), our problem is of finding \(\mathscr {F}\) which can be transformed into
where
Hence, we obtain
It is easy to see that \(\mathcal{K} : \mathcal{L}^{2}({\varOmega}) \to\mathcal{L}^{2}({\varOmega}) \) is a compact operator, then problem (10) is ill-posed. To give the ill-posedness problem, we propose an illustrative example. Assume that \(\textrm{g} = 0\), we choose the final data \(\mathscr {H}^{l} (\textrm{x} ) = \frac{w_{l} (\textrm{x} )}{\sqrt{{a}_{l}}}\), then the corresponding source terms
By Lemma (2.6), we get \(\| \mathscr{F}^{l} (\textrm{x} ) \|_{\mathcal{L}^{2}({\varOmega}) } \ge\frac{ \sqrt{{a}_{l}}}{\overline{\mathbb {M}_{\upalpha,\upbeta}^{+} } \| \mathcal{Q} \|_{\infty}} \), hence \(\lim_{l \to\infty}\| \mathscr{F}^{l} (\textrm{x} ) \| _{\mathcal{L}^{2}({\varOmega}) } \to\infty\). But \(\| \mathscr{H}^{l} \|_{\mathcal{L}^{2}({\varOmega})} = \frac{1}{\sqrt{{a}_{p} }} \), or \(\lim_{l \to\infty}\| \mathscr{H} ^{l} (\textrm{x} ) \|_{\mathcal{L}^{2}({\varOmega }) } \to0\). Some of the above observations imply that our problem (1) satisfying (3) is ill-posed in the sense of Hadamard.
Next, we get conditional stability in the following theorem.
Theorem 2.1
Let\(\mathcal{Q}: [0,{T} ] \to\mathbb{R}\)for all\(t \in [0,{T} ]\). Assume that\(\mathcal{P} \)is a positive constant and
Then we obtain
Proof
By Hölder’s inequality and (11), we obtain
Applying Lemma 2.6 and (11), we get
Hence,
We finish the proof. □
3 Fractional Landweber regularization method and convergence rate
From Sect. 2, we know that problem (1) satisfying (3) is ill-posed. Therefore, we need a regularization method. Now, we propose a fractional Landweber regularization method to solve the ill-posed problem (1) satisfying (3). The convergence rates for the regularized solution under two parameter choice rules are also considered.
From [14], \(\mathcal{K} \mathscr{F} = \varTheta\) is equivalent to
where \(0 < {c} < \Vert \mathcal{K} \Vert ^{-2}\) and \(\mathcal{K}^{*}\) is the adjoint operator of \(\mathcal{K} \).
Applying the fractional Landweber method given by [15], we propose the following regularized solution with exact data \(\mathscr {H} \):
If the observation data \(\mathscr{H} \) is noised by \(\mathscr{H} ^{\upepsilon} \), then we have
here \(\varTheta^{\upepsilon} := \mathscr{H}^{\upepsilon} - \sum_{p=1}^{\infty} ({E_{\upalpha,1}} (- \frac{{a}_{p} }{\upbeta^{\upalpha}} {T}^{\upalpha\upbeta}) \textrm{g} _{p} ) w_{p}(\textrm{x} )\), \(\uptheta\in( \frac{1}{2}, 1 ] \) is the fractional order, and \(\mathtt{m} > 0 \) is the iterative step and is a regularization parameter. Here, we note that when \(\uptheta= 1\), the fractional Landweber method becomes a standard Landweber regularization.
Lemma 3.1
Let\({a}_{p} >0 \), \(\uptheta\in(\frac{1}{2},1]\), \(\mathtt {m} > 0\)and
we get
Proof
We define \(\psi(y) := y^{-2} [ 1 - ( 1 -y^{2})^{\mathtt{m} } ]^{2\uptheta} \), where \(y^{2} := {c} [ \frac{1}{\upbeta^{\upalpha}} \int _{0}^{ {T}} ( {T}^{\upbeta} - \uptau^{\upbeta} )^{\upalpha - 1} \boldsymbol {E} (- \frac{{a}_{p} }{\upbeta ^{\upalpha }} ( {T}^{\upbeta} - \uptau^{\upbeta} ) ^{\upalpha } ) \mathcal{Q} (\uptau) \,d (\uptau ^{\upbeta}) ]^{2}\). It is easy to see that the function \(\psi(y)\) is continuous in \([0, + \infty )\) when \(y \in(0,1)\) and
For \(\uptheta\in(\frac{1}{2},1]\) and \(y \in(0,1)\), applying Lemma 3.3 of [15], we get \(\psi(y) \le\mathtt{m} \). That infers that inequality (14) is correct. □
3.1 A priori parameter choice rule and convergence estimate
Let us choose \(\mathtt{m} := \mathtt{m} (\upepsilon)\) such that \(\| \mathscr{F}_{\mathtt{m} ,\uptheta} ^{\upepsilon} ( \cdot) - \mathscr{F} ( \cdot) \|_{\mathcal{L}^{2}({\varOmega}) } \to 0\) as \(\upepsilon\to0\). Using an a priori regularization parameter choice rule, we propose the convergence rate for the fractional Landweber regularized solution \(\mathscr{F}_{\mathtt{m} ,\uptheta} ^{\upepsilon} \) to \(\mathscr{F} \).
Theorem 3.1
Let\(\mathcal{Q}: [0,{T} ] \to\mathbb{R}\)for all\(0 \le t \le{T} \)and\(\mathscr{H} \in\mathcal{L}^{2}({\varOmega}) \). Assume that (4) and bound condition (12) hold.
If we choose
then we get
here\(\lfloor \mathtt{m} \rfloor\)represents the largest integer not larger thanm.
Proof
From the triangle inequality, we obtain
First, we give an estimate for the first term \(\| \mathscr{F}_{\mathtt {m} ,\uptheta} ^{\upepsilon} ( \cdot) - \mathscr{F}_{\mathtt{m} ,\uptheta} ( \cdot) \|_{\mathcal{L}^{2}({\varOmega}) }\). Applying Lemma 3.1, we get
Hence,
On the other hand, we estimate the second term
We note that \(\uptheta\in(\frac{1}{2},1]\), Lemma 2.6, then
Apply Lemma 2.4
Thus
Choose the regularization parameter m by
then we have
This ends the proof. □
3.2 A posteriori parameter choice rule and convergence estimate
Now, we consider an a posteriori regularization choice rule called Morozov’s discrepancy principle [9], we choose the regularization parameter m such that
where \(\| \varTheta^{\upepsilon } \|_{\mathcal{L}^{2}({\varOmega}) } \ge\upvartheta \upepsilon\), ϑ, which makes (17) hold at the first iterator time, is a constant independent of ε.
Choose \(\upvartheta > 1\), and the bound for m is given and depends on ε and \(\mathcal{P}\).
Lemma 3.2
Ifmsatisfies (17), we can get the following inequality:
Proof
From (17), we have
By \(\uptheta\in(\frac{1}{2},1]\) and \(0 < {c} [ \frac {1}{\upbeta^{\upalpha}} \int_{0}^{ {T}} ( {T}^{\upbeta} - \uptau^{\upbeta} )^{\upalpha - 1} \boldsymbol {E} (- \frac{{a}_{p} }{\upbeta^{\upalpha }} ( {T}^{\upbeta } - \uptau^{\upbeta} ) ^{\upalpha } ) \mathcal{Q} (\uptau) \,d (\uptau ^{\upbeta}) ]^{2} < 1\), we get
where
In view of Lemma 2.6 and (11), we obtain
By Lemma 2.4, this implies that
From the above results we have
This yields
□
Theorem 3.2
We recall thatmin Lemma3.2and bound condition (12) hold. Then we have
Proof
From the triangle inequality, we obtain
In view of (16) and Lemma 3.2, we deduce that
Now we give the bound for the second term. Same as above, we have
Applying Hölder’s inequality, we have
where
From \(\uptheta\in(\frac{1}{2},1]\) and \(0 < {c} [ \frac {1}{\upbeta^{\upalpha}} \int_{0}^{ {T}} ( {T}^{\upbeta} - \uptau^{\upbeta} )^{\upalpha - 1} \boldsymbol {E} (- \frac{{a}_{p} }{\upbeta^{\upalpha }} ( {T}^{\upbeta } - \uptau^{\upbeta} ) ^{\upalpha } ) \mathcal{Q} (\uptau) \,d (\uptau ^{\upbeta}) ]^{2} < 1\), we deduce that
Hence, we obtain
In view of (17), we have
Hence, we get
This ends the proof. □
4 Fractional Tikhonov regularization method and convergence rate
Now, we propose another method to solve the ill-posed problem (1) satisfying (3), that is, a fractional Tikhonov regularization method. Besides, the convergence analysis between the exact solution \(\mathscr{F} \) and the fractional Tikhonov regularized solution \(\mathscr{F}_{\textrm{n} ,\uptheta} ^{\upepsilon} \) are also considered.
From [15], the fractional Tikhonov regularization solution is given by
If the observation data \(\mathscr{H} \) are noised by \(\mathscr{H} ^{\upepsilon} \), then we have
where the regularization parameter \(\textrm{n} > 0 \), and \(\uptheta \in( \frac{1}{2}, 1 ] \) is the fractional order. We note that when \(\uptheta= 1\), the fractional Tikhonov method becomes a standard Tikhonov regularization.
Lemma 4.1
Let\(\uptheta\in(\frac{1}{2},1]\), \({n} > 0\), we get
Proof
We define \(\psi(y) := y^{-2} ( \frac{y^{2}}{y^{2} + \textrm{n} } )^{2\upchi} \), where
By proving similarly Lemma 3.1 in [15], we get
□
4.1 A priori parameter choice rule and convergence estimate
Let us choose a regularization parameter n, which depends on ε so that if \(\upepsilon\to0\), then we get \(\| \mathscr{F}_{\textrm{n} ,\uptheta} ^{\upepsilon} ( \cdot) - \mathscr{F} ( \cdot) \|_{\mathcal{L}^{2}({\varOmega}) }\) tends to 0. The convergence rate for the regularized solution \(\mathscr {F}_{\textrm{n} ,\uptheta} ^{\upepsilon} \) to the exact solution \(\mathscr{F} \) can be got under an a priori regularization parameter choice rule.
Theorem 4.1
Let\(\mathscr{H} \in\mathcal{L}^{2}({\varOmega}) \)and\(\mathcal{Q}: [0,{T} ] \to\mathbb{R}\)for all\(0 \le t \le {T} \). Assume that assumption (4) and a priori bound condition (12) hold.
If we choose
which is the regularization parameter, then we get
here\(\lfloor {n} \rfloor\)represents the largest integer not larger thann.
Proof
From the triangle inequality, we obtain
First, we obtain
Moreover, from (11) we obtain
Using the inequality \(c_{1} ^{\varsigma} - c_{2} ^{\varsigma} \le(c_{1} - c_{2})^{\varsigma}\), for \(0 \le c_{2} \le c_{1}\), \(0 \le\varsigma \le1\), we deduce that
Applying Lemma 2.6, we get
From a priori bound on the final data \(\| \mathscr{F} \|_{\mathcal{D}((-\mathscr{B} )^{-\kappa})} \le \mathcal{P}\) for any \(\kappa> 0\), \(\uptheta\in(\frac{1}{2},1]\) and Lemma 2.5, we have
Substituting the above inequality into (24) and applying (25), we get
Choose the regularization parameter n by
then we have
The proof is completed. □
4.2 A posteriori parameter choice rule and convergence estimate
Now, based on Morozov’s discrepancy principle [9], we consider the choice of an a posteriori regularization. Let us choose the regularization parameter n such that
where \(\| \varTheta^{\upepsilon } \|_{\mathcal{L}^{2}({\varOmega}) } \ge \upchi \upepsilon>0 \). Choose \(\upchi > 1\), and the bound for n is given and depends on ε and \(\mathcal{P}\).
Lemma 4.2
Ifnsatisfies (27), we can get the following inequality:
Proof
From (27), we have
In view of Lemma 2.6 and \(\uptheta\in(\frac{1}{2},1]\), it implies that
By bound condition (12) and Lemma 2.5, we obtain
Combining the above equations, we deduce that
This yields
□
Theorem 4.2
We recall thatnin Lemma4.2and bound condition (12) hold. Then we have
Proof
From the triangle inequality, we obtain
First, the estimate for \(\| \mathscr{F}_{\textrm{n} ,\uptheta} ^{\upepsilon} ( \cdot) - \mathscr{F}_{\textrm{n} ,\uptheta} ( \cdot ) \|_{\mathcal{L}^{2}({\varOmega}) }\) is given. By (25), we get
Substituting (28) into the above equation, we get
In the next step, we prove the bound for the second term. Similar to the above equation, we have
Applying Hölder’s inequality, we have
where
By bound condition (12), we get
Hence, we deduce that
In view of (4), (17), and Lemma (2.6), we have that
From the above, we can conclude that
This ends the proof. □
5 Numerical example
In this section, we present a numerical example to illustrate the proposed method.
Step 1: Set up some essentials for example. Let \(\alpha, \beta \in(0,1)\), \({T} =1\) be a fixed value, \(\boldsymbol {\varOmega }= [0,\pi]\) and \(\mathscr{B} =- \Delta\) on \(\mathcal{L}^{2}(0,\pi)\). Then we have the eigenvalues \({a}_{p} = p^{2}\), \(p = 1,2,\dots\), and the corresponding eigenfunction \(w_{p}(\textrm{x}) = \sqrt{\frac{2}{\pi }}\sin(p\textrm{x})\).
To perform the calculation in this example, we use the Matlab software, we use the code made by JC Medina [19] to compute the integral via Simpson’s rule in the interval \([a,b]\), and the Mittag-Leffler function by Igor Podlubny [22].
Next, we choose the exact data function g, \(\mathscr {F}\), and \(\mathcal {Q}\) as follows:
Step 2: Partitioning of axes: Let \(N_{\textrm{x}},N_{t}\) be given positive integers, a uniform Cartesian grid is given by
Step 3: Model of noise: From (9) and (33), the value of the final time is given by
Then we consider the noise model satisfying (see an example in Fig. 1)
where the noise level \(\upepsilon\longrightarrow0^{+}\) and the function \(\operatorname{randn}(\cdot)\) generates arrays of random numbers whose elements are normally distributed.
The function \(\mathscr{H}\) and \(\mathscr{H}^{\upepsilon}\) for \(\textrm{x} \in[0,\pi]\), \(\upalpha= \upbeta= 0.5\), and \(\upepsilon = 0.1\)
Step 4: The regularization results:
Case 1: Applying the fractional Landweber regularization method, we have the following solution with the observation data \(\mathscr{H} ^{\upepsilon}\):
here \(\varTheta:= \mathscr{H} - \sum_{p=1}^{N(p)} ({E_{\upalpha ,1}} (- \frac{{a}_{p} }{\upbeta^{\upalpha}} ) \textrm{g} _{p} ) w_{p}(\textrm{x} )\), and \(\varTheta^{\upepsilon} := \mathscr{H}^{\upepsilon} - \sum_{p=1}^{N(p)} ({E_{\upalpha,1}} (- \frac{{a}_{p} }{\upbeta^{\upalpha }} ) \textrm{g} _{p} ) w_{p}(\textrm{x} )\), \(\uptheta\in( \frac{1}{2}, 1 ] \) is the fractional order, \(\mathtt{m} > 0 \) is the iterative step, and \(N(p)\) is a truncation parameter of Fourier series.
Case 2: Applying the fractional Tikhonov regularization method, we have the following solution with the observation data \(\mathscr{H} ^{\upepsilon}\):
where the regularization parameter \(\textrm{n} > 0 \), and \(\uptheta \in( \frac{1}{2}, 1 ] \) is the fractional order.
The absolute error estimation Error between the exact source function and the regularized source function in two cases is as follows:
The results of this section are presented in Table 1, Figs. 2, 3, and 4. In Table 1, we present the error estimation between the exact and regularized source functions. We also show the graph of the source functions for \(\upalpha = \upbeta= 0.9\) and \(\upepsilon\in\{0.1, 0.01, 0.001\}\), respectively. From the error table and the figures above, we can see that the smaller the ε, the better the computed approximation. In particular, the regularized source function approaches exact source function as ε tends to zero.
The source functions \(\mathcal{F}(\textrm{x},t) =\mathscr {F}(\textrm{x}) \mathcal{Q}(t)\), \(\mathcal{F}_{1}^{\upepsilon}(\textrm{x},t) =\mathscr{F}_{\mathtt{m},\uptheta}^{1,\upepsilon}(\textrm{x}) \mathcal{Q}(t)\), and \(\mathcal{F}_{2}^{\upepsilon}(\textrm{x},t) =\mathscr{F}_{\mathtt{n},\uptheta}^{2,\upepsilon}(\textrm{x}) \mathcal{Q}(t)\) for \((\textrm{x}, t) \in(0,\pi) \times(0,1)\), \(\upepsilon= 0.1\), \(\upalpha=\upbeta =0.9\)
The source functions \(\mathcal{F}_{1}^{\upepsilon}(\textrm{x},t) =\mathscr{F}_{\mathtt{m},\uptheta}^{1,\upepsilon}(\textrm{x}) \mathcal{Q}(t)\) and \(\mathcal{F}_{2}^{\upepsilon}(\textrm{x},t) =\mathscr {F}_{\mathtt{n},\uptheta}^{2,\upepsilon}(\textrm{x}) \mathcal{Q}(t)\) for \((\textrm{x}, t) \in(0,\pi) \times(0,1)\), \(\upepsilon= 0.01\), \(\upalpha=\upbeta =0.9\)
The source functions \(\mathcal{F}_{1}^{\upepsilon}(\textrm{x},t) =\mathscr{F}_{\mathtt{m},\uptheta}^{1,\upepsilon}(\textrm{x}) \mathcal{Q}(t)\) and \(\mathcal{F}_{2}^{\upepsilon}(\textrm{x},t) =\mathscr {F}_{\mathtt{n},\uptheta}^{2,\upepsilon}(\textrm{x}) \mathcal{Q}(t)\) for \((\textrm{x}, t) \in(0,\pi) \times(0,1)\), \(\upepsilon= 0.001\), \(\upalpha=\upbeta =0.9\)
6 Conclusion
The paper considers the regularization problem for the time-fractional diffusion equation with the hyper-Bessel operator. Firstly, through an example, we proved that the backward problem is not well posed (in the sense of Hadamard). Secondly, by the fractional Landweber and Tikhonov methods, we showed the results of the convergence rates for the regularized solution to the exact solution by using a priori and a posteriori regularization parameter choice rules.
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Luc, N.H., Huynh, L.N., Baleanu, D. et al. Identifying the space source term problem for a generalization of the fractional diffusion equation with hyper-Bessel operator. Adv Differ Equ 2020, 261 (2020). https://doi.org/10.1186/s13662-020-02712-y
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DOI: https://doi.org/10.1186/s13662-020-02712-y