Regularization of a final value problem for a linear and nonlinear biharmonic equation with observed data in $L^{q}$ space

1.   Introduction

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}\; (N = 1, 2, 3)$ a sufficiently smooth boundary $\partial \Omega$. In this study, we consider a nonlinear biharmonic equation, as follows:

satisfying the following boundary conditions:

and

wherein the condition (1.2) is the Navier boundary condition; and the condition (1.3) is the mixed Dirichlet-Neumann boundary condition. The function $u = u(x, t)$ represents a concentration of contaminant at a position $x$ and at time $t$. The data $f, g \in L^{q}(\Omega)$ and $G \in L^{\infty}(0, T; L^{q}(\Omega))$ are defined later on. However, in actual conditions, there are always included errors in the measurement methods of a physical process, so we have the following conditions:

The biharmonic equation plays an important role in engineering and physics. It arises in the deformation of thin plates, the motion of fluids, free boundary problems and nonlinear elasticity, see ^[1,2,3,4,5]. Therefore, the biharmonic equation has a long history of research. It has been studied by many authors at early time. The most highlighted studies on numerical methods for the biharmonic equation are described in ^{[5,6,7,8,9,10]}. In particular, Smith ^[6] presented a numerical method for solving the biharmonic difference equation using finite difference methods. Ehrlich ^[7] has improved the iteration scheme to lead the Smith's result to be a special case of his study. Recently, Tuan et al. ^[11] have studied an approximate solution for a nonlinear biharmonic equation with discrete random data. Especially, in applications to radar imaging, Matevossian et al. ^[12,13] have focused on the solution of the biharmonic equation with the Dirichlet, Neumann and Cauchy boundary value problems for the Poisson equation using the scattering model.

Regarding the regularization for biharmonic equations, the authors in ^[14] considered a nonlinear biharmonic equation, and proved that problem (1.1) under the conditions (1.2) and (1.3) is ill-posed in the sense of Hadamard, and showed the error estimates. The corresponding regularized solutions in their study are strongly converged to the exact solution in $L^{2}(\Omega)$ under some priori assumptions on the solution. Besides, there are many other studies on linear homogeneous biharmonic equations; however, most of previous studies are focused on the regularization for biharmonic equations in $L^{q}(\Omega)$ with $q = 2$; and the convergent rate in $L^{q}(\Omega)$, with $q \ne 2$ is still not well implemented (Nam et al. ^[14]). Therefore, it can be stated that our study in this paper is one of the first results regarding the inverse problem for the biharmonic equation, once the observed data is obtained in the $L^{q}(\Omega)$ space with $q \ne 2$. The main objective of this study is to establish regularized solutions for problem $(1.1)$ under the conditions (1.2) and (1.3) and showed the regularized solution is converged to the exact solution; in the linear case refered to (3.13), and in the nonlinear case referred to (3.73).

For evaluation cases in $L^{q}(\Omega)$ spaces, the most obstacle is unable to use Parseval's equality; therefore, we applied the embedding between $L^{q}(\Omega)$ and Hilbert scales spaces $\mathcal{H}^{\rho}(\Omega)$ to overcome this limitation; and Lemma 2.1 will be used throughout this article. The manuscript is proceeding as follows:

● The first part deals with the inverse problem with a defined source function. In this subsection, we introduce the mild solution of problem (1.1) under the conditions (1.2) and (1.3) with the observed data $f_{\delta}, g_{\delta} \in L^{q}(\Omega)$, and $G_{\delta} \in L^{\infty}(0, T; L^{q}(\Omega))$. Then, applying the Fourier series truncation method, we estimate the error between the regular and exact solution in the $L^{\frac{2N}{N-4\rho}}(\Omega)$.

● The second part of the manuscript investigates the inverse initial value problem for problem (1.1) under the conditions (1.2) and (1.3) with a nonlinear source function. In this section, the main results to be obtained are theorems: (ⅰ) The existence and the well-posedness of regularized solutions using Banach fixed point theorem; and (ⅱ) The convergent rate between the regularized solution and the exact solution through the estimation of $\big\| \widetilde{V}^{\delta}_{\mathcal{N}_{\delta }}(\cdot, t) - u(\cdot, t)\big\|_{L^{\frac{2N}{N-4\rho}}(\Omega)}$.

Hence, this manuscript is organized as follows. In Section 2, some preliminaries such as definition and Lemmas are given. Section 3 introduces some results on regularization of problem (1.1) under the conditions (1.2) and (1.3) in the linear and non-linear cases. Numerical examples is described in Section 4 associate with observed data in $L^{q}(\Omega)$.

2.   Preliminaries

We begin this section by introducing some preliminary definitions and basic lemmas that are needed for our analysis.

Definition 2.1. Assume that $-\Delta$ has the eigenvalues $\lambda_{k}, \; k \in {{\mathbb{N}}}^{*}$:

and the corresponding eigenelements $\mathit{e}_{k}(x)$, which form an orthonormal basis in $L^{2}(\Omega)$.

Definition 2.2. Let $\big < \cdot, \cdot\big >$ be an inner product in ${L}^{2}(\Omega)$. The notation $\|\cdot\|_{X}$ stands for in the norm in the Banach space. We denote by $L^{q}(0, T; X)$, $1 \le q < \infty$, the Banach space of real-valued functions $u:(0, T) \to X$ measurable, providing that

while

Definition 2.3. (see ^[8]) For any $\sigma \geq 0$, we also define the space

then $\mathcal{H}^{\sigma}(\Omega)$ is a Hilbert space endowed with the norm

Lemma 2.1. (see ^[15]) The following inclusions hold true:

3.   Results on regularization of the biharmonic equation in $L^{q}$ spaces

First of all, we present the formula of a mild solution of problem (1.1) as follows.

3.1.   Mild solution of problem (1.1)

The solution of problem (1.1) can be written in the following Fourier series form:

We have a particular solution of problem (1.1) in the form

Substituting the result into (3.1), we will have the formal solution of problem (1.1). The following steps, we are going to find the mild solution of problem (1.1) when the source function is linear and nonlinear.

3.2.   Mild solution for a linear source function

By applying the Fourier truncation method, we provide a regularization solution as follows:

whereby $\mathcal{N}_{tr}$ is a parameter regularization which will be defined later. From (3.3), it allows us to deduce that the mild solution to problem (1.1) in the following form:

Theorem 3.1. Let assume that $f_{\delta}, g_{\delta}, G_{\delta} \in L^{q}(\Omega) \times L^{q}(\Omega) \times L^{\infty}(0, T; L^{q}(\Omega))$ are observed data such that

Let $u \in L^{\infty}(0, T; \mathcal{H}^{\sigma+\gamma}(\Omega))$ for any $\gamma > 0$. By choosing $\mathcal{N}_{tr} = \bigg(\dfrac{1-\alpha}{T\sqrt{\mathcal{C}_{1}}} \log \big(\delta^{-1}\big) \bigg)^{N}$ for any $0 < \alpha < 1$, then we have $\big\| u_{\delta}^{\mathcal{N}_{tr}}(\cdot, t) - u(\cdot, t) \big\|_{L^{\frac{2N}{N-4\sigma}}(\Omega)}$ is of order

Proof. Because of the Sobolev embedding $L^{q}(\Omega) \hookrightarrow \mathcal{H}^{\frac{N(q-2)}{4q}}(\Omega)$, we have

with $\mathcal{C}_{1}$ depends on $N, p$. Our goal in this theorem is to assess the convergence error of $\big\|u(\cdot, t)-u_{\delta}^{\mathcal{N}_{tr}} (\cdot, t)\big\|_{\mathcal{H}^{\sigma}(\Omega)}$. Next, we first introduce the following function:

Using the triangle inequality, we receive

Next, we evaluate (3.9) through two steps as follows:

Step 1: Estimate of $\big\|u_{\delta}^{\mathcal{N}_{tr}}(\cdot, t) - \widetilde{U}^{\mathcal{N}_{tr}}(\cdot, t)\big\|_{\mathcal{H}^{\sigma}(\Omega)}$, we find that

whereby

First of all, estimating the ${A}_{1}(x, t)$, it is easy to check that $\cosh (x) \leq \exp (x)$ and $\sinh (x) \leq \exp (x), \\ \forall x > 0$, this implies that

Therefore, it gives

In the fact that $\lambda_{k} \leq \mathcal{C}_{2} k^{\frac{2}{N}}$, and noting that $\sigma \geq \frac{N}{4} - \frac{N}{2q}$, we can verify that for $k \leq \mathcal{N}_{tr}$,

in which $\mathcal{C}_{3} = 2\mathcal{C}_{2}^{2\sigma - \frac{N(q-2)}{2q}}$ and $\mathcal{C}_{4} = \dfrac{T^{2}}{2\lambda_{1}} \mathcal{C}_{2}^{2\sigma - \frac{N(q-2)}{2q}}$.

Combining (3.13) and (3.14), we have

with $\mathcal{C}_{5} = 2\max \big\{\mathcal{C}_{3}, \mathcal{C}_{4}\big\}$. It follows from (3.7) that

Next, considering the term $\|{A}_{2}(., t)\|_{\mathcal{H}^{\sigma}(\Omega)}$, applying the Parseval's equality, we can see that

From (3.17), using the Hölder's inequality, we receive

Because of $\lambda_{k} \leq \mathcal{C}_{2} k^{\frac{2}{N}}$, and noting that $\sigma \geq \dfrac{N}{4} - \dfrac{N}{2q}$, we can verify that for $k \leq \mathcal{N}_{tr}$,

From (3.18), we noticed that

Due to condition (3.7) and from the estimation in (3.20), one has

From all the estimation above, we received

Step 2: Estimate of $\big\|\widetilde{U}^{\mathcal{N}_{tr}}(\cdot, t) - u(\cdot, t)\big\|_{\mathcal{H}^{\sigma}(\Omega)}$, from (3.4) and (3.8), and using the Parseval's inequality, we deduce that

From (3.23), for any $\gamma > 0$, we received

In case $k > \mathcal{N}_{tr}$, there exists a postive constant $\mathcal{C}_{6} > 0$ such that $\lambda_{k}^{-2\gamma} \leq \mathcal{C}_{6} k^{-\frac{4\gamma}{N}}$, we have

Combining (3.22) to (3.25), we conclude that

By choosing $\mathcal{N}_{tr} = \bigg(\dfrac{1-\alpha}{T\sqrt{\mathcal{C}_{2}}} \log \Big(\dfrac{1}{\delta}\Big) \bigg)^{N}$, we need the following results:

The provision of this theorem is completed.

3.3.   Mild solution for a nonlinear source function

In this section, we will study the initial inverse problem for nonlinear of source term.

with the final condition

We assume that $\nu \in L^{2}(\Omega)$, let $\mathcal{P}_{i}(z-t), \; i = \{1, 2, 3, 4\}$ is an operator defined as follows:

From the way to set the operator (3.30), the mild solution of the problem (3.28) under the condition (3.29) is as follows:

We regularized the mild solution (3.31) by Fourier method. Assume that we have $\ell \in L^{2}(\Omega)$, then we define $\ell_{\delta, \mathcal{N}_{tr}}$ as follows:

without loss of generality, we can completely assume that $f_{\delta}, g_{\delta} \in L^{q}(\Omega)$ such that $\big\|f-f_{\delta}\big\|_{L^{q}(\Omega)} + \big\|g-g_{\delta}\big\|_{L^{q}(\Omega)} \leq \delta$. Therefore, we build the structure of regularized solution and symbols it is $\widetilde{V}_{\mathcal{N}_{\delta}}^{\delta}$.

The next theorem will provide details about the existence and the well-posedness of regularized solutions.

Theorem 3.2. Let the terminal data $\ell \in L^{q}(\Omega)$, then the nonlinear integral equation (3.33) has a unique solution $\widetilde{V}_{\mathcal{N}_{\delta}}^{\delta}(x, t) \in L^{\infty}(0, T; L^{\frac{2N}{N-4\rho}}(\Omega))$, then we have the following estimate:

where $m \geq \sqrt{\mathcal{N}_{\delta}} + 2\mathcal{L}_{f} (\mathcal{N}_{\delta})^{\rho} \max \{T, 1\} \sqrt{\mathcal{N}_{\delta}}$.

Proof. Let any $\ell \in \mathcal{H}^{\sigma_{1}}(\Omega)$, suppose that $\sigma_{1} \geq \sigma$, we get

From (3.35) and in view of (3.12), we receive

By a similar argument above (3.36), we can find also that

and we can see that

For $m > 0$, we denote by $L_{m}^{\infty}(0, T; L^{\frac{2N}{N-4\rho}}(\Omega))$ the function space $L^{\infty}(0, T; L^{\frac{2N}{N-4\rho}}(\Omega))$ with the following norm:

Next, we define a nonlinear map $\mathcal{M}: L_{m}^{\infty}(0, T; L^{\frac{2N}{N-4\rho}}(\Omega)) \hookrightarrow L_{m}^{\infty}(0, T; L^{\frac{2N}{N-4\rho}}(\Omega))$ by

● {Case 1:} $u = 0$, we have $\mathcal{M}(u = 0) = \mathcal{P}_{1}(T-t)\mathbb{T}_{\mathcal{N}_{\delta}}f_{\delta} + \mathcal{P}_{2}(T-t)\mathbb{T}_{\mathcal{N}_{\delta}}g_{\delta}$. From Lemma 2.1 and $0 \leq \rho \leq \frac{N}{4}$, with the Sobolev embedding $\mathcal{H}^{\rho}(\Omega) \hookrightarrow L^{\frac{2N}{N-4\rho}}(\Omega)$, there exists a constant $\mathcal{C}_{1}$ depends on $N, \rho$ such that

From (3.41), use evaluation results in (3.36), taking square root on the both side, we choose $\sigma_{1} = \rho$ and $\sigma = \frac{N(2-q)}{4q}$, it gives

For $1 < q < 2$, using Lemma 2.1, we find that $L^{q}(\Omega) \hookrightarrow \mathcal{H}^{\frac{N(2-q)}{4q}}(\Omega)$. Therefore, there exist a constant $\mathcal{C}_{8}(N, \rho)$ such that

Combining (3.41) to (3.43), this leads to

whereby $\mathcal{C}_{9}(N, \rho) = \mathcal{C}_{8}(N, \rho) \mathcal{C}_{7}(N, \rho)$. Combining with above arguments, we deduce that

● {Case 2:} In this case, we take two function $u_{1}, u_{2} \in L_{m}^{\infty}(0, T; L^{\frac{2N}{N-4\rho}}(\Omega))$, from (3.40), it is easy to see that

Taking any $m > 0$, this implies that

If we reuse estimates of (3.37) and (3.38) with $\sigma_{1} = \rho$ and $\sigma = 0$, and in the two reviews below, we have used the Sobolev embedding $L^{\frac{2N}{N-4\rho}}(\Omega) \hookrightarrow L^{2}(\Omega)$. So, $\mathcal{D}_{1}(r, t)$ and $\mathcal{D}_{2}(r, t)$ can be bounded as follows:

and by prove similarly, we obtain

Combining (3.40) to (3.49), we have

From (3.50), make a simple transformation, we get

Using the fact that, we get

From (3.47) and (3.52), we follow that

Form (3.53), we can see that

We combine estimation from (3.53) to (3.54), and Sobolev embedding as in $\mathcal{H}^{\rho}(\Omega) \hookrightarrow L^{\frac{2N}{N-4\rho}}(\Omega)$, one obtains

Therefore, we have

For any $u_{1}, u_{2} \in L^{\infty}_{m}(0, T; L^{\frac{2N}{N-4\rho}}(\Omega))$, we conclude that

From (3.57), if we choose $m \geq \sqrt{\mathcal{N}_{\delta}} + 2\mathcal{L}_{f} (\mathcal{N}_{\delta})^{\rho} \max \{T, 1\} \sqrt{\mathcal{N}_{\delta}}$, we can see that $\mathcal{M}$ is a contraction mapping from $L^{\infty}_{m}(0, T; L^{\frac{2N}{N-4\rho}}(\Omega)) \hookrightarrow L^{\infty}_{m}(0, T; L^{\frac{2N}{N-4\rho}}(\Omega))$, we conclude that $\mathcal{M}$ has a fixed point $\widetilde{V}_{\mathcal{N}_{\delta}}^{\delta} \in L^{\infty}_{m}(0, T; L^{\frac{2N}{N-4\rho}}(\Omega))$ throught the Banach fixed point. For the estimation (3.57), let assume that $u_{1} = \widetilde{V}^{\delta}_{\mathcal{N}_{\delta}}$ and $u_{2} = 0$, and we denote $\widetilde{V}^{\delta}_{\mathcal{N}_{\delta}} = \mathcal{M} \big(\widetilde{V}^{\delta}_{\mathcal{N}_{\delta}} \big)$, and $\mathcal{M} u_{2} = \mathcal{P}_{1}(T-t)\mathbb{T}_{\mathcal{N}_{\delta}}f_{\delta} + \mathcal{P}_{2}(T-t)\mathbb{T}_{\mathcal{N}_{\delta}}g_{\delta}$, we get

Using the estimation (3.44), thus, we obtain that

In the theory below, we are going to show the error estimate between the regularized and exact solutions in the space of $L^{q}(\Omega)$ type.

Theorem 3.3. Assume that problem (1.1) under the condition (1.2) has a unique solution $u \in L^{\infty}(0, T; \mathcal{H}^{n+\rho}(\Omega))$ for any $n > 0$ and $0 < \rho < \dfrac{N}{4}.$ In addition, we assume that there exists a positive constant ${M}$ such that

where $\varsigma > 0$, taking the noisy data $f_{\delta}, g_{\delta} \in L^{q}(\Omega)$ such that

By choosing $\mathcal{N}_{\delta}$ such that

Specifically in this section, we choose $\mathcal{N}_{\delta}$ as follows:

then we have

Proof. In this provision, we provide the upper bound for the term $\big\| \widetilde{V}^{\delta}_{\mathcal{N}_{\delta}}(\cdot, t) - u(\cdot, )\big\|_{\mathcal{H}^{\rho}(\Omega)}$, using the triangle inequality, then we get

From definition (3.33), we can know that

From (3.33) and (3.67), we see that

Since (3.66) and (3.68), we receive

We rated $\big\|\widetilde{V}_{\mathcal{N}_{\delta}}^{\delta}(\cdot, t) - u(\cdot, t) \big\|_{L^{2}(\Omega)}$ through four steps as follows:

Step 1: Estimate of $\mathcal{B}_{1}$, we have

Step 2: Estimate of $\mathcal{B}_{2}$, for $1 < q < 2$, we follow Lemma 2.1 in combination wiith the Sobolev embedding, one has

This implies that

Step 3: Estimate of $\mathcal{B}_{3}$, by using the similar argument as in (3.48), we can find that

Step 4: Estimate of $\mathcal{B}_{4}$, by using the similar argument as in (3.49), one obtains

Combining (3.66), estimation of Steps 1–4, we conclude that

Multiplying both sides by $\exp \big(t \sqrt{\mathcal{N}_{\delta}} \big)$, we have

Applying the Grönwall's inequality, we get

This implies that

Next, we have the following inequality:

It is easy to see that

Now, for $0 < \varsigma < \rho$, we immediately have a rating the first term of (3.80) as follows:

$\mathcal{E}_{2}$ can be bounded as follows:

From the observation above, we have

4.   Numerical example

In this section, we carry out a numerical example in order to verify our proposed theory. In other words, we consider the stable property of the regularized solution based on the Fourier truncation method. First of all, we introduce some definitions to support the numerical implementation as follows. By choosing $\Omega \times (0, T) : = (0, \pi) \times (0, 1)$, we have the eigenvalues $\lambda_{k}$ and the corresponding eigenvector ${e_k}$ which are the complete orthonormal system of eigenfunctions forming an orthogonal basis such that $-\Delta e_k = \lambda_k e_k\ {\rm{ and }}\ e_k|_{\partial \Omega} = 0 \ {\rm{ for }}\ k \in \mathbb{N}$. Here, we choose $\lambda_k = k^2 \pi^2, \; e_k(x) = \sqrt{2}\sin(k \pi x).$ We are going to find a function $\mathcal{X}$ satisfied

under the boundary conditions

and the boundary conditions at $T = 1$ as

Finally, we use the finite difference method with the following partitions of temporal and spatial variable. For $x \in [0, 1]$ and $t \in [0, 1]$, let us consider the partitions $\mathcal{D}_{\Omega} \times \mathcal{D}_{T}$ as follows:

In Python software, the solutions can be re-write by the following form matrix:

In this example, by choosing the solution $u(x, t) = \cosh(1-t)\sin(\pi x)$ to test the proposed results with the following input data:

During the measurement of electromagnetic fields in applications of environmental and geophysical imaging, the exact data is approximated by the function $f_{\delta}, \; g_{\delta}, \; G_{\delta}$ as follows:

where

By applying the Fourier truncation method, we provide a regularization solution as follows:

where $\mathcal{N}_{tr}$ is a parameter regularization.

We use the following estimation to evaluate the error between the regularized and exact solution at a certain time $t$:

Table 1 and Figures 1–5 show the error estimate between the exact and regularized solutions at five observation times $t \in \{0.1, 0.3, 0.5, 0.7, 0.9\}$ with $\delta \in \{0.5, 0.05, 0.005\}$, respectively. Figure 6 presents the 3D graphs of the exact and regularized solutions, it shows they are quite similar. Overall, it shows that the error becomes smaller as the noise $\delta$ is decreased. It also shows the regularized solution is converged to the exact solution in this example.

5.   Conclusions

In this study, we focused on the final value problem of an inverse problem for both linear and nonlinear bi-harmonic equations. The regularized method for the biharmonic equation is proposed using the Fourier series truncation method and the terminal input data in $L^{q}(\Omega)$ for $q \ne 2$. The error between the exact and regularized solutions is estimated in $L^{q}(\Omega)$ using the embedding between $L^{q}(\Omega)$ and Hilbert scale spaces $\mathcal{H}^{\rho}(\Omega)$. The proposoed method has been verified by a numerical example; wherein, the regularized solution is well converged to the exact solution. It shows that our proposal method is capable of solving the final value problem of an inverse problem for both linear and nonlinear biharmonic equations.

Acknowledgements

The authors would like to thank for the support from the National Research Foundation of Korea under grant number NRF-2020K1A3A1A05101625, and from the Institute of Construction and Environmental Engineering at Seoul National University.

Conflict of interest

The authors declare no conflicts of interest.