- Research
- Open Access
- Published:
On backward problem for fractional spherically symmetric diffusion equation with observation data of nonlocal type
Advances in Difference Equations volume 2021, Article number: 445 (2021)
Abstract
The main target of this paper is to study a problem of recovering a spherically symmetric domain with fractional derivative from observed data of nonlocal type. This problem can be established as a new boundary value problem where a Cauchy condition is replaced with a prescribed time average of the solution. In this work, we set some of the results above existence and regularity of the mild solutions of the proposed problem in some suitable space. Next, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given by the fractional Tikhonov method and convergence rate between the regularized solution and the exact solution under a priori parameter choice rule and under a posteriori parameter choice rule.
1 Introduction
In recent decades, the study of noninteger diffusion equations has received great attention from mathematicians around the world. These models have many applications in various types of research fields, for example, thermal diffusion in fractal domains [1] and protein dynamics [2], finance [3], systems biology [4], physics [5] and medicine [6], and besides, there are also some references as follows [7–11], and [12]. In this work, we consider the following problem:
Here Caputo fractional derivative \(D_{t}^{\beta }\) is defined as follows:
and the source function \(G(r,t) \in L^{\infty }([0,R];r^{2})\), the final data \(f(r) \in L^{2}([0,R],r^{2})\) are given. Note that when the fractional order β is equal to 1, the fractional derivative \(D_{t}^{\beta }u(r,t)\) is equal to the first-order derivative \(\frac{du}{dt}\) (see in [13]), and thus problem (1.1) reproduces the classical diffusion problem. In practice, the input data \((f,G)\) is noisy by the observed data \(f^{\varepsilon }\), \(G^{\varepsilon }\) which satisfy
Our problem is called inverse problem and its solution is not stable. This property is called ill-posed in the sense of Hadamard. In other words, easier to understand, if ϵ is small, it will lead to large errors for the corresponding solution if using an unapproximate model for observed data \(f^{\varepsilon }\), \(G^{\varepsilon }\). The question mentioned in this paper is: Find an approximation method for the solution of the problem with noisy input data \(f^{\varepsilon }\), \(G^{\varepsilon }\).
Before discussing the main results, we would like to outline a few previous papers that mentioned problem (1.1).
-
If \(\beta =1\), \(\xi _{2}=0\), \(\xi _{1}=1\), and \(G=0\), then the last condition in (1.1) becomes the final condition
$$ u(r,T)= f(r). $$(1.4)In such a case, problem (1.1) is called backward problem. Then the authors [14] used a modified Tikhonov regularization method for solving problem. In [15], the authors used a spectral method for regularizing the problem.
-
If \(\beta =1\), \(\xi _{2}=0\), \(\xi _{1}=1\), and \(G(r,t) = \varphi (t)f(r)\), WeiCheng et al. [16] applied a spectral method to approximate the backward problem and obtained Hölder type estimate with a suitable choice of regularization parameter.
-
If \(\beta \neq 1\) and \(\xi _{2}=0\), \(\xi _{1}=1\) and \(G(r,t) = 0\), then the authors [17] proposed the quasi-boundary regularization method to solve problem (1.1). They showed convergence estimates between the regularization solution and the exact solution presented under the a priori and a posteriori regularization parameter choice rules.
-
If \(\beta \neq 1\) and \(\xi _{2}=0\), \(\xi _{1}=1\) and \(G(r,t) = f(r)\), Yang et al. [18] investigated problem (1.1) and provided the estimate of Hölder type.
Let us mention some interesting papers with many various methods for the case \(G(r,t) \neq 0\), for example, the truncation method [19], iterated fractional Tikhonov regularization method [20], and the references therein. Besides, regarding other regularization methods and applications, readers can view the following references: [21–28].
Our novel point in this paper is to replace the final condition (1.4) with the nonlocal condition \(\xi _{1} u(r,T) + \xi _{2} \int _{0}^{T} u(r,t)\,dt = f(r)\) as introduced in the last condition of our problem. This condition is proposed in the paper by Dokuchaev [29]. Very recently, Tuan and co-authors used this condition to solve some nonlocal problem, for example [30–32], and [33]. Motivated by this above reason, in this paper, we apply the fractional Tikhonov method to solve problem (1.1). To the best of authors’ knowledge, there are not any results concerning problem (1.1). Our paper investigates problem (1.1), and the main results of this work are as follows:
-
We give the stability and the regularity of the mild solution.
-
We show the ill-posedness and the conditional stability of solution in \(L^{2}([0,R];r^{2})\).
-
We propose a regularized method and prove the convergence rate under a priori parameter choice rule and a posteriori parameter choice rule.
Let us say that in an analytical sense, our problem seems to be more complicated than the models studied before.
This paper is organized as follows. Section 2 gives some preliminaries that are needed throughout the paper. In Sect. 3, we show the sought solution of problem (1.1), and an example describes the ill-posedness of the problem. In Sect. 4, we study the fractional Tikhonov method to solve problem (1.1) and show the convergence rate under a priori parameter choice rule and a posteriori parameter choice rule. Finally, we add the conclusion for this paper.
2 Preliminaries
In this paper, we denote by \(L^{2}([0,R];r^{2})\) the Hilbert space of Lebesgue measurable functions \(u(r,t)\) with weight \(r^{2}\) on \([0,R]\)
with the scalar product \(\|u \|_{L^{2}([0,R],r^{2})} = ( \int _{0}^{R} r^{2} |u(r) |^{2}\,dr )^{\frac{1}{2}} \), and the \(L^{\infty }(0,T;Y)\) consists of all measurable functions \(u:[0,T] \to Y\) with
where Y is a real Banach space with the norm \(\|\cdot \|_{Y}\). We define the space
then \(\mathcal{H}^{s}([0,R];r^{2})\) is a Hilbert space with the norm
For any constant \(\alpha > 0\) and \(\beta \in \mathbb{R}\), the Mittag-Leffler function is defined as follows:
where \(\alpha > 0\) and \(\beta \in \mathbb{R}\) are arbitrary constants.
Lemma 2.1
([18])
Assuming that \(0 < \beta _{0} < \beta _{1} < 1\), then there exist constants \(\overline{D}_{1}\) and \(\overline{D}_{2}\) depending only on β, \(\beta _{1}\) such that for all \(\beta \in [\beta _{0},\beta _{1}]\) there hold
Lemma 2.2
([19])
For \(\lambda _{j} \geq \lambda _{1} > 0\), \(\lambda _{j}\) is the eigenvalues satisfying \(\lambda _{1} < \lambda _{2} < \cdots < \lambda _{j} \to \infty \) as \(j \to \infty \), then there exist constants \(\overline{D}_{3}\) and \(\overline{D}_{4}\) depending only on β, T, \(\lambda _{1}\) such that
Proof
This proof can be found in [20]. □
Lemma 2.3
Let \(G \in L^{\infty }(0,T;L^{2}([0,T],r^{2}))\), then the exists a positive constant \(\overline{D}_{5}\) such that
Proof
This lemma provision can be found in [20]. □
Lemma 2.4
For any \(j \geq 1\), we have the following estimate:
in which \(\underline{E} = ( \frac{\overline{D}_{3}R^{2}}{\pi ^{2} } + \frac{\xi _{2}\overline{D}_{3}R^{2}}{\pi ^{2}} \frac{T}{\frac{R^{2}}{\pi ^{2}}+ T^{\beta }} )\), \(\overline{E} = ( \frac{\xi _{1}\overline{D}_{4}R^{2}}{\pi ^{2}} + \frac{\xi _{2} \overline{D}_{4} R^{2}}{\pi ^{2}} \frac{T^{1-\beta }}{1-\beta } )\).
Proof
From Lemma 2.2, we need to show that
This leads to
Next, due to the fact, we also get
which implies that
 □
In this section, we need the solution of the direct problem of (1.1)
From [20], by using the Fourier expansion, we know that
where \(\psi _{j}(r) = \frac{\sqrt{2} j \pi }{\sqrt{R^{3}}} j_{0} ( \frac{j \pi r}{R} )\), and \(j_{0}(y)\) denotes the 0th order spherical Bessel functions of the first kind. Besides, we know that \(\{\psi _{j}(r)\}_{j=1}^{\infty }\) from an orthonormal basis in \(L^{2}([0,T],r^{2})\).
3 The mild solution of problem (1.1)
Theorem 3.1
Let \(f \in L^{2}([0,R];r^{2}) \) and \(G \in L^{\infty }(0,T;L^{2}([0,R];r))\). Let us further assume that
Then problem (1.1) has a unique solution u given as follows:
Then we get the following regularity:
Here \(\mathcal{M}\), MÌ… are defined later. Let \(f \in \mathcal{H}^{s}([0,R];r^{2})\) and \(G \in L^{2}(0,T;\mathcal{H}([0,R];r^{2})) \cap L^{\infty }(0,T; \mathcal{H}^{s-\gamma }([0,R];r^{2}))\) for \(s > \gamma + 1\), then \(u \in L^{p}(0,T;\mathcal{H}^{s}([0,R];r^{2}))\) and the regularity result holds
Proof
From (2.15) and using the nonlocal condition in problem (1.1), we obtain that
By integrating both sides from 0 to T for equation (2.15), we immediately have the following equality:
From two observations (3.5) and (3.6), we get the following equality:
Our next aim is to express the formula of the function ℓ in terms of two input data f and G. In view of the nonlocal condition as in the last condition of problem (1.1)
we find the following identity for the Fourier coefficient of the function â„“:
Therefore, by taking Fourier series for the term \(u_{j}(t)\), the formula of the mild solution to problem (1.1) can be given by
Using the inequality \((a+b+c+d)^{2} \leq 4 (a^{2} + b^{2} + c^{2} + d^{2} )\), for \(a,b,c,d \geq 0\), we have
Now, we give the regularity result of a mild solution. First of all, from Lemma 2.4, it gives
From now on, in short, we denote \(\mathcal{M}^{2}(\beta ,\underline{E},R,\pi ,t, \overline{D}_{2}) = ( \frac{\overline{D}_{2}}{\underline{E}\Gamma (1-\beta )} )^{2} (\frac{R^{2}}{\pi ^{2}t^{\beta }} )^{2}\). Next, let us evaluate for \(\|\mathcal{A}_{i}(\cdot ,t)\|^{2}_{L^{2}([0,T];r^{2})} ,i = \overline{1,4}\), one by one.
Step 1: Estimate of \(\|\mathcal{A}_{1}(\cdot ,t) \|^{2}_{L^{2}([0,T];r^{2})}\). By using the estimation for (3.11), we have
Step 2: Applying Lemma 2.3 and the estimation of (3.11), \(\|\mathcal{A}_{2}(\cdot ,t) \|^{2}_{L^{2}([0,T];r^{2})}\) can be bounded as follows:
Step 3: Similarly, the estimate of \(\|\mathcal{A}_{3}(\cdot ,t) \|^{2}_{L^{2}([0,T];r^{2})}\) is given by
Step 4: By using Lemma 2.3, the estimate of \(\|\mathcal{A}_{4}(\cdot ,t) \|^{2}_{L^{2}([0,T];r^{2})}\) is given by
Combining (3.12), (3.13), (3.14), and (3.15), we have the estimate
Part B: Similar to part A, let us divide this part into the following steps.
Step 1: Estimate for \(\|\mathcal{A}_{1}(\cdot ,t)\|_{\mathcal{H}^{s}([0,R];r^{2})}\), we have
Denoting \(\mathcal{F}^{2}_{1} = (\underline{E}^{2})^{-1} \frac{\overline{D}_{4}^{2} R^{4}}{\pi ^{4}}\), taking the square root on both sides, we have
Step 2: Estimate for \(\|\mathcal{A}_{2}(\cdot ,t)\|_{\mathcal{H}^{s}([0,R];r^{2})}\), for any \(0 < \gamma < 1\), we receive
Denoting \(\mathcal{F}_{2} = \overline{D}_{6} (\frac{\pi }{R} )^{-2\gamma }\). Therefore, we can find that
Using the Hölder inequality, we get the following estimate:
From (3.20) and (3.21), one has the following bound for the second term \(\mathcal{A}_{2}(\cdot ,t)\):
From (3.22), we conclude that
Step 3: Estimate for \(\|\mathcal{A}_{3}(\cdot ,t)\|_{\mathcal{H}^{s}([0,R];r^{2})}\), by using (3.19), we receive
By applying the Hölder inequality and denoting \(0 < \gamma < \min \{1,\frac{1}{2\beta } \}\), we find the following bound:
From (3.24), with \(\mathcal{F}_{3}^{2} = \xi _{2}^{2}\mathcal{F}_{1}^{2}T\overline{D}_{6}^{2} ( \frac{T^{1-2\gamma \beta }}{1-2\gamma \beta } )\), by two above observations, we deduce that
Step 4: Estimate for \(\|\mathcal{A}_{4}(\cdot ,t)\|_{\mathcal{H}^{s}([0,R];r^{2})}\), using (3.19) and by applying the Hölder inequality, \(\|\mathcal{A}_{4}(\cdot ,t)\|_{\mathcal{H}^{s}([0,R];r^{2})}\) can be bounded as follows:
Combining four steps as above, we conclude that
From (3.28), by choosing \(1 < p < \frac{1}{\beta }\), then the integral \(\int _{0}^{T} t^{-\beta p }\,dt \) is convergent, we have a comment as follows:
this implies that
 □
4 Ill-posedness and conditional stability of problem (1.1)
4.1 The ill-posedness of problem (1.1)
Theorem 4.1
The inverse problem (1.1) is ill-posed in the case \(t=0\).
Proof
A linear operator \(\mathcal{K}:L^{2}([0,R];r^{2}) \to L^{2}([0,R];r^{2}) \) is as follows:
where
It is obvious that \(\ell (r,z)=\ell (z,r)\), we know \(\mathcal{K}\) is a self-adjoint operator. Next, we are going to prove its compactness and consider the finite rank operators as follows:
From (4.2), using the inequality \((a+b)^{2} \leq 2(a^{2}+b^{2})\), \(a,b \geq 0\), we have
Therefore, \(\|\mathcal{K}_{N}\ell - \mathcal{K}\ell \|_{L^{2}([0,R];r^{2})}\) in the sense of operator norm in \(L(L^{2}([0,R];r^{2});L^{2}([0,R]; r^{2}))\) as \(N \to \infty \). Also, \(\mathcal{K}\) is a compact operator. Next, the SVDs for the linear self-adjoint compact operator \(\mathcal{K}\) is \(\mathcal{K} \ell = f\), and by Kirsch [34], we conclude that the problem is ill-posed. Next, we continue to give an example for ill-posedness. In here, we assume \(f \in L^{2}([0,R];r^{2})\) and \(G \in L^{\infty }(0,T;L^{2}([0,R];r^{2}))\) and \(f = G = 0\). In here, we choose
Let us choose input final data \(f=0\), we know that an error in the \(L^{2}([0,R];r^{2})\) norm between two input final data is as follows:
On the other hand, because of \(\beta \in (0,1)\), one has
Combining (4.5) and (4.6) yields that
Setting \(u_{k}(r,0)\) is the solution of (1.1), we obtain
Using the inequality \(|a - b| > |a| - |b|\), we know that
From (4.9), we get
Applying the inequality \((a+b)^{2} \le 2(a^{2}+b^{2})\), \(\forall a,b \ge 0\) and Hölder’s inequality, we get
Combining (4.4), (4.5), and (4.6), we obtain
From the aforementioned inequality, we get the following estimate:
Hence, we come up with the estimate
which allows us to give that
Using the inequality \(\sqrt{a^{2}+b^{2}} \le a+b\), \(\forall a, b \ge 0\), one has
Therefore, we have that
Thus, problem (4.8) is, in general, ill-posed in the Hadamard sense in the \(L^{2}([0,R];r^{2})\)-norm. □
4.2 The conditional stability of the solution for problem (1.1)
Theorem 4.2
Let us assume that \(\mathcal{C}\) is a positive constant such that \(u(\cdot ,0)\in \mathcal{H}^{s}([0,R];r^{2})\) for some \(s>0\) satisfying the following a priori bound condition:
Then we have
where
Proof
By using Parseval’s equality, we obtain that
The term \(Q_{1}\) can be estimated by using Lemma 2.4 as follows:
The term \(\mathcal{Q}_{2}\) can be bounded as follows:
5 The fractional Tikhonov method and convergence rate
5.1 The fractional Tikhonov method
In this section, we apply the fractional Tikhonov method given by Morigi [35]. From now on, we denote
we propose the following regularized solution with exact data \((f,G)\):
in which
However, if the measured data \(\{f,G\}\) are noised by \(\{f_{\epsilon },G_{\epsilon }\}\), then we get
in which
where \(\delta \in (\frac{1}{2};1 ]\) and \(\alpha > 0\). Noting that when \(\delta = 1\), the fractional Tikhonov method becomes a standard Tikhonov regularization.
Lemma 5.1
Let \(\delta \in (\frac{1}{2};1]\), we have
with \(\mathcal{Z}\) depending on δ.
Proof
The proof of lemma can be found in [36]. □
5.2 An a priori parameter choice rule
Let us consider the operator
for \(\nu \in L^{2}([0,R];r^{2})\) and \(0 \leq t \leq T\).
By applying the fractional Tikhonov method, we can see that
By choosing the regularization parameter α, the following theorem gives that the choice α is valid by using suitable assumptions. In order to give error estimate, let us assume that \(\|\Xi \|_{\mathcal{H}^{s}([0,R];r^{2})} \leq \mathcal{C}\) for any \(s > 0\), where \(\mathcal{C}\) is a positive constant. Before going to the main theorem, we have auxiliary lemmas as follows.
Lemma 5.2
For some positive constant, we get
Proof
This lemma is proven similarly [36]. □
Theorem 5.3
Let \(f \in L^{2}([0,R];r^{2})\) and \(G \in L^{\infty }([0,T];L^{2}([0,R];r^{2}))\), inside Θ performed as in the digital formula (5.2), and if we choose the parameter regularization
then it gives:
-
If \(0 < s < 1\), then we have
$$ \bigl\Vert u_{\alpha ,\delta }^{\epsilon }(\cdot ,0) - u(\cdot ,0) \bigr\Vert _{L^{2}([0,R];r^{2})} \textit{ is of order } \epsilon ^{\frac{s}{s+1}}. $$(5.9) -
If \(s \geq 1\), then we have
$$ \bigl\Vert u_{\alpha ,\delta }^{\epsilon }(\cdot ,0) - u(\cdot ,0) \bigr\Vert _{L^{2}([0,R];r^{2})} \textit{ is of order } \epsilon ^{\frac{1}{2}}. $$(5.10)
Proof
From the triangle inequality, we have
First of all, we have the estimate \(\|u_{\alpha ,\delta }^{\epsilon }(\cdot ,0) - u_{\alpha ,\delta }( \cdot ,0) \|_{L^{2}([0,R];r^{2})}\). Now, using the inequality \((a+b)^{2} \leq 2 ( a^{2} + b^{2} ) \) gives
From (5.12), applying Lemma 2.3 and the estimate of (1.3), it is easy to see that
Hence, we conclude that
Moreover, with 〈Ξ, \(\psi _{j}\rangle \) defined in (5.2), we also get
From the definition of \(\|\Xi (\cdot )\|_{\mathcal{H}^{s}([0,R];r^{2})} \leq \mathcal{C}\), for any \(s>0\), \(\delta \in (\frac{1}{2},1]\), we obtain
Applying Lemma 5.2 and Lemma 2.4, where we used \(|\Theta _{\beta ,j}^{\xi _{1},\xi _{2}}(\pi ,R,T) |^{2} \geq \frac{\underline{E}^{2}}{j^{4}}\), we get
Combining (5.15), (5.16), and (5.17), we obtain
From (5.14) and (5.18), combining the inequality \(\sqrt{a^{2} + b^{2}} \leq a + b\), \(\forall a,b \geq 0\), we deduce that
Choose the regularization parameter α as
then we have:
-
If \(0 < s < 1\), then we have the following estimate:
$$ \bigl\Vert u_{\alpha ,\delta }^{\epsilon }(\cdot ,0) - u(\cdot ,0) \bigr\Vert _{L^{2}([0,R];r^{2})} \text{ is of order } \epsilon ^{\frac{s}{s+1}}. $$(5.21) -
If \(s \geq 1\), then we have the following estimate:
$$ \bigl\Vert u_{\alpha ,\delta }^{\epsilon }(\cdot ,0) - u(\cdot ,0) \bigr\Vert _{L^{2}([0,R];r^{2})} \text{ is of order } \epsilon ^{\frac{1}{2}}. $$(5.22)
The proof is completed. □
5.3 An a posteriori parameter choice rule
In this subsection, considering the choice of the a posteriori regularization parameter in Morozov’s discrepancy principle, [37] we choose the regularization parameter α such that
where \(\|\Xi ^{\epsilon }\|_{L^{2}([0,R];r^{2})}\geq \zeta \epsilon \), \(\zeta > 1\) depends on ϵ.
Lemma 5.4
For some positive constants k, α, \(\mathcal{N}\), ς, we get
Proof
This lemma is proven similarly [36]. □
Lemma 5.5
From (5.23), if we can find that ζ is satisfied, then we have the estimate of α as follows:
Proof
From (5.23), it gives
We have the estimate ζϵ through two steps as follows, one by one.
Step 1: Estimate of \(\mathcal{X}_{1}\), to do this, we recall Ξ, \(\Xi ^{\epsilon }\) from expressions (5.2) and (5.4), and we have
Step 2: Estimate of \(\mathcal{X}_{2}\), using again the a priori bound condition of Ξ, we obtain
From (5.28), using Lemma 2.4 and Lemma 5.4 implies that
From the analytics assessment on the side, we get
This yields
 □
Theorem 5.6
Assume that (1.3) holds, recalling the α in Lemma 5.5, then we have the following estimate:
Proof
From the triangle inequality, we get
From (5.14), we obtain
Substituting (5.31) into above equation, one has
Next, we give the estimate \(\| u_{\alpha ,\delta }(\cdot ,0) - u(\cdot ,0) \|_{L^{2}([0,R];r^{2})}\) as follows:
From (5.36), applying the Hölder inequality, we get
Because of \(\|\Xi \|_{\mathcal{H}^{s}([0,R];r^{2})} \leq \mathcal{C}\), we obtain
Next, using Lemma 2.4, \(\mathcal{I}_{1}\) can be bounded as follows:
Combining (5.37) to (5.39), we conclude that
From (5.33) and (5.40), we know that
This ends the proof of this theorem. □
6 Conclusion
In this paper, we focus on the spherically symmetric backward time-fractional diffusion equation with the nonlocal integral condition. By using some properties of the Mittag-Leffler function, we show two results as follows. First of all, we show the properties of the well-posedness and regularity of the mild solution to this problem. Next, we present that our problem is ill-posed. In addition, we construct a regularized solution and present the convergence rate between the regularized and exact solutions by the fractional Tikhonov method under a priori parameter choice rule and under a posteriori parameter choice rule.
Availability of data and materials
Not applicable.
References
Alsaedi, A., Ahmad, B., Kirane, M.: Maximum principle for certain generalized time and space fractional diffusion equations. Q. Appl. Math. 73(1), 163–175 (2015)
Bianchi, D., Buccini, A., Donatelli, M.: Iterated fractional Tikhonov regularization. Inverse Probl. 31(5), 055005 (2015)
Sabatelli, L., Keating, S., Dudley, J., Richmond, P.: Waiting time distributions in financial markets. Eur. Phys. J. B 27, 273–275 (2002)
Yuste, S.B., Lindenberg, K.: Subdiffusion-limited reactions. Chem. Phys. 284, 169–180 (2002)
Liu, J.J., Yamamoto, M.: A backward problem for the time-fractional diffusion equation. Appl. Anal. 89(11), 1769–1788 (2010)
Hall, M.G., Barrick, T.R.: From diffusion-weighted MRI to anomalous diffusion imaging. Magn. Reson. Med. 59, 447–455 (2008)
Lazreg, J.E., Abbas, S., Benchohra, M., Karapınar, E.: Impulsive Caputo–Fabrizio fractional differential equations in b-metric spaces. Open Math. 19, 363–372 (2021). https://doi.org/10.1515/math-2021-0040
Adiguzel, R.S., Aksoy, U., Karapınar, E., Erhan, I.M.: On the solutions of fractional differential equations via Geraghty type hybrid contractions. Appl. Comput. Math. 20(2), 313–333 (2021)
Adiguzel, R.S., Aksoy, U., Karapınar, E., Erhan, I.M.: On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.6652
Adiguzel, R.S., Aksoy, U., Karapınar, E., Erhan, I.M.: Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions. Rev. R. Acad. Cienc. Exactas FÃs. Nat., Ser. A Mat. 115, 155 (2021). https://doi.org/10.1007/s13398-021-01095-3
Nghia, B.D., Luc, N.H., Binh, H.D., Long, L.D.: Regularization method for the problem of determining the source function using integral conditions. Adv. Theory Nonlinear Anal. Appl. 5(3), 351–362 (2021)
Abdelouaheb, A.: Asymptotic stability in Caputo–Hadamard fractional dynamic equations. Res. Nonlinear Anal. 4(2), 77–86 (2021)
Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1990)
Cheng, W., Ma, Y.-J., Fu, C.L.: A regularization method for solving the radially symmetric backward heat conduction problem. Appl. Math. Lett. 30, 38–43 (2014)
Cheng, W., Fu, C., Qin, F.: Regularization and error estimate for a spherically symmetric backward heat equation. J. Inverse Ill-Posed Probl. 19(3), 369–377 (2011)
Cheng, W., Fu, C.L.: Identifying an unknown source term in a spherically symmetric parabolic equation. Appl. Math. Lett. 26, 387–391 (2013)
Yang, F., Wang, N., Li, X.-X., Huang, C.-Y.: A quasi-boundary regularization method for identifying the initial value of time-fractional diffusion equation on spherically symmetric domain. J. Inverse Ill-Posed Probl. 27(5), 609–621 (2019). https://doi.org/10.1515/jiip-2018-0050
Yang, F., Zhang, M., Li, X.-X., Ren, Y.-P.: A posteriori truncated regularization method for identifying unknown heat source on a spherical symmetric domain. Adv. Differ. Equ. 2017, 263 (2017). https://doi.org/10.1186/s13662-017-1276-1
Yang, F., Pu, Q., Li, X.-X.: The fractional Tikhonov regularization methods for identifying the initial value problem for a time-fractional diffusion equation. J. Comput. Appl. Math. 380, 112998 (2020). https://doi.org/10.1016/j.cam.2020.112998
Shuping, Y., Xiangtuan, X., Yan, N.: Iterated fractional Tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equation. Appl. Numer. Math. 160, 217–241 (2021)
Alqahtani, B., Aydi, H., Karapınar, E., Rakocevic, V.: A solution for Voltera fractional integral equations by hybrid contractions. Mathematics 7, 694 (2019)
Afshari, H., Kalantari, S., Karapınar, E.: Solution of fractional differential equations via couple fixed point. Electron. J. Differ. Equ. 2015, 286 (2015)
Salim, A., Benchohra, B., Karapınar, E., Lazreg, J.E.: Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations. Adv. Differ. Equ. 2020, 601 (2020)
Karapınar, E., Fulga, A., Rashid, M., Shahid, L., Aydi, H.: Large contractions on quasi-metric space with an application to nonlinear fractional differential-equations. Mathematics 2019(7), 444 (2019)
Tuan, N.H., Baleanu, D., Thach, T.N., O’Regan, D., Can, N.H.: Approximate solution for a 2-D fractional differential equation with discrete random noise. Chaos Solitons Fractals 133, 109650 (2020)
Tuan, N.H., Zhou, Y., Can, N.H.: Identifying inverse source for fractional diffusion equation with Riemann–Liouville derivative. Comput. Appl. Math. 39(2), 1–16 (2020)
Phuong, N.D., Hoan, L.V.C., Karapınar, E., Singh, J., Binh, H.D., Can, N.H.: Fractional order continuity of a time semi-linear fractional diffusion-wave system. Alex. Eng. J. 59(6), 4959–4968 (2020)
Tuan, N.H., Au, V.V., Can, N.H.: Regularization of initial inverse problem for strongly damped wave equation. Appl. Anal. 97(1), 69–88 (2018)
Dokuchaev, N.: On recovering parabolic diffusions from their time-averages. Calc. Var. Partial Differ. Equ. 58(1), Paper No. 27 (2019)
Tuan, N.H., Triet, N.A., Luc, N.H., Phuong, N.D.: On a time fractional diffusion with nonlocal in time conditions. Adv. Differ. Equ. 2021, 204 (2021)
Thach, T.N., Can, N.H., Tri, V.V.: Identifying the initial state for a parabolic diffusion from their time averages with fractional derivative. Math. Methods Appl. Sci., 1–16 (2021). https://doi.org/10.1002/mma.7179
Luc, N.H., Dumitru, B., Agarwal, R.P., Long, L.D.: Identifying the source function for time fractional diffusion with non-local in time conditions. Comput. Appl. Math. 40, 159 (2021). https://doi.org/10.1007/s40314-021-01538-y
Phuong, N.D., Baleanu, D., Phong, T.T., Long, L.D.: Recovering the source term for parabolic equation with nonlocal integral condition. Math. Methods Appl. Sci., 1–16 (2021). https://doi.org/10.1002/mma.7331
Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problem. Springer, Berlin (1996)
Morigi, S., Reichel, L., Sgallari, F.: Fractional Tikhonov regularization with a nonlinear penalty term. J. Comput. Appl. Math. 324, 142–154 (2017)
Tuan, N.H., Huynh, L.N., Baleanu, D., Can, N.H.: On a terminal value problem for a generalization of the fractional diffusion equation with hyper-Bessel operator. Math. Methods Appl. Sci. (2019). https://doi.org/10.1002/mma.6087
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, vol. 375. Kluwer Academic, Dordrecht (1996)
Acknowledgements
The authors are thankful to the area editor for giving valuable comments and suggestions.
Funding
No funding.
Author information
Authors and Affiliations
Contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Long, L.D., Van, H.T.K., Binh, H.D. et al. On backward problem for fractional spherically symmetric diffusion equation with observation data of nonlocal type. Adv Differ Equ 2021, 445 (2021). https://doi.org/10.1186/s13662-021-03603-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-021-03603-6
MSC
- 35K05
- 35K99
- 47J06
- 47H10x
Keywords
- Backward in time
- Fractional spherically symmetric diffusion equation
- Fractional Tikhonov method
- Ill-posed problem
- Convergence estimates
- Regularization