Numerical Determination of a Time-Dependent Boundary Condition for a Pseudoparabolic Equation from Integral Observation

Abstract

The third-order pseudoparabolic equations represent models of filtration, the movement of moisture and salts in soils, heat and mass transfer, etc. Such non-classical equations are often referred to as Sobolev-type equations. We consider an inverse problem for identifying an unknown time-dependent boundary condition in a two-dimensional linear pseudoparabolic equation from integral-type measured output data. Using the integral measurements, we reduce the two-dimensional inverse problem to a one-dimensional problem. Then, we apply appropriate substitution to overcome the non-local nature of the problem. The inverse ill-posed problem is reformulated as a direct well-posed problem. The well-posedness of the direct and inverse problems is established. We develop a computational approach for recovering the solution and unknown boundary function. The results from numerical experiments are presented and discussed.

1. Introduction

Third-order pseudoparabolic partial differential equations are distinguished by the presence of mixed third-order derivatives, specifically second-order spatial and first-order temporal terms. These equations are utilized to model processes such as electrical conduction in heterogeneous media [1], two-phase flow in porous media [2], heat conduction in two-temperature systems, fluid flow through porous media, diffusion of a fluid in fractured porous media, the movement of moisture and salts in soils, and the regularization of ill-posed transport problems, see, e.g., [3,4,5,6,7].

Initial boundary-value problems for third-order pseudoparabolic equations, including linear, semilinear, nonlinear, and fractional-order modifications, have been well studied in the literature both analytically and numerically. For example, the uniqueness, nonexistence, and instability of solutions to a linear pseudoparabolic equation were investigated in [8]. Results on existence and uniqueness are presented in [9] for linear pseudoparabolic PDEs, and in [10] for nonlinear pseudoparabolic equations with a non-local boundary condition. The local existence and uniqueness of weak solutions were studied in [11] for nonlinear pseudoparabolic equations with Robin–Dirichlet boundary conditions, and in [12,13] for a fractional pseudoparabolic equation.

Furthermore, the ADI method for pseudoparabolic problems was developed in [14], a three-layer difference method for linear pseudoparabolic models with time delay was constructed and analyzed in [15,16], and spectral discretization and finite element two-grid methods for a nonlinear Dirichlet pseudoparabolic equation were investigated in [2] and [17], respectively.

An important class of pseudoparabolic equations are the Benjamin–Bona–Mahony (BBM) equations, describing the propagation of one-dimensional, unidirectional small-amplitude long waves in nonlinear dispersive media, see, e.g., [18,19]. It has been proved [18] that the evolutionary problem for the non-damped BBM equation is well-posed. In contrast, the traveling-wave non-dumped BBM problem is not well-posed [19].

In inverse problems for partial differential equations, the unknowns include not only the solution but also the parameters and functions of the model, such as the coefficients, right-hand side of the differential equation, boundary conditions, and the initial condition. These are data that cannot be directly measured and must be determined through mathematical methods using additional overspecified data, obtained from observations. For this reason, inverse problems have wide practical applications. Such problems are ill-posed [20,21,22,23,24,25,26,27], and their solution is far more challenging than solving the corresponding direct problems.

The physical and mathematical justification of various formulations of inverse problems for pseudoparabolic equations were thoroughly discussed in [28]. In engineering, determining boundary conditions is often crucial for designing and optimizing systems: for instance, in thermal engineering—identifying unknown boundary heat flux or temperature conditions in layered materials or structures; in soil science and civil engineering—estimating moisture or contaminant flow at interfaces in layered soils, crucial for irrigation management and groundwater safety; in material science—determining stress or displacement at boundaries in viscoelastic or thermoviscoelastic materials under load; in chemical engineering—understanding reaction–diffusion systems, such as the distribution of reactants in reactors with unknown input conditions, see, e.g., [24,29]. In the context of such engineering applications, we encourage readers to refer to the various comprehensive studies on parabolic and pseudoparabolic inverse problems, published in Applied Mathematics in Science and Engineering, where these topics are discussed.

Scientific publications investigating inverse problems for third-order pseudoparabolic equations are numerous. In [30], the authors studied an inverse problem for identifying the space-dependent part of the right-hand side of a pseudoparabolic equation for given measurements at the final time. They established the existence, uniqueness, and stability of the solution and developed a numerical method, based on the cubic B-spline functions and minimization of the Tikhonov regularization function. The solvability of the inverse source problem for a linear pseudoparabolic equation was discussed in [31]. The well-posedness of an inverse source problem for a linear pseudoparabolic equation with a memory was investigated in [32]. The Galerkin method was applied in [3] to prove the existence of a solution for the inverse coefficient problem for a pseudoparabolic equation and derive sufficient conditions for the blow-up of the solution. Additionally, the authors constructed a numerical algorithm, based on the finite difference method for solving direct and inverse problems. In [33], the authors established the existence and uniqueness of a solution and proposed an implicit finite-difference method for solving the pseudoparabolic inverse problem with periodic boundary conditions for recovering a time-dependent source term. The well-posedness of an inverse space-dependent source problem in a time-fractional pseudoparabolic equation with an involution operator was discussed in [34]. The existence, uniqueness, and regularity of the inverse problem for recovering the leading coefficient of a multi-dimensional pseudoparabolic equation was investigated in [35].

Boundary condition identification inverse problems for parabolic partial differential equation problems have been investigated in many papers, see, e.g., [36,37,38,39,40,41,42,43]. To the best of our knowledge, there has been a limited number of papers investigating inverse problems for recovering boundary conditions in pseudoparabolic equations. The solvability of the boundary condition determination inverse problem for pseudoparabolic problem was investigated in [44]. The inverse problem for identification of a time-dependent boundary coefficient in a pseudoparabolic equation was studied in [45]. The authors established the correctness of the problem.

In this work, we develop a numerical method for recovering a time-dependent boundary function in a Dirichlet–Neumann two-dimensional pseudoparabolic problem. We extend the method, proposed in [37] for the heat equation. Apart from the distinctions in the differential equation, the boundary conditions are also treated differently, to avoid the involvement of the third derivative and to overcome their non-local nature.

The contributions of this study are as follows:

■

We formulate an initial boundary-value problem for a new third-order, two-dimensional pseudoparabolic equation and pose a time-dependent inverse problem for determining an unknown boundary condition based on integral observations;

■

Using the integral condition, we reduce the two-dimensional inverse problem to an equivalent direct one-dimensional problem;

■

We develop an efficient numerical method based on a non-standard local approach to solve the one-dimensional non-local problem;

■

We propose effective computational algorithms for the implementation of this numerical method;

■

We present results from computational simulations that demonstrate the effectiveness of the numerical method.

The rest of this paper is organized as follows. In the next section, we formulate the direct and inverse problems and the establish well-posedness of the direct problem. In Section 3, we reduce the two-dimensional problem to a one-dimensional problem. Section 4 is devoted to the construction and implementation of the numerical method. In Section 5, we present the results from numerical simulations. The paper ends with some concluding remarks.

2. The Direct and Inverse Problems

In this section, we formulate the direct and inverse pseudoparabolic problems and establish the well-posedness of the direct problem.

2.1. Setup of the Direct and Inverse Problems

We consider the two-dimensional pseudoparabolic equation:

$${\displaystyle \frac{\partial }{\partial t}}\left(u-\epsilon {\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}\right)=\mu {\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+f(x,y,t),$$

(1)

where $\mu >0$, $\epsilon >0$ are constants, with the initial condition

$$u(x,y,0)=u_0(x,y),0\le x,y\le 1$$

(2)

and boundary conditions

$$\begin{array}{ccc}& & \mu {\displaystyle \frac{\partial u(0,y,t)}{\partial x}}=g_0(y,t),0\le t\le T,0\le y\le 1,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & \mu {\displaystyle \frac{\partial (1,y,t)}{\partial x}}=g_1(y,t),0<t\le T,0\le y\le 1,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & u(x,1,t)=p(x,t),0<t\le T,0\le x\le 1,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & u(x,0,t)=q(x,t)=q_0(x,t)r\left(t\right),0<t\le T,0\le x\le 1,\hfill \end{array}$$

(6)

and the non-local condition

$${\int }_0^1{\int }_0^{1}u(x,y,t)dxdy=\phi \left(t\right),0<t\le T.$$

(7)

In previous studies, see, e.g., [4,6,8,10,33,44], the third-order pseudoparabolic equation has typically been presented in one of the following forms: either (1) with $\mu =0$ for the one-dimensional case, or

$${\displaystyle \frac{\partial }{\partial t}}\left(u-▵u\right)=▵u+f(·),▵u={\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}},$$

for the two-dimensional version.

A large number of phenomena in modern physics and technology can be described in terms of non-local problems [10,14,36,37,39]. The non-local integral condition (7) mainly arises when the full mass or heat are measured.

When $u_0$, $g_0$, $g_1$, p, q and $\phi \left(t\right)$ are known functions, the problem (1)–(6) is called a direct (forward) problem. Our main interest is concentrated on inverse problems (1)–(7) in which $u_0$, $g_0$, $g_1$, p and $\phi \left(t\right)$ are known functions, while the functions u and $r\left(t\right)$ are unknown.

2.2. Well-Posedness of the Direct Problem

We establish the well-posedness of the direct problem, as this will serve as the foundation for demonstrating the well-posedness of the inverse problem.

We start with a weak formulation of the direct problem (1)–(6). Let $\psi (x,y,t)$ be a function from the Sobolev space $W_2^{1,2,1}\left(Q_T\right)$, $Q_T=\mathsf{\Omega }\times [0,T]$, $\mathsf{\Omega }=[0,1]\times [0,1]$, having first weak derivatives with respect to x and t, and second with respect to y. We multiply Equation (1) by an arbitrary function $\psi \in W_2^{1,2,1}\left(Q_T\right)$, where $\psi (x,y,T)=0$, $\psi (x,0,t)=0$, $\psi (x,1,t)=0$, and integrate the result on the domain $Q_T$, to derive

$$\begin{array}{ccc}& & \underset{0}{\overset{T}{\int }}\left[-\underset{\mathsf{\Omega }}{\int \int }\left(u{\displaystyle \frac{\partial }{\partial t}}\left(\psi -\epsilon {\displaystyle \frac{{\partial }^2\psi }{\partial y^2}}\right)+\mu {\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{\partial \psi }{\partial y}}+u{\displaystyle \frac{{\partial }^2\psi }{\partial y^2}}\right)dxdy\right.\hfill \\ & & \left.-\underset{0}{\overset{1}{\int }}\left(g_1(y,t)\psi (1,y,t)-g_0(y,t)\psi (0,y,t)\right)dy\right]dt\hfill \\ & & =\underset{\mathsf{\Omega }}{\int \int }\left(u_0(x,y)-\epsilon {\displaystyle \frac{{\partial }^2{u}_0}{\partial y^2}}(x,y)\right)\psi (x,y,0)dxdy+\underset{0}{\overset{T}{\int }}\underset{\mathsf{\Omega }}{\int \int }f(x,y,t)\psi (x,y,t)dxdydt.\hfill \end{array}$$

Theorem 1.

Assume that $u_0(x,y)\in W^{1,2}\left(\mathsf{\Omega }\right)$, $f(x,y,t)\in L_{\infty }\left((0,T);L_2\left(\mathsf{\Omega }\right)\right)$. Then, there exists a unique weak solution $u(x,y,t)$ of the problem (1)–(6).

The proof follows the same line of reasoning as in [44].

3. Reduction of the Two-Dimensional Problems to One-Dimensional Problems

Following the main idea in the papers in [37,46], we use the transformation

$$v(y,t)={\int }_0^{1}u(x,y,t)dx,0\le y\le 1.$$

(8)

Then, the following initial-boundary problem for $v(y,t)$ arises:

$${\displaystyle \frac{\partial }{\partial t}}\left(v-\epsilon {\displaystyle \frac{{\partial }^{2}v}{\partial y^2}}\right)={\displaystyle \frac{{\partial }^{2}v}{\partial y^2}}+R(y,t),(y,t)=Q_T=\mathsf{\Omega }\times (0,T),\mathsf{\Omega }=(0,1),$$

(9)

with initial condition

$$v(y,0)=v_0\left(y\right),y\in \mathsf{\Omega },$$

(10)

boundary condition

$$v(1,t)=v_1\left(t\right),t\in (0,T]$$

(11)

and non-local condition

$${\int }_0^{1}v(y,t)dy=\phi \left(t\right),t\in (0,T],$$

(12)

where

$$\begin{array}{cc}& {\displaystyle R(y,t)=g_1(y,t)-g_0(y,t)+\underset{0}{\overset{1}{\int }}f(x,y,t)dx,}\\ & {\displaystyle v_0\left(y\right)={\int }_0^1{u}_0(x,y,0)dx,v_1\left(t\right)={\int }_0^{1}p(x,t)dx.}\end{array}$$

Finally, the problem for the unknown function $v(y,t)$ is defined by (9)–(11), and instead of a right-boundary condition, we have the non-local condition (12). When R, $v_0$, $v_1$ and $\phi \left(t\right)$ are known functions, then the problem (9)–(12) is a so-called direct (forward) problem. Solving this direct problem and using (6), we find

$$r\left(t\right)=v(0,t){\left[\underset{0}{\overset{1}{\int }}{q}_0(x,t)dx\right]}^{-1},$$

(13)

which immediately solves the inverse problem (1)–(7), formulated in the previous section.

Theorem 2.

Let the conditions of Theorem 1 be fulfilled and ${\displaystyle {\int }_0^1{q}_0(x,t)dx>0}$. Then, there exists a unique weak solution to the problem (9)–(12), and it is a unique solution of the inverse problem (1)–(7).

Proof. 

The existence and uniqueness of weak solution of the problem (9)–(12) follows from the results in the papers in [45,47]. The uniqueness of the solution of the inverse problem (1)–(7) is a direct corollary of formula (13). □

Furthermore, we will develop an approach aimed at avoiding the non-locality in the problem (9)–(12).

4. Numerical Solution of Problem (9)–(12)

In this section, we require additional smoothness of the solution of the problem (9)–(12). For this reason, we assume that the functions $f(x,y,t)$, $u_0(x,y)$, $g_0(x,y)$, and $g_1(x,y)$ have a first continuous derivative with respect to y, which provide sufficient smoothness of the solution for the further discretizations.

4.1. Local Approach

First, we propose the idea of the method, which aims to avoid the potential non-locality. Following [37,39], we reformulate the problem (9)–(12), setting

$$w(y,t)={\displaystyle \frac{\partial v}{\partial y}}(y,t),(y,t)\in Q_T.$$

(14)

Then, in view of (9) and (10), w solves the equation

$${\displaystyle \frac{\partial }{\partial t}}\left(w-\epsilon {\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\right)={\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}+S(y,t),S(y,t)={\displaystyle \frac{\partial R(y,t)}{\partial y}},(y,t)\in Q_T,$$

(15)

with the initial condition

$$w(y,0)=w_0\left(y\right)={\displaystyle \frac{\partial v_0\left(y\right)}{\partial y}},x\in \mathsf{\Omega }$$

(16)

and boundary conditions obtained as follows.

Taking the limit $y\to 1$ in the Equation (9) and using (11), we find

$${\displaystyle \frac{\partial w}{\partial y}}(1,t)+\epsilon {\displaystyle \frac{{\partial }^{2}w}{\partial t\partial y}}(1,t)={\displaystyle \frac{d{v}_1}{dt}}-R(1,t).$$

(17)

Next, differentiating both sides of (12) with respect to t and using (9), (14), we find

$${\displaystyle \frac{d\phi }{dt}}=\epsilon {\displaystyle \frac{d}{dt}}\left(w(1,t)-w(0,t)\right)+w(1,t)-w(0,t)+{\int }_0^{1}R(y,t)dy.$$

(18)

The method for solving the inverse problem for recovering the function $r\left(t\right)$ and the solution $u(x,y,t)$ is described in Algorithm 1.

Algorithm 1 Method for recovering $r\left(t\right)$ and $u(x,y,t)$

Require: $f(x,y,t)$, $\epsilon >0$, $g_0(y,t)$, $g_1(y,t)$, $p(x,t)$, $q_0(x,t)$, $u_0(x,y)$, $\phi \left(t\right)$. Ensure: $r\left(t\right)$, $u(x,y,t)$. Main steps: 1. Solve (15)–(18) to find $w(y,t)$. 2. Find $v(0,t)$ from (14). 3. Find $r\left(t\right)$ from (13). 4. Solve the direct problem (1)–(6) to find $u(x,y,t)$.

Remark 1.

In step 4 of Algorithm 1, we may use (8) instead of solving the problem (1)–(6), but this would introduce non-locality and, consequently, in the numerical implementation using implicit approximation (8) would produce a full coefficient matrix that is not diagonally dominant. On the other hand, applying an explicit approximation might lead to instability.

4.2. Numerical Realization

In this section, we discuss the numerical implementation of Algorithm 1.

In the domain $Q_T$, we introduce the mesh ${\overline{w}}_{h\tau }={\overline{w}}_h\times w_{\tau }$, where

$${\overline{w}}_h=\{y_j=jh,j=0,1,\dots ,J,h=1/J\},{\overline{w}}_{\tau }=t_n=n\tau ,n=0,1,\dots ,N,n=T/N\}.$$

First, we discretize the problem (15)–(18). Furthermore, we use the notations $W_j^n\approx w(y_j,t_n)$, $V_j^n\approx v(y_j,t_n)$. Using finite difference (first-order in time and second-order in space) [48], we approximate Equation (15) using a fully implicit scheme:

$${\displaystyle \frac{W_j^{n+1}-W_j^n}{\tau }}=W_{\overline{y}y,j}^{n+1}+{\displaystyle \frac{\epsilon }{\tau }}\left(W_{\overline{y}y,j}^{n+1}-W_{\overline{y}y,j}^n\right)+S_j^{n+1},j=1,2,\dots ,J-1,$$

(19)

where $W_{\overline{y}y,j}^n=(W_{j-1}^n-2W_j^n+W_{j+1}^n)/h^2$ and $S_j^n=S(y_j,t_n)$.

The boundary condition (17) is approximated by the central finite difference in space

$$W_{\mathring{y},J}^{n+1}+{\displaystyle \frac{\epsilon }{\tau }}\left(W_{\mathring{y},J}^{n+1}-W_{\mathring{y},J}^n\right)={\displaystyle \frac{1}{\tau }}\left(v_1^{n+1}-v_1^n\right)-R_J^{n+1},$$

(20)

where $W_{\mathring{y},j}^n=(W_{j+1/2}^n-W_{j-1/2}^n)/\left(h\right)$, $W_{j\pm 1/2}^n=W(y_{j\pm 1/2},t_n)$, $y_{j\pm 1/2}=y_j\pm {\displaystyle \frac{h}{2}}$, $v_1^n=v_1\left(t_n\right)$ and $R_j^n=R(y_j,t_n)$. Note that the terms $W_{\mathring{y},J}^{n+1}$ and $W_{\mathring{y},J}^n$ involve the solution $W_{J+1/2}^{n+1}$ and $W_{J+1/2}^n$ at outer spatial node $y_J+h/2$, which must be eliminated.

Rewrite Equation (15) in an equivalent divergent form

$${\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{\partial w}{\partial y}}+\epsilon {\displaystyle \frac{{\partial }^{2}w}{\partial t\partial y}}\right)={\displaystyle \frac{\partial w}{\partial t}}-S(y,t),$$

and approximate it in the boundary node $y_J$ as follows. First, at half interval $[y_J-h/2,y_J]$, we have

$${\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{\partial w}{\partial y}}+\epsilon {\displaystyle \frac{{\partial }^{2}w}{\partial t\partial y}}\right)\approx {\displaystyle \frac{2}{h}}\left[{\left({\displaystyle \frac{\partial w}{\partial y}}+\epsilon {\displaystyle \frac{{\partial }^{2}w}{\partial t\partial y}}\right)}_{y=J}-{\left({\displaystyle \frac{\partial w}{\partial y}}+\epsilon {\displaystyle \frac{{\partial }^{2}w}{\partial t\partial y}}\right)}_{y=J-1/2}\right].$$

Then, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \frac{2}{h}}\left(W_{\mathring{y},J}^{n+1}+{\displaystyle \frac{\epsilon }{\tau }}\left(W_{\mathring{y},J}^{n+1}-W_{\mathring{y},J}^n\right)-W_{\mathring{y},J-1/2}^{n+1}-{\displaystyle \frac{\epsilon }{\tau }}\left(W_{\mathring{y},J-1/2}^{n+1}-W_{\mathring{y},J-1/2}^n\right)\right)\\ \hfill ={\displaystyle \frac{W_J^{n+1}-W_J^n}{\tau }}-S_J^{n+1}.\end{array}\end{array}$$

(21)

Expressing the term ${\displaystyle W_{\mathring{y},J}^{n+1}+\frac{\epsilon }{\tau }\left(W_{\mathring{y},J}^{n+1}-W_{\mathring{y},J}^n\right)}$ from (20) and substituting in (21), we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \frac{v_1^{n+1}-v_1^n}{\tau }}-R_J^{n+1}-W_{\mathring{y},J-1/2}^{n+1}-{\displaystyle \frac{\epsilon }{\tau }}\left(W_{\mathring{y},J-1/2}^{n+1}-W_{\mathring{y},J-1/2}^n\right)\\ \hfill =h{\displaystyle \frac{W_J^{n+1}-W_J^n}{2\tau }}-{\displaystyle \frac{h}{2}}{S}_J^{n+1}.\end{array}\end{array}$$

(22)

Furthermore, the boundary condition (18) is discretized by directly replacing the derivatives with finite differences

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \frac{{\phi }^{n+1}-{\phi }^n}{\tau }}={\displaystyle \frac{\epsilon }{\tau }}\left(W_J^{n+1}-W_J^n-W_0^{n+1}+W_0^n\right)+W_J^{n+1}-W_0^{n+1}+\underset{0}{\overset{1}{\int }}R(y,t_{n+1})dy.\end{array}\end{array}$$

(23)

Depending on the function $R(y,t)$, the integral in (23) can be computed approximately, for example, using the trapezoidal rule.

Therefore, at each time layer, in order to find the function $w(y,t)$ numerically, we need to solve the system (19), (22), (23) completed with the initial condition (16). This system can be written in the form

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & A_0{W}_0^{n+1}-A_0{W}_J^{n+1}={\displaystyle \frac{{\phi }^n-{\phi }^{n+1}}{\tau }}+{\displaystyle \frac{\epsilon }{\tau }}(W_0^n-W_J^n)+\underset{0}{\overset{1}{\int }}R(y,t_{n+1})dy,\hfill \\ & -A{W}_{j-1}^{n+1}+\left({\displaystyle \frac{1}{\tau }}+2A\right)W_j^{n+1}-A{W}_{j+1}^{n+1}={\displaystyle \frac{W_j^n}{\tau }}-{\displaystyle \frac{\epsilon }{\tau }}{W}_{\overline{y}y,j}^n+S_j^{n+1},j=1,2,\dots ,J-1,\hfill \\ & -2A{W}_{J-1}^{n+1}+\left({\displaystyle \frac{1}{\tau }}+2A\right)W_J^{n+1}={\displaystyle \frac{2(v_1^{n+1}-v_1^n)}{h\tau }}-{\displaystyle \frac{2}{h}}{R}_J^{n+1}+{\displaystyle \frac{2\epsilon }{h\tau }}{W}_{\mathring{y},J-1/2}^n+{\displaystyle \frac{W_J^n}{\tau }}+S_J^{n+1},\hfill \end{array}\end{array}$$

(24)

where ${\displaystyle A_0=1+\frac{\epsilon }{\tau }}$ and ${\displaystyle A=\frac{A_0}{h^2}}$.

Remark 2.

The numerical scheme (24) only includes a non-local term $W_J^{n+1}$ in the boundary condition at $y=0$, but the coefficient matrix remains sparse (almost tridiagonal). Moreover, it is strictly diagonally dominant with positive main diagonal elements and negative off-diagonal entries. This simple non-locality can be overcome by using implicit–explicit approximation, for example the Saul’yev schemes in [46]. Our previous investigations [49] showed the advantages of the fully implicit scheme compared to the Saul’yev method when solving inverse problems.

Next, we need to compute the values $V_0^n$, $n=0,1,\dots ,N$. Integrating (14) in the interval $[0,1]$ and approximating the left-hand side, we obtain

$$V_0^{n+1}=v_1^{n+1}-h\left({\displaystyle \frac{1}{2}}{W}_0^{n+1}+\sum _{j=1}^{J-1}{W}_j^{n+1}+{\displaystyle \frac{1}{2}}{W}_J^{n+1}\right).$$

(25)

Thus, we may determine $r^n$, $n=0,1,\dots ,N$ from (13). Depending on the function $q_0(x,t)$, the integral in (13) can be calculated exactly or numerically.

It remains to find the solution of the direct problem (1)–(6). We introduce a uniform mesh in the x-direction

$${\overline{w}}_k=\{x_i=ik,i=0,1,\dots ,I,k=1/I\}$$

and denote by $U_{i,j}^n$ the numerical solution, which approximates u at grid node $(x_i,y_j,t_n)$. We construct a second-order in space, first-order in time fully implicit finite difference scheme. To obtain second-order discretizations on the Neumann boundaries $x=0$ and $x=1$, we use central finite difference approximations and then, as before, the artificial nodes are eliminated from the approximation of the Equation (1), written at the corresponding boundary nodes. The resulting numerical scheme is

$$\begin{array}{ccc}& & {\displaystyle \frac{U_{i,j}^{n+1}-U_{i,j}^n}{\tau }}-{\displaystyle \frac{\epsilon }{\tau }}\left(U_{\overline{y}y,i,j}^{n+1}-U_{\overline{y}y,i,j}^n\right)=\mu U_{\overline{x}x,i,j}^{n+1}+U_{\overline{y}y,i,j}^{n+1}+f_{i,j}^{n+1},i=1,\dots ,I,j=1,\dots ,J,\hfill \\ & & {\displaystyle \frac{U_{0,j}^{n+1}-U_{0,j}^n}{\tau }}-{\displaystyle \frac{\epsilon }{\tau }}\left(U_{\overline{y}y,0,j}^{n+1}-U_{\overline{y}y,0,j}^n\right)=2\mu {\displaystyle \frac{U_{1,j}^{n+1}-U_{0,j}^{n+1}}{h^2}}+U_{\overline{y}y,0,j}^{n+1}\hfill \\ & & -{\displaystyle \frac{2}{h}}{\left(g_0\right)}_j^{n+1}+f_{0,j}^{n+1},j=1,2,\dots ,J,\hfill \\ & & {\displaystyle \frac{U_{I,j}^{n+1}-U_{I,j}^n}{\tau }}-{\displaystyle \frac{\epsilon }{\tau }}\left(U_{\overline{y}y,I,j}^{n+1}-U_{\overline{y}y,I,j}^n\right)=2\mu {\displaystyle \frac{U_{I-1,j}^{n+1}-U_{I,j}^{n+1}}{h^2}}+U_{\overline{y}y,I,j}^{n+1}\hfill \\ & & +{\displaystyle \frac{2}{h}}{\left(g_1\right)}_j^{n+1}+f_{I,j}^{n+1},j=1,2,\dots ,J,\hfill \\ & & U_{i,J}^{n+1}=p_i^{n+1},i=0,1,\dots ,I,\hfill \\ & & U_{I,j}^{n+1}={\left(q_0\right)}_j^{n+1}{r}^{n+1},j=0,1,\dots ,J,\hfill \\ & & U_{i,j}^0=u_0(x_i,y_j),i=0,1,\dots ,I,j=0,1,\dots ,J.\hfill \end{array}$$

(26)

The numerical approach for solving the inverse problem (1)–(7) for recovering the function $r\left(t\right)$ and solution $u(x,y,t)$ is described in Algorithm 2.

Algorithm 2 Numerical method for recovering $r\left(t\right)$ and $u(x,y,t)$

Require: $f(x,y,t)$, $\epsilon >0$, $g_0(y,t)$, $g_1(y,t)$, $p(x,t)$, $q_0(x,t)$, $u_0(x,y)$, $\phi \left(t\right)$ Ensure: $r^n$, $U_{i,j}^n$, $i=0,1,\dots ,I$, $j=0,1,\dots ,J$, $n=0,1,\dots ,N$. Main steps: 1. Solve (24) to find $W_j^n$, $j=0,1,\dots ,J$, $n=0,1,\dots ,N$. 2. Compute $V_0^n$, $n=0,1,\dots ,N$ from (25). 3. Compute $r^n$, $n=0,1,\dots ,N$ from (13). 4. Solve (26) to find $U_{i,j}^n$, $i=0,1,\dots ,I$, $j=0,1,\dots ,J$, $n=0,1,\dots ,N$.

5. Computational Experiments

In this section, we demonstrate the effectiveness of the developed numerical approach. We use a test problem with an exact solution. Let $\mu =1$ and

$$\begin{array}{cc}& {\displaystyle f(x,y,t)=\left(Psin\frac{\pi x}{2}+Qcos\frac{\pi x}{2}\right)e^{\lambda t},P=\left(\lambda +\frac{{\pi }^2}{4}\right),Q=P+\frac{\epsilon \lambda {\pi }^2}{4},\lambda \in \mathbb{R},}\\ & {\displaystyle g_0(y,t)=\frac{\pi }{2}{e}^{\lambda t},g_1(y,t)=0,p(x,y)=e^{\lambda t}sin\frac{\pi x}{2},q(x,y)=e^{\lambda t}\left(sin\frac{\pi x}{2}+1\right),}\\ & {\displaystyle u_0(x,y)=sin\frac{\pi x}{2}+cos\frac{\pi y}{2}.}\end{array}$$

Then, the exact solution of the problem (1)–(6) is ${\displaystyle u(x,t,t)=e^{\lambda t}\left(sin\frac{\pi x}{2}+cos\frac{\pi y}{2}\right)}$.

We consider three test problems (TP):

• TP1: ${\displaystyle q_0(x,t)=sin\frac{\pi x}{2}+1}$, ${\displaystyle r\left(t\right)=e^{\lambda t}}$;
• TP2: ${\displaystyle q_0(x,t)=e^{\lambda t/2}\left(sin\frac{\pi x}{2}+1\right)}$, ${\displaystyle r\left(t\right)=e^{\lambda t/2}}$;
• TP3: ${\displaystyle q_0(x,t)=\frac{1}{2}{e}^{\lambda t/2}(t^3+1)\left(sin\frac{\pi x}{2}+1\right)}$, ${\displaystyle r\left(t\right)=\frac{2e^{\lambda t/2}}{t^3+1}}$.

The errors and order of convergence are computed in maximum and $L_2$ norms

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle E_{i,j}^n=U_{i,j}^n-u(x_i,y_j,t_n),e^n=r^n-r\left(t_n\right),}\hfill \\ & {\displaystyle {\mathcal{E}}_{\infty }={\mathcal{E}}_{\infty }^{I,J}=\underset{1\le n\le N}{max}\underset{0\le i\le I}{max}\underset{0\le j\le J}{max}\left|E_{i,j}^n\right|,{\mathcal{E}}_2={\mathcal{E}}_2^{I,J}=\sqrt{hk\tau \sum _{i=0}^I\sum _{n=0}^J\sum _{n=1}^N{\left(E_{i,j}^n\right)}^2},}\hfill \\ & {\displaystyle {ϵ}_{\infty }=e_{\infty }^N=\underset{1\le n\le N}{max}\left|e^n\right|,{ϵ}_2={ϵ}_2^N=\sqrt{\tau \sum _{i=0}^N{\left(e^n\right)}^2},}\hfill \\ & {\displaystyle C{R}_{\infty }={log}_2\frac{{\mathcal{E}}_{\infty }^{I,J}}{{\mathcal{E}}_{\infty }^{2I,2J}},C{R}_2={log}_2\frac{{\mathcal{E}}_2^{I,J}}{{\mathcal{E}}_2^{2I,2J}},c{r}_{\infty }={log}_2\frac{{ϵ}_{\infty }^N}{{ϵ}_{\infty }^{2N}},c{r}_2={log}_2\frac{{ϵ}_2^N}{{ϵ}_2^{2N}}.}\hfill \end{array}\end{array}$$

Furthermore, all computations are performed for $I=J$, and in Examples 1, 2, we set $T=1$, $\lambda ={\displaystyle \frac{1}{2}}$.

Example 1

(Exact measurements). In these tests, we obtain the measurement $\varphi \left(t\right)$ in (7) from the exact solution. In

Table 1. Errors and convergence rates of the numerically recovered function r and solution U, Algorithm 2, TP1, $\tau =h$, Example 1.

N	${\mathcal{E}}_{\mathit{\infty }}$	${\mathit{C}\mathit{R}}_{\mathit{\infty }}$	${\mathcal{E}}_2$	${\mathit{C}\mathit{R}}_2$	${\mathit{e}}_{\mathit{\infty }}$	${\mathit{c}\mathit{r}}_{\mathit{\infty }}$	${\mathit{e}}_2$	${\mathit{c}\mathit{r}}_2$
20	1.9260 × 10−3		6.8559 × 10−4		7.0016 × 10−4		6.2560 × 10−4
40	1.0998 × 10−3	0.8083	2.9158 × 10−4	1.2334	5.4992 × 10−4	0.3485	4.1032 × 10−4	0.6085
80	6.7881 × 10−4	0.6962	1.4217 × 10−4	1.0363	3.3940 × 10−4	0.6962	2.3047 × 10−4	0.8322
160	3.7145 × 10−4	0.8698	7.1236 × 10−5	0.9970	1.8573 × 10−4	0.8698	1.2156 × 10−4	0.9229
320	1.9372 × 10−4	0.9392	3.5774 × 10−5	0.9937	9.6858 × 10−5	0.9392	6.2359 × 10−5	0.9630
640	9.8852 × 10−5	0.9706	1.7926 × 10−5	0.9969	4.9426 × 10−5	0.9706	3.1573 × 10−5	0.9819

,

Table 2. Errors and convergence rates of the numerically recovered function r and solution U, Algorithm 2, TP2, $\tau =h$, Example 1.

N	${\mathcal{E}}_{\mathit{\infty }}$	${\mathit{C}\mathit{R}}_{\mathit{\infty }}$	${\mathcal{E}}_2$	${\mathit{C}\mathit{R}}_2$	${\mathit{e}}_{\mathit{\infty }}$	${\mathit{c}\mathit{r}}_{\mathit{\infty }}$	${\mathit{e}}_2$	${\mathit{c}\mathit{r}}_2$
20	1.9260 × 10−3		6.8559 × 10−4		6.1249 × 10−4		5.4551 × 10−4
40	1.0998 × 10−3	0.8083	2.9158 × 10−4	1.2334	4.2829 × 10−4	0.5161	3.4838 × 10−4	0.6469
80	6.7881 × 10−4	0.6962	1.4217 × 10−4	1.0363	2.6433 × 10−4	0.6962	1.9420 × 10−4	0.8432
160	3.7145 × 10−4	0.8698	7.1236 × 10−5	0.9970	1.4464 × 10−4	0.8698	1.0214 × 10−4	0.9270
320	1.9372 × 10−4	0.9392	3.5774 × 10−5	0.9937	7.5433 × 10−5	0.9392	5.2332 × 10−5	0.9648
640	9.8852 × 10−5	0.9706	1.7925 × 10−5	0.9969	3.8493 × 10−5	0.9706	2.6481 × 10−5	0.9827

and

Table 3. Errors and convergence rates of the numerically recovered function r and solution U, Algorithm 2, TP3, $\tau =h$, Example 1.

N	${\mathcal{E}}_{\mathit{\infty }}$	${\mathit{C}\mathit{R}}_{\mathit{\infty }}$	${\mathcal{E}}_2$	${\mathit{C}\mathit{R}}_2$	${\mathit{e}}_{\mathit{\infty }}$	${\mathit{c}\mathit{r}}_{\mathit{\infty }}$	${\mathit{e}}_2$	${\mathit{c}\mathit{r}}_2$
20	1.9260 × 10−3		6.8559 × 10−4		1.1417 × 10−3		9.1968 × 10−4
40	1.0998 × 10−3	0.8083	2.9158 × 10−4	1.2334	6.5394 × 10−4	0.8040	5.4396 × 10−4	0.7576
80	6.7881 × 10−4	0.6962	1.4217 × 10−4	1.0363	3.5819 × 10−4	0.8685	2.9554 × 10−4	0.8801
160	3.7145 × 10−4	0.8698	7.1236 × 10−5	0.9970	1.8780 × 10−4	0.9315	1.5387 × 10−4	0.9416
320	1.9372 × 10−4	0.9392	3.5774 × 10−5	0.9937	9.6171 × 10−5	0.9655	7.8485 × 10−5	0.9713
640	9.8852 × 10−5	0.9706	1.7925 × 10−5	0.9969	4.8662 × 10−5	0.9828	3.9631 × 10−5	0.9858

, we present the errors and order of convergence of the recovered function $r^n$ and the solution U, computed by Algorithm 2, $\tau =h$ for TP1–TP3, respectively. We can observe that the order of convergence of the boundary function $r^n$ and the solution U is first-order in time for both maximal and $L_2$ norms. The precision of the recovered $r^n$ is very similar across all three examples, while the corresponding precision of the numerical solution U remains identical. For this reason, in the numerical test evaluating the spatial order of the convergence of U, it is sufficient to present the computational results for just one of the test problems. In

Table 4. Errors and convergence rates of the numerically recovered solution U, Algorithm (2), TP2, $\tau =h^2$, Example 1.

I	${\mathcal{E}}_{\mathit{\infty }}$	${\mathit{C}\mathit{R}}_{\mathit{\infty }}$	${\mathcal{E}}_2$	${\mathit{C}\mathit{R}}_2$
10	8.0085 × 10−3		2.2029 × 10−3
20	1.9952 × 10−3	2.0050	5.1666 × 10−4	2.0921
40	4.9836 × 10−4	2.0012	1.2533 × 10−4	2.0435
80	1.2456 × 10−4	2.0004	3.0875 × 10−5	2.0212
160	3.1069 × 10−5	2.0033	7.6578 × 10−6	2.0115
320	7.7632 × 10−6	2.0008	1.9070 × 10−6	2.0057

, we report the errors and convergence order for the recovered solution U, computed using Algorithm 2 with $\tau =h^2$ for TP2. The results demonstrate that for the considered test example with smooth initial data and solution, the convergence rate of the solution U is second-order in space, for both the maximum and $L_2$ norms.

Example 2

(Perturbed measurements). Now, we use the same measurements as in Example 1, but adding noise

$$\begin{array}{c}\hfill {\psi }^n=\phi \left(t\right)+2\rho ({\sigma }^n-0.5),\end{array}$$

where ρ is the noise level and ${\sigma }^n$ are random values, uniformly distributed in the interval $[0,1]$. We apply degree 5 polynomial curve fitting to smooth the data.

In

Table 5. Errors of the numerically recovered solution U, Algorithm 2, TP2, $\tau =h$, Example 2.

$\mathsf{\rho }$	${\mathcal{E}}_{\mathit{\infty }}$	${\mathcal{E}}_2$	${\mathit{e}}_{\mathit{\infty }}$	${\mathit{e}}_2$
	TP1
0.002	2.5199 × 10−3	2.9115 × 10−4	1.2600 × 10−3	4.3896 × 10−4
0.03	2.8296 × 10−2	3.5931 × 10−3	1.4148 × 10−2	4.1707 × 10−3
0.05	4.6707 × 10−2	5.9746 × 10−3	2.3354 × 10−2	6.8635 × 10−3
0.1	9.2736 × 10−2	1.1930 × 10−2	4.6368 × 10−2	1.3599 × 10−2
	TP2
0.002	2.5199 × 10−3	2.9115 × 10−4	9.8127 × 10−4	3.6082 × 10−4
0.03	2.8296 × 10−2	3.5931 × 10−3	1.1018 × 10−2	3.5048 × 10−3
0.05	4.6707 × 10−2	5.9746 × 10−3	1.8188 × 10−2	5.7771 × 10−3
0.1	9.2736 × 10−2	1.1930 × 10−2	3.6111 × 10−2	1.1461 × 10−2
	TP3
0.002	2.5199 × 10−3	2.9115 × 10−4	9.8127 × 10−4	4.9846 × 10−4
0.03	2.8296 × 10−2	3.5931 × 10−3	1.1018 × 10−2	5.2720 × 10−3
0.05	4.6707 × 10−2	5.9746 × 10−3	1.8188 × 10−2	8.7391 × 10−3
0.1	9.2736 × 10−2	1.1930 × 10−2	3.6111 × 10−2	1.7412 × 10−2

, we present the computational results for different levels of noise and $I=J=80$, $\tau =h$. We can observe that the restoration has satisfactory precision, even for perturbed data. As in the previous example, the accuracy of the solution U for the different test problems is one and the same. In Figure 1, Figure 2 and Figure 3, we depict the exact and the recovered function r in the case of noise levels $\rho =0.03$, $\rho =0.1$ for TP1–TP3, respectively. In Figure 4 and Figure 5, we plot the numerical solution of TP2, computed for the recovered $r^n$ and the corresponding error in the whole computational domain for $\rho =0.03$ and $\rho =0.1$, respectively. As can be expected, the higher error appears at the boundary $y=0$, where the function $r\left(t\right)$ is determined in the corresponding boundary condition.

Example 3

(Discontinuous and perturbed measurements). Although, in this work, we consider the case of a continuous function $\phi \left(t\right)$, i.e., $u(x,y,t)\in W_2^{1,2,1}\left(Q_T\right)$, we present computational results in the case of data discontinuous with respect to t. The test problem is TP1, but now we take

$$\lambda =\left\{\begin{array}{cc}{\displaystyle \frac{1}{4},}\hfill & t\le T/2,\hfill \\ {\displaystyle \frac{1}{2},}\hfill & t>T/2.\hfill \end{array}\right.$$

The computations are performed for $T=3$, $I=J=80$, $\tau =h$. We consider perturbed data with a level of noise $\rho =0.03$. In Figure 6, we plot exact and recovered function r, computed with and without polynomial curve fitting to smooth the measured data. In Figure 7 and Figure 8, we depict the solution and the error, obtained using recovered r for smoothed and non-smoothed measurements, respectively.

We can observe that the accuracy of the numerical solution of the direct problem, solved for the recovered function r, is similar when using smoothed and non-smoothed measurements. Moreover, the algorithm performs successfully, but in order to enhance the accuracy at point $t=T/2$, we need numerical discretizations (24) and (26) adapted to account for the discontinuity.

The computational simulations discussed in this section to validate the developed method were implemented using MATLAB R2022a on a computer with an Intel^Ⓡ Core i5-7500 Processor, DDR4 2133 MHz.

6. Conclusions

In this work, we examined a pseudoparabolic equation, commonly used for simulating filtration, moisture movement, and salt transport in soils. This study extends previous investigations, which primarily focused on one-dimensional cases. While the addition of a second spatial dimension introduces complexity, it provides a more realistic and accurate model for engineering applications. Currently, studies on direct and inverse problems for third-order parabolic equations are largely developed for the one-dimensional case. This applies specifically to the class of pseudoparabolic problems with non-local constraints on the solution.

We investigated an inverse problem for an unknown, time-varying boundary condition based on integral observation in a third-order, two-dimensional partial differential equation. In the initial stage of our approach, we use a solution transformation to reduce the original two-dimensional problem to an equivalent one-dimensional direct problem. By introducing a specialized local substitution, we developed an efficient numerical method to solve the one-dimensional direct problem. Computational experiments confirmed the effectiveness of the numerical algorithm in implementing this method.

The computational results showed that, for exact measurements, the order of convergence of the function $r^n$ is first-order in time, and for the solution U, it is first-order in time and second-order in space, both in maximum and $L_2$ norms. In the case of noisy data, the boundary function r and the solution were recovered with optimal precision.

In future work, we plan to extend the investigation to address inverse problems for unknown source terms and initial conditions in the considered third-order partial differential equation model. We will apply a similar approach to obtain a 1D retrospective inverse pseudoparabolic problem. Furthermore, we intend to provide a theoretical study with energy estimates. Additionally, we plan to extend the results to pseudohyperbolic problems.