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An inverse problem of reconstructing the timedependent coefficient in a onedimensional hyperbolic equation
Advances in Difference Equations volume 2021, Article number: 452 (2021)
Abstract
In this paper, for the first time the inverse problem of reconstructing the timedependent potential (TDP) and displacement distribution in the hyperbolic problem with periodic boundary conditions (BCs) and nonlocal initial supplemented by overdetermination measurement is numerically investigated. Though the inverse problem under consideration is illposed by being unstable to noise in the input data, it has a unique solution. The Crank–Nicolsonfinite difference method (CNFDM) along with the Tikhonov regularization (TR) is applied for calculating an accurate and stable numerical solution. The programming language MATLAB builtin lsqnonlin is used to solve the obtained nonlinear minimization problem. The simulated noisy input data can be inverted by both analytical and numerically simulated. The obtained results show that they are accurate and stable. The stability analysis is performed by using Fourier series.
Introduction
The reconstruction of the unknown coefficients in the inverse problem of the hyperbolic problem has various applications in science and engineering. In the last few decades, various authors have reconstructed the unknown coefficients in the inverse problem of the hyperbolic wave equations. For example, Bakushinsky and Leonov [2] recovered the spacedependent coefficient from integral data, while authors of [5, 24, 25] determined the timedependent source coefficients. Cannon and Dunninger [7] reconstructed a force function from overspecified data. Further, Cannon and DuChateau [6] determined both time and spacedependent coefficients. Stefanov and Uhlmann [27] recovered a source term in anisotropic media. Additionally, Bellassoued and Yamamoto [3] studied the inverse problem to determine the unknown term in the hyperbolic model with variable terms. Boumenir and Tuan [4] showed the process for reconstructing the unknown coefficient in the inverse problem of the wave equation from a finite number of special lateral measurements. Yamamoto [31] considered an inverse problem for identifying spacedependent function from Neumann BCs and showed a TR and reconstruction formula. Yang et al. [32] recovered heat source term in the inverse problem of heat conduction equation using Tikhonov regularization. Huntul [12] identified the unknown timedependent coefficient in the thirdorder equation from nonlocal integral observation.
Recently, the inverse problems of the wave equations for recovering timedependent potential from overdetermination integral condition have been investigated by Tekin [29] while the timedependent force function has been studied by Hussein and Lesnic [19]. The authors of [13–15, 17] studied the inverse problems for identifying the timewise potential terms in third and fourthorder equations. Huntul et al. [18] investigated an inverse problem to reconstruct the timewise potential terms in a wave equation as an overdetermination condition. In [10, 11, 20], authors recovered the time and spacedependent source functions. Huntul and Tamsir [16] investigated an inverse problem to recover a timewise heat source from the integral condition. Still, the inverse problem of reconstructing the timewise potential coefficient numerically for the hyperbolic problems with integral and periodic BCs is inadequate in the literature.
In this paper, the timedependent potential is reconstructed for the first time numerically in a onedimensional hyperbolic problem with periodic and integral BCs from the overdetermination estimation. Azizbayov [1] has already proved that this problem is locally uniquely solvable, but no numerical realization has been carried out till now, which is the main contribution of proposed work.
The paper is organized as follows: The research problem is stated in Sect. 2. Sect. 3 discretizes the direct problem using CNFDM. Stability analysis is presented in Sect. 4, while the minimization technique is given in Sect. 5. Some numerical results are tabulated in Sect. 6, while the conclusions are highlighted in Sect. 7.
Formulation of the inverse problem
We consider an inverse problem of reconstructing an unknown timedependent potential coefficient \(\alpha (t)\) in the onedimensional hyperbolic equation
where \(z=z(r,t)\) is an unknown displacement, \(s(r,t)\) is a known source term which is as distributed force. The nonlocal initial conditions (ICds)
the periodic BCs
the integral boundary condition
and the overdetermination condition
where \(R_{0}\in (0,1)\) is some fixed point, the functions \(P_{1}(t)\), \(P_{2}(t)\), \(\xi (r)\), \(\zeta (r)\), and \(q(t)\) are given.
The numerical solution of the inverse problem hyperbolic wave equation (1)–(5) is written as \(\{\alpha (t),z(r,t) \}\) such that \(\alpha (t)\in C[0,T]\) and \(z(r,t)\in C^{2}(\overline{D}_{T})\). Along with inverse problem (1)–(5), the following auxiliary inverse problem is considered. It is needed to find \(\{\alpha (t),z(r,t) \}\in C[0,T]\times C^{2}(\overline{D}_{T})\) from (1)–(3), and
The upcoming theorems are taken from [1], and they read as follows.
Theorem 1
Suppose that \(\xi (r),\zeta (r)\in C[0,1]\), \(P_{i}(t)\in C[0,T]\), \(i=1,2\), \(q(t)\in C^{2}[0,T]\), \(q(t)\neq 0\), \(s(r,t)\in C(\overline{D}_{T})\), \(\int _{0}^{1} s(r,t)\,dr=0\), \(t\in [0,T]\), and the consistency conditions, given below, are satisfied:
Then the following arguments are true:

1.
Each classical solution \((\alpha (t),z(r,t))\) of (1)–(5) is a solution of (1)–(3), (6), and (7);

2.
Each solution \((\alpha (t),z(r,t))\) of (1)–(3), (6), and (7) by virtue of
$$ \biggl(T \bigl\Vert P_{2}(t) \bigr\Vert _{C[0,T]}+ T \bigl\Vert P_{1}(t) \bigr\Vert _{C[0,T]}+ \frac{T}{2} \bigl\Vert \alpha (t) \bigr\Vert _{C[0,T]} \biggr)T< 1 $$(10)
We impose the below conditions to the functions ξ, ζ, s, \(P_{1}\), \(P_{2}\), and q [1]:

(A1)
\(\xi '''(r)\in L_{2}(0,1)\), \(\xi (r)\in C^{2}[0,1]\), \(\xi (0)=\xi (1)\), \(\xi '(0)=\xi '(1)\), \(\xi ''(0)=\xi ''(1)\);

(A2)
\(\zeta ''(r)\in L_{2}(0,1)\), \(\zeta (r)\in C^{1}[0,1]\), \(\zeta (0)=\zeta (1)\), \(\zeta '(0)=\zeta '(1)\);

(A3)
\(s(r,t), s_{r}(r,t)\in C(\overline{D}_{T})\), \(s_{rr}(r,t)\in L_{2}( \overline{D}_{T})\), \(s(0,t)=s(1,t)\), \(s_{r}(0,t)=s_{r}(1,t)\), \(t\in [0,T]\);

(A4)
\(P_{1}(t), P_{2}(t)\in C[0,T]\), \(q(t)\in C^{2}[0,T]\), \(q(t)\neq 0\), \(t\in [0,T]\).
Theorem 2
Let conditions (A1)–(A4) be satisfied, and suppose that
where
Then inverse problem (1)–(3), (6), (7) has a unique solution in the ball \(K=K_{R}\).
The uniqueness of solution for inverse problem (1)–(5) has been proved by Azizbayov [1]; it is given as follows.
Theorem 3
Let the conditions
and all the assumptions of Theorem 2be satisfied. Then (1)–(5) has a unique solution in \(K=K_{R} (\z\_{E^{3}_{T}}\leq A_{1}(t) +A_{2}(t)+2 )\) of the space \(E^{3}_{T}\).
Numerical solution of the forward problem
First of all, we consider forward problem (1)–(4) with all initial and boundary conditions, then we calculate solution using the CNFDM scheme in this section.
Theorem 4
Let \(\alpha (t)\), \(\xi (r)\), \(\zeta (r)\), \(P_{1}(t)\), \(P_{2}(t)\), \(s(r,t)\) be known functions, and the CNFDM scheme is utilized for time discretization. Then the numerical solution \(z(r,t)\) is given in equations (19) and (21).
Proof
We denote \(z(r_{i},t_{j})=z_{i,j}\), \(P_{1}(t)=P_{1}^{j}\), \(P_{2}(t)=P_{2}^{j}\), \(s(r_{i},t_{j})=s_{i,j}\), and \(\alpha (t_{j})=\alpha _{j}\), where \(r_{i}=i\Delta r\), \(t_{j}=j\Delta t\), \(\Delta r=\frac{1}{M}\), and \(\Delta t=\frac{T}{N}\), \(i=0(1)M\) and \(j=0(1)N\). Then the \((\Delta r^{2},\Delta t^{2})\) CNFDM [26] discretizes (1) as follows:
Equation (11) yields
where
The discretization of nonlocal ICs (2) is
and the periodic BC (3) is
Finally, discretization of integral BC (4) is given as follows:
For \(i=0\) and M, from equations (15) and (16), we get
and
The above equations (12), (17), and (18) can be reformulated and converted at time \(t_{j+1}\) into the \((M+1)\times (M+1)\) system:
where
At time \(t_{1}\), using (14) in (12), we get
Now, at time \(t_{1}\), equations (17), (18), and (20) can be reformulated into the \((M+1)\times (M+1)\) system:
where
□
Stability analysis
The von Neumann stability [8, 9, 22, 23, 28, 30] is carried out for the discretized system of hyperbolic wave equation.
Theorem 5
Show that the proposed numerical scheme is stable.
Proof
Taking \(s(r,t)=0\) and assuming local constant \(\alpha _{j}=\alpha _{j+1}=\bar{b}_{1}\) for known level in (12), we get
where
Now we consider one Fourier mode out of the full solution \(z_{i,j} = \hat{z}^{j} e^{i \omega \phi }\) as trial solutions at a given point \(r_{i}\), where \(\phi =\theta h\) is the phase angle, i is the node number, and \(\omega =\sqrt{1}\). Using trial solutions in the above equation and simplifying the terms, we get
which can be written as
where
Under the transformation \(\hat{z}=\frac{1+\rho }{1\rho }\) in equation (24), we get
The discretized system (22) will be stable if
Using the values of \(\beta _{1}\), \(\beta _{2}\), \(\beta _{3}\) and simplifying the terms, we get
It is clear from (27) that \(\beta _{1}+\beta _{2}+\beta _{3} \ge 0\). From (28) and (29), we get \(\beta _{1}\beta _{3} \ge 0\) and \(\beta _{1}\beta _{2}+\beta _{3} \ge 0\) if \((\Delta r)^{2} \le \frac{1}{\bar{b}_{1}} (2 \sin ^{2} ( \frac{\phi }{2} ) 1 )\), and the discretized system of (1) will be stable. □
Numerical solution of the inverse problem
In this section, we want to find accurate and stable identification of \(z(r,t)\) and \(\alpha (t)\) which satisfies the nonlinear and illposed inverse problem (1)–(5). This is achieved by minimizing the objective function
where z solves the forward problem (1)–(4) for given \(\alpha (t)\). The discrete form of equation (30) is
The objective function \(\mathbb{F}\) (31) is minimized by the MATLAB subroutine lsqnonlin [21].
Numerical results and discussion
The accuracy is measured by RMS:
Here, for simplicity, we take \(T= 1\), and 10^{2} and −10^{2} as upper and lower bounds, respectively, for coefficient \(\alpha (t)\).
The inverse problem (1)–(5) is solved with exact and noisy measurement data (5). We numerically simulated the perturbed data as follows:
where \(\epsilon _{j}\) are random variables with mean zero and standard deviation σ given by
where p denotes the percentage of noise. In the case of noisy data (34), we replace \(q(t_{j})\) with \(q^{\epsilon }(t_{j})\) in (31).
Test 1
The proposed problem (1)–(5) is considered with some smooth potential term
the exact solution
and the rest of the data are as follows:
The periodic BCs
and the overdetermination condition
It is observed that Theorem 3 guarantees the uniqueness of the solution because their conditions have been fulfilled. The accuracy of problem (1)–(4) has been assessed with data (37) and (38) when \(\alpha (t)\) is given by (36). Fig. 1 depicts the approximate displacement measurement in (5) in comparison to the analytical solution (40) obtained by using the Crank–Nicolson FDM with \(M=N\in \{ 10,20,40,80 \}\). The exact (37) and approximate solutions for the displacement \(z(r, t)\) are illustrated in Fig. 2. The absolute errors between those solutions are also included and it can be seen that these errors decrease as the CNFDM grid becomes finer. A good agreement observed between the analytical (40) and the approximate \(q(t)\) solutions as the mesh size decreases, see Table 1.
Next, we fix \(\Delta r=\Delta t=0.025\), and the timedependent potential coefficient \(\alpha (t)\) determination is started, when the value of p equal zero in equation (5), as put in (35). The initial guess (IG) for \(\underline{\alpha }\) is taken as follows:
Fig. 3(a) is depicted by the objective function \(\mathbb{F}\) (31), and a monotonically decreasing convergence is observed by making 15 time repetitions for attaining a less order tolerance \(O(10^{30})\). Numerical outcomes for the potential \(\alpha (t)\) are observed in Fig. 3(b) and also obtaining with \(\operatorname{RMS}(\alpha ) = 0.0868\). The perturbed data has been used to examine the stability of the approximate solution. The noise \(p\in \{\frac{1}{100}\%,\frac{1}{10}\% \}\) is included to simulate the input noisy data via equation (34) for \(q(t)\). The potential term \(\alpha (t)\) is depicted in Figs. 4 and 5. From Figs. 4(a) and 5(a) it can be observed that as p is increased the approximate results for \(\alpha (t)\) start to build up oscillations. To retrieve stability, we penalize \(\mathbb{F}\) (30) by adding \(\beta \\alpha (t)\^{2}_{L^{2}[0,T]}\) to it, where \(\beta >0\) is Tikhonov’s regularization parameter. Then a discretized form of Tikhonov functional is
For \(p\in \{0.01\%,0.1\% \}\) noise, Figs. 4(b) and 5(b) show the analytical (36) and the approximate \(\alpha (t)\) achieved by minimizing \(\mathbb{F}_{\beta }\) (42) for various β. The \(\operatorname{RMS}(\alpha )\) values are \(\{3.1204,0.0790,0.0776 \}\) for the value of \(p=\frac{1}{100}\%\), and \(\{34.2297,0.1573,0.1100 \}\) for the value of \(p=\frac{1}{10}\%\), respectively, with \(\beta \in \{0,10^{10},10^{9} \}\) and \(\beta \in \{0,10^{9},10^{8} \}\). It is noticed that the approximate \(\alpha (t)\) obtained with \(\beta =0\) demonstrates instability; however, on inclusion of regularization with \(\beta =10^{10}\) to 10^{−8}, a stable solution is obtained which is consistent in accuracy with \(p=0.01\%\) and 0.1% violating the input data (34). The absolute error norms between the exact (37) and approximate solutions with \(\beta =0, 10^{9}\) and 10^{−8} are illustrated in Fig. 6. It is observed that the displacement \(z(r,t)\) component is accurate and stable when penalty term \(\beta >0\) is added as in (42) to stabilize the solution.
Test 2
The smooth timewise potential \(\alpha (t)\) given by (36) has been recovered in Test 1. Now, consider a nonlinear numerical test problem which is given by
Since the conditions of Theorem 3 hold, the solution uniqueness is guaranteed. The IG for \(\underline{\alpha }(t)\) is considered as follows:
We take fixed values \(M=N=40\) as in the previous test problem, and value of \(p = 0\) in \(q(t)\), as also in equation (35). Fig. 7(a) is depicted by the objective function \(\mathbb{F}\) (31), and a monotonically decreasing convergence is observed by making 15 time repetitions for attaining a less order tolerance \(O(10^{30})\). Numerical outcomes for the potential \(\alpha (t)\) are observed in Fig. 7(b) and also obtaining with \(\operatorname{RMS}(\alpha ) = 0.0872\).
The \(\mathbb{F}\) (31) is depicted in Fig. 7(a), where a monotonically decreasing convergence is achieved in 15 iterations for achieving \(O(10^{30})\). Fig. 7(b) depicts the exact (43) and approximate \(\alpha (t)\), obtaining with \(\operatorname{RMS}(\alpha ) = 0.0872\).
Next, we take \(p\in \{\frac{1}{100}\%,\frac{1}{10}\% \}\) to the measured data \(q(t)\), as also in equation (35), for checking the stability of problem. The regularized function \(\mathbb{F}_{\beta }\) (42) is depicted in Fig. 8 with various parameters. The approximate outcomes of potential \(\alpha (t)\) are illustrated in Figs. 9 and 10. As in the previous test problem, it is observed that the numerical solutions for \(\beta =0\) tabulated in Figs. 9(a) and 10(a) are unstable due to high oscillations and unbounded, obtaining with \(\mathrm{RMS} (\alpha )=3.3676\) for the value of \(p=\frac{1}{100}\%\) and \(\mathrm{RMS} ( \alpha )=33.9465\) for the value of \(p=\frac{1}{10}\%\). However, the addition of some restrictions on \(\beta >0\) in \(\mathbb{F}_{\beta }\) (42) improves the stability of the numerical solutions, as drawn further in Figs. 9(b) and 10(b), respectively. It is observed that the addition of \(\beta \in \{10^{12},10^{11},10^{10} \}\) for \(\frac{1}{100}\%\) and \(\beta \in \{10^{11},10^{10},10^{9} \}\) for \(\frac{1}{10}\%\) provides a perfect and stable approximate solution for \(\alpha (t)\), getting with \(\operatorname{RMS} (\alpha ) \in \{0.2368,0.1245,0.2728 \}\) and \(\operatorname{RMS}(\alpha ) \in \{0.8454,0.5282,0.6936 \}\). The comparison between analytical (37) and approximate \(z(r,t)\) in terms of absolute errors with and without descriptor parameters are depicted in Fig. 11, where \(\beta >0\) has the effect of reducing the unstable performance of the restored displacement. Refer to Table 2 for additional information on the \(\operatorname{RMS}(\alpha )\) values, the minimum value of \(\mathbb{F}\) or \(\mathbb{F}_{\beta }\) at final iteration without and with regularization for both numerical test problems. For the term \(\alpha (t)\), the same results may be derived regarding the stable reconstruction.
Concluding remarks
For the first time, the reconstruction of a potential coefficient \(\alpha (t)\) and the displacement distribution \(z(r,t)\) from the overdetermination condition in the hyperbolic equation using nonlocal ICs and periodic BCs have been computationally addressed. The CNFDM was applied to solve the direct problem. The stability analysis of the discretized system of the wave equation has been discussed using the von Neumann method. The TR was used to overcome the instability due to the illposed issue. For both numerical test problems, the RMS values for noise without and with regularization were compared. It has been noticed that on the inclusion of regularization with \(\beta =10^{12}\) to \(\beta =10^{7}\), a stable solution is obtained which is consistent in accuracy with 0.01% and 0.1% noise. Finally, future study into the extension of the suggested approach for determining the timewise potential coefficient in a twodimensional wave equation is an intriguing prospect.
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Acknowledgements
The authors are indebted to the anonymous referees for their valuable comments and suggestions that helped improve the paper. Also, we thank Dr Muhammad Amin for his assistance in proofreading of the manuscript.
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Huntul, M.J., Abbas, M. & Baleanu, D. An inverse problem of reconstructing the timedependent coefficient in a onedimensional hyperbolic equation. Adv Differ Equ 2021, 452 (2021). https://doi.org/10.1186/s13662021036081
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DOI: https://doi.org/10.1186/s13662021036081
Keywords
 Hyperbolic equation
 Inverse problem
 Periodic boundary
 Integral boundary
 Tikhonov regularization
 Optimization