Abstract
In this paper, we study the inverse problem for a class of abstract ultraparabolic equations which is well known to be illposed. We employ some elementary results of semigroup theory to present the formula of solution, then show the instability cause. Since the solution exhibits unstable dependence on the given data functions, we propose a regularization method to stabilize the solution, then obtain the error estimate. A numerical example shows that the method is efficient and feasible. This work slightly extends the earlier results in Zouyed and Rebbani (J. Inverse IllPosed Probl. 22(4):449466, 2014).
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1 Introduction
Let us denote by \(\Vert \cdot \Vert \) the norm and by \(\langle \cdot,\cdot\rangle\) the inner product in \(L^{2} (0,\pi)\), i.e.,
In this paper, we consider the following problem: determine a function \(u: [0,T ]\times[0,T ]\to L^{2} (0,\pi)\) solution to the Cauchy problem
with corresponding perturbed data functions \((\psi^{\varepsilon },\varphi^{\varepsilon} )\) satisfying
where \(\psi^{\varepsilon}\) and \(\varphi^{\varepsilon}\) play roles as perturbed functions and \(\varepsilon>0\) represents a bound between the exact function \((\varphi,\psi)\) and the perturbed one \((\varphi ^{\varepsilon},\psi^{\varepsilon} )\) over \(L^{2} (0,\pi)\) and the given function f is called the source function.
Ultraparabolic equations arise in several areas of science, such as mathematical biology in population dynamics [1] and probability in connection with multiparameter Brownian motion [2], and in the theory of boundary layers [3]. From those applications, ultraparabolic equations have gained considerable attention in many mathematical aspects (see, e.g., [1, 4–9] and the references therein).
In the mathematical literature, various types of ultraparabolic problems have been solved. There have been some papers dealing with the existence and uniqueness of solution for many kinds of ultraparabolic equations, e.g., [1, 10, 11]. As the pioneer in numerical methods for such equations, Akrivis et al. [5] numerically approximated the solution of a prototype ultraparabolic equation by applying a fixedstep backward Euler scheme and a secondorder boxtype finite difference method. Some extension works for the numerical angle that should be mentioned are [12, 13] by Ashyralyev and Yilmaz, and Marcozzi, respectively. We also remark that, in general, ultraparabolic equations do not possess properties that are closely fundamental to many kinds of parabolic equations including strong maximum principles, a priori estimates, and so on.
In the phase of illposed ultraparabolic problems, the authors Zouyed and Rebbani very recently proposed in [7] the modified quasiboundary value method to regularize the solution of problem (1) in the homogeneous backward case \(f\equiv0\). In particular, via the instability terms in the form of the solution of (1) (cf. [4, Theorem 1.1]), they established an approximate problem by replacing \(\mathcal{A}_{\alpha}=\mathcal{A} (I+\alpha\mathcal{A}^{1} )\) for the operator \(\mathcal{A}\) and taking the perturbation α into final conditions of the illposed problem, and obtained the convergence order \(\alpha^{\theta}\), \(\theta\in(0,1 )\). Motivated by that work, this paper is devoted to investigating a new regularization method.
In the past, many approaches have been studied for solving illposed problems, especially the backward heat problems. For example, Lattès and Lions [14], Showalter [15] and Boussetila and Rebbani [16] used the quasireversibility method; in [17] Ames and Epperson applied the least squares method with Tikhonovtype regularization; Clark and Oppenheimer [18], Denche and Bessila [19] and Trong et al. [20] used the quasiboundary value method. Moreover, some other methods that should be listed are the mollification method by Hao [21] and the operatorsplitting method studied by Kirkup and Wadsworth [22]. To the best of the authors’ knowledge, although there are many works on several types of parabolic backward problems, the theoretical literature on regularizing the inverse problems for ultraparabolic equations is very scarce. Therefore, proposing a regularization method for problem (1) is the scope of this paper.
Our work presented in this paper has the following features. At first, for ease of the reading, we summarize in Section 2 some wellknown facts in a semigroup of operators and present the formula of the solution of (1). Secondly, in Section 3 we construct the regularized solution based on our method, then obtain the error estimate. Finally, a numerical example is given in Section 4 to illustrate the efficiency of the result.
2 Preliminaries
The operator −Δ is a positive selfadjoint unbounded linear operator on \(L^{2} (0,\pi)\). Therefore, it can be applied to some elementary results in [4, 7, 15, 23–26]. Particularly, the formula of the solution of problem (1) can be obtained by Lorenzi [4], and the authors in [23, 24] gave a detailed description on fundamental properties of the generalized operator. In this section, we thus recall those results which we want to apply to our main results in this paper. We list them and skip their proofs for conciseness.
In fact, we shall study in this section the generalized formula of the solution by the following operator equation in terms of semigroup theory.
where \(\mathcal{A}\) is a positive selfadjoint unbounded linear operator on the Hilbert space ℋ.
We denote by \(\{ E_{\lambda},\lambda>0 \} \) the spectral resolution of the identify associated to \(\mathcal{A}\). Let us denote
the \(C_{0}\)semigroup of contractions generated by \(\mathcal{A}\) (\(\mathcal{L} (\mathcal{H} )\) stands for the Banach algebra of bounded linear operators on ℋ). Then
for all \(u\in\mathcal{D} (\mathcal{A} )\). In this connection, \(u\in\mathcal{D} (\mathcal{A} )\) iff the integral (3) exists, i.e.,
For this family of operators \(\{ S (t ) \} _{t\ge0}\), we have:

1.
\(\Vert S (t )\Vert \le1\) for all \(t\ge0\);

2.
the function \(t\mapsto S (t )\), \(t>0\) is analytic;

3.
for every real \(r\ge0\) and \(t>0\), the operator \(S (t )\in\mathcal{L} (\mathcal{H},\mathcal{D} (\mathcal{A}^{r} ) )\);

4.
for every integer \(k\ge0\) and \(t>0\), \(\Vert S^{k} (t )\Vert =\Vert \mathcal{A}^{k}S (t )\Vert \le c (k )t^{k}\);

5.
for every \(x\in\mathcal{D} (\mathcal{A}^{r} )\), \(r\ge0\), we have \(S (t )\mathcal{A}^{r}x=\mathcal{A}^{r}S (t )x\).
Remark 1
In the sequel, let us denote
and make some conditions on the given functions as follows:

(A1)
\(\varphi\in C ( [0,T ];\mathcal{D} (\mathcal{A} ) )\cap C^{1} ( [0,T ];\mathcal{H} )\);

(A2)
\(\psi\in C ( [0,T ];\mathcal{D} (\mathcal{A} ) )\cap C^{1} ( [0,T ];\mathcal{H} )\);

(A3)
\(\varphi(0 )=\psi(0 )\);

(A4)
\(f\in C ( [0,T ]\times[0,T ];\mathcal{H} )\cap C^{1} (D_{1}\times D_{2};\mathcal{H} )\).
In the following theorems, we show the formula of the solution of problem (2) by employing Theorem 1.1 in [4] with \(a_{1} (t )=a_{2} (s )=1\) and following the steps in [7].
Theorem 2
Under conditions (A1)(A4), the problem
admits a unique solution u presented by the following formula. For any \((t,s )\in D_{1}\),
and for any \((t,s )\in D_{2}\),
Moreover, the solution u belongs to the space \(C ( [0,T ]\times[0,T ];\mathcal{D} (\mathcal{A} ) )\cap C^{1} ( [0,T ]\times[0,T ];\mathcal {H} )\).
Theorem 3
Under conditions (A1)(A4), if the problem
admits a solution u, then this solution can be presented by
Proof
We put \(\tau=Tt\), \(\xi=Ts\) and write
the function \(v (\tau,\xi): [0,T ]\times[0,T ]\to\mathcal{H}\) satisfies problem (4), namely
Thanks to Theorem 2, \(v (\tau,\xi)\) is given by
It follows that
Thus, we obtain
by the maps \(\zeta=T\eta\) in the integrals. We can see by the initial conditions of (5) that
which leads to
By virtue of semigroup properties, we get
Substituting (7) into (6), we thus have
□
Theorem 4
Under conditions (A1), (A2) and (A4), if problem (2) with \(\varphi(T )=\psi(T )\) admits a solution u, then this solution can be given by
Proof
Now we put \(\tau=Tt\) and \(\xi=Ts\), then write
the function \(v (\tau,\xi): [0,T ]\times[0,T ]\to\mathcal{H}\) satisfies problem (5), namely
Using Theorem 3, the solution \(v (\tau,\xi)\) can be presented by
It follows that
Hence, we obtain
which completes the proof. □
Now we return to the consideration of problem (1). All of our results in this paper apply to more general problems, for which the boundary conditions are generalized in Robintype, for example,
or we can consider, in general, the operator equations with the selfadjoint operator \(\mathcal{A}\) having a discrete spectrum on an abstract Hilbert space ℋ and satisfying the condition that \(\mathcal{A}\) generates a compact contraction semigroup on ℋ, like problem (2) considered above. However, for the sake of simplicity, we confine our attention to problem (1) in which the homogeneous Dirichlet boundary conditions at the endpoints of \([0,\pi]\) are given. In this problem, we have \(\mathcal{H}=L^{2} (0,\pi)\) and \(\mathcal{D} (\mathcal{A} )=H_{0}^{1} (0,\pi)\cap H^{2} (0,\pi)\), so there exists an orthonormal basis of \(L^{2} (0,\pi)\), \(\{ \phi_{n} \} _{n\in\mathbb{N}}\) satisfying (see, e.g., [27, p.181])
The operator thus has a discrete spectrum \(\sigma(\mathcal{A} )= \{ \lambda_{n} \} _{n\ge1}\) with \(\lambda_{n}=n^{2}\) and gives the orthonormal eigenbasis \(\phi _{n}=\sqrt{\frac{2}{\pi}}\sin(nx )\) for \(n\in\mathbb{N}\), \(n\ge1\). Then, thanks to those theorems above, the solution has the form
where
We can see that the instability is caused by all of the exponential functions. In fact, let us see the case \((t,s )\in D_{1}\) in (8). Since the discrete spectrum increases monotonically as n tends to infinity, the rapid escalation of \(e^{ (Tt )n^{2}}\) and \(e^{ (T\eta)n^{2}}\) is mainly the instability cause. Even though these exact given functions \((\psi_{n},f_{n} )\) may tend to zero very fast, performing classical calculation is impossible. It is because the given data may be diffused by a variety of reasons such as roundoff errors, measurement errors. A small perturbation in the data can arbitrarily generate a large error in the solution. A regularization method is thus required.
3 Theoretical results
In this section, assuming that the problem has an exact solution u satisfying various corresponding assumptions, we construct the regularized solution depending continuously on the data such that it converges to the exact solution u in some sense. Moreover, the accuracy of regularized solution is estimated.
The solution of (1) can be given by
We shall replace all instability terms by the better ones, particularly \((\varepsilon+e^{pn^{2}} )^{\frac{tT}{p}}\) and \((\varepsilon +e^{pn^{2}} )^{\frac{sT}{p}}\), where \(p\ge1\) is a real number. Then the regularized solution corresponding to the exact data is
for any \((t,s )\in D_{1}\), and
for any \((t,s )\in D_{2}\).
We also denote the regularized solution corresponding to the perturbed data by
for any \((t,s )\in D_{1}\), and
for any \((t,s )\in D_{2}\).
Now we shall show two elementary inequalities in the following lemmas.
Lemma 5
For \(0\le t\le T\le p\), we have
Proof
It is obvious that \((\varepsilon+e^{n^{2}p} )^{\frac{tT}{p}}\leq \varepsilon^{\frac{tT}{p}}\) since \(\varepsilon+e^{n^{2}p}\ge\varepsilon\). □
Lemma 6
For all \(x>0\), \(0<\alpha<1\), we have
Proof
The proof of this lemma is based on the fact that \(x^{\alpha}< (x+1 )^{\alpha}\). Therefore, we have
which leads to
□
In the sequel, we only prove the case \((t,s )\in D_{1}\) in our main result because of the similarity. The results are about the regularized solution depending continuously on the corresponding data and the convergence of that solution to the exact solution. Now we shall use two elementary lemmas above to support the proof of the main results.
Lemma 7
Under conditions (A1), (A2), (A4) and assuming that \(\varphi(T )=\psi (T )\), then the function \(u^{\varepsilon}\) given by (10)(11) depends continuously on \((\varphi,\psi)\) in \(L^{2} (0,\pi)\).
Proof
Let \(u_{1}^{\varepsilon}\) and \(u_{2}^{\varepsilon}\) be two solutions of (10)(11) corresponding to the data \((\varphi ^{1},\psi^{1} )\) and \((\varphi^{2},\psi^{2} )\), respectively. By using Parseval’s relation, for \((t,s )\in D_{1}\), we have
Similarly, for any \((t,s )\in D_{2}\), we get
□
Theorem 8
Under conditions (A1), (A2) and (A4), if problem (1) with \(\varphi(T )=\psi(T )\) admits a unique solution u satisfying
and
where \({ u_{n} (t,s )=\int_{0}^{\pi}u (x,t,s )\sin(nx )\,dx}\), let \((\varphi^{\varepsilon},\psi^{\varepsilon} )\) be perturbed functions satisfying conditions (A1)(A2), respectively, and let \(v^{\varepsilon}\) be the regularized solution, given by (12)(13), corresponding to the perturbed data \((\varphi^{\varepsilon},\psi ^{\varepsilon} )\), then for \((t,s )\in D_{1}\) we have
and for \((t,s )\in D_{2}\),
Proof
For any \((t,s )\in D_{1}\), we have
Using the triangle inequality, in order to get the error estimate, we have to estimate \(\Vert v^{\varepsilon} (\cdot,t,s )u^{\varepsilon} (\cdot,t,s )\Vert \) and \(\Vert u^{\varepsilon} (\cdot,t,s )u (\cdot,t,s )\Vert \). Indeed, we get
Next, \(\Vert u^{\varepsilon} (\cdot,t,s )u (\cdot,t,s )\Vert \) can be estimated as follows. We put
then we have
Therefore, we conclude that
Now using Parseval’s relation again, we thus obtain
Thanks to Lemma 6 and assumption (14), we have
Combining (16)(17), we obtain
Similarly, we obtain the error estimate
for the case \((t,s )\in D_{2}\) with assumption (15).
Hence, we complete the proof. □
Remark 9
From Theorem 8, we can see that \(v^{\varepsilon} (\cdot,t,s )\) strongly converges to \(u (\cdot,t,s )\) in \(L^{2} (0,\pi)\) for any \((t,s )\in[0,T ]\times[0,T ]\) as ε tends to zero. One advantage of this method is that the endpoints of time \([0,T ]\times[0,T ]\), for example, \((t,s )= (0,0 )\) and \((t,s )= (T,T )\), nearly have the same rate of convergence in some cases. Indeed, the convergence speed at \((t,s )= (0,0 )\) is \(\varepsilon^{\frac{pT}{p}}\) and it is of order ε for \((t,s )= (T,T )\). Then, if p is very large for any fixed \(T>0\), the order \(\varepsilon ^{\frac{pT}{p}}\) may approach ε. This creates the globally stable behavior of the error in numerical sense. On the other hand, the natural acceptance of (14)(15) can be obtained at \((t,s )= (0,0 )\). Namely, by letting \(p=T\) the conditions become
Moreover, the error is of order \(\mathcal{O} (\varepsilon^{\frac {pT}{p}} )\) for all \((t,s )\in[0,T ]\times[0,T ]\). If \(p>T\), this error is faster than the order \(\ln(\varepsilon^{1} )^{q}\), \(q>0\) as \(\varepsilon\to0\) which is studied in many works, such as [18–20, 23]. Combining the strong points above, the reader can infer that our method is feasible.
4 A numerical example
In order to see how well the method works, we consider problem (1) by choosing
and the domain \([0,\pi]\times[0,1 ]^{2}\). For those given functions, the problem has a unique solution
Now let us take perturbation on data functions as follows. For \(m\in \mathbb{N}\), we define
Thus, the solution corresponding to the perturbed data functions is
It is easy to see that \((\varphi_{m},\psi_{m} )\) converges to \((\varphi,\psi)\) over the norm \(L^{2} (0,\pi)\) as \(m\to\infty\). To observe the illposedness, we can compute, for example, \(u_{ex} (x,\frac{1}{2},\frac{1}{2} )=e^{\frac{3}{2}}\sin x\) and
Therefore, we get
as \(m\to\infty\). This divergence is also shown in Figure 1 with \(m=2\) and \(m=3\).
Now we compute the regularized solution based on (12)(13) as follows.
for any \((t,s )\in D_{1}\), and
for any \((t,s )\in D_{2}\).
To obtain numerical results, we use a uniform grid of meshpoints \((x,t,s )= (x_{j},t_{k},s_{m} )\), where
We thus seek the discrete solutions \(u_{ex}^{j,k,l}=u_{ex} (x_{j},t_{k},s_{l} )\) and \(v_{m}^{j,k,l}=v_{m} (x_{j},t_{k},s_{l} )\) given by (18) and (19)(20), respectively.
By fixing \(K=100\), \(M=80\) and \(p=10\), the numerical results are shown in Table 1 and illustrated in Figures 2 and 3. Figure 2 is the graphical representations for curved surfaces of the exact solution \((t,s )\mapsto u_{ex} (\frac{\pi}{2},t,s )\equiv e^{2ts}\) and of the approximate solution \((t,s )\mapsto v_{m} (\frac{\pi }{2},t,s )\) determined in (19)(20) with \(m=10^{10}\). In Figure 3, we have drawn the exact solution \(x\mapsto u_{ex} (x,0,0 )\equiv\sin x\) and the approximate solution \(x\mapsto v_{m} (x,0,0 )\), where m are \(5\times10^{9}\), \(7\times10^{9}\) and 10^{10}, respectively, in order to see the convergence at \((t,s )= (0,0 )\) as m becomes very large, namely the bound ε in theoretical result tends to zero. As in Figures 2 and 3, we can conclude that the regularized solution converges to the exact one as the error becomes smaller and smaller. Moreover, convergence is, particularly, observed from the absolute (abs.) errors in Table 1. Hence, our numerical results are all reasonable for the theoretical result.
5 Conclusion
In this work, a regularization method has been successfully applied to the inverse ultraparabolic problem. This method is to replace the instability terms appearing in the formula of the solution which is employed by semigroup theory. Therefore, such a way forms the socalled regularized solution which strongly converges to the exact solution in \(L^{2}\)norm. We also obtain the error estimate which is of order \(\varepsilon^{\frac{pT}{p}}\), \(p>T\). By a numerical example, application of the method is flexible and calculation of successive approximations is direct and straightforward. This work is more general than [7], a recent work of Zouyed and Rebbani, in both error estimate and the considered problem.
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The authors wish to express their sincere thanks to the anonymous referees and the handling editor for many constructive comments leading to the improved version of this paper.
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VAK, LTL organized and wrote this manuscript. VAK, LTL and TTH contributed to all the steps of the proofs in this research together. NHT participated in the discussion and corrected the main results. All authors read and approved the final manuscript.
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Tuan, N.H., Khoa, V.A., Lan, L.T. et al. On the regularization of solution of an inverse ultraparabolic equation associated with perturbed final data. J Inequal Appl 2015, 13 (2015). https://doi.org/10.1186/s136600140526y
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DOI: https://doi.org/10.1186/s136600140526y