Computational Solution of an Inverse Boundary-Value Problem for Heat Transfer in a Composite Material

Abstract

In the numerical simulation of composite material models, it is often necessary to recover boundary values of the solution to parabolic problems from integral constraints. In this work, we consider an inverse problem of determining a Dirichlet boundary condition for a heat equation with multiple interfaces and integral overspecification on a part of the spatial domain. After establishing the well-posedness of the direct problem, we propose an efficient numerical method for identifying an unknown Dirichlet boundary condition. The method decomposes the global inverse problem into a sequence of local subproblems, solved independently within each layer, including the accurate reconstruction of the solution at the interfaces. The approach relies solely on explicit schemes, employing an unconditionally stable Saulyev-type discretization and a novel interface treatment that avoids matrix inversion. Results from numerical experiments are presented and discussed.

1. Introduction

Multilayered and composite media are often modeled using parabolic partial differential equations with discontinuous coefficients that reflect the heterogeneity of the materials. Such models arise in a wide range of applications, including but not limited to heat conduction in composite structures [1,2,3], thermal propagation through skin layers in burn modeling [4], transport of contaminants, chemicals, or gases in stratified porous media [5,6], and they also appear in fields such as financial mathematics [7].

In many practical settings, direct measurements at the boundary are not accessible, and boundary values must instead be recovered from indirect data, such as interior observations or integral constraints, leading to inverse boundary value problems. For example, in industrial heat transfer processes, it is often impractical or even impossible to measure temperature or heat flux directly on the boundary during transient regimes. Instead, measurements are collected inside the domain, and inverse methods are employed to reconstruct the unknown boundary conditions [8]. In many cases, sensors provide average temperature values over spatial regions rather than pointwise data, and such integral measurements can be used to recover boundary conditions or material properties [9,10].

This study focuses on the numerical solution of an inverse problem for determining a Dirichlet boundary condition in a heat equation with multiple interfaces, motivated by the modeling of heat conduction in composite media. Such inverse problems are typically ill-posed [11,12,13,14], making them challenging to solve even numerically. Moreover, the presence of multiple interfaces and the limited spatial resolution of integral measurements further complicate the stable and accurate reconstruction of the unknown boundary data, requiring the use of advanced numerical techniques.

The direct interface parabolic problem, i.e., with all input data known, has been well studied in the literature. Analytical approaches such as the separation of variables method are often applied to pure-diffusion multilayer problems [1,15]. Stability analysis of one- and two-dimensional multilayer diffusion-reaction problems with a large number of layers is presented in [16,17]. Semi-analytical solution techniques for time-dependent diffusion problems in multi-layered structures, incorporating temporally variable external boundary conditions and general transmission conditions at internal layer interfaces, are developed in [18]. The Laplace transform has also been used to obtain a semi-analytical solution of the advection–dispersion–reaction interface problem modeling solute transport in layered porous media [19]. Finite volume schemes for solving multilayer diffusion problems are developed in [20].

Numerical methods for solving nonlocal problems with integral boundary conditions for classical one- and two-dimensional heat equations defined on a single domain are presented in [21,22]. The authors emphasize the challenges associated with addressing such problems.

Inverse problems involving the recovery of unknown boundary conditions for classical (single-domain) parabolic equations have been studied extensively; see, for example, refs. [23,24,25,26,27,28]. In [23], three regularization methods are presented for the identification of an unsteady Dirichlet boundary condition in the heat equation from interior point measurements. A semigroup approach is employed in [24] to investigate the recovery of a Dirichlet boundary condition in a time-dependent diffusion equation using pointwise interior data, and a similar semigroup-based technique is applied in [28] to analyze the inverse problem of identifying the Dirichlet boundary condition in a quasilinear parabolic equation using measurements of both the solution and its spatial derivative. Reference [25] considers a diffusion flow model in a chemical reactor and addresses an inverse problem for a Robin boundary condition, formulated via decomposition with respect to the unknown coefficient and minimization of a functional. Reference [27] deals with the identification of transient orthotropic thermal conductivity components in a two-dimensional parabolic heat equation from nonlocal flux data and Dirichlet boundary conditions, proving uniqueness and computing the solution via a nonlinear least-squares approach. This work is extended in [26] to include more general nonlocal over-specification conditions.

Inverse problems involving integral data have also been widely investigated. In [29,30], the inverse problem of identifying a time-dependent source in the Dirichlet boundary condition for a two-dimensional heat equation with mixed Neumann and Dirichlet boundary conditions and integral observation is considered. The authors develop numerical methods based on reformulating the problem into a one-dimensional problem with a nonlocal boundary condition, using standard implicit schemes, backward time-centered space schemes, and the Saulyev method. In [31], an inverse source problem for a two-dimensional parabolic equation subject to both Neumann and Dirichlet boundary conditions with integral measurements is studied. The existence and uniqueness of the solution are analyzed. A comprehensive theoretical foundation and the mathematical challenges of inverse problems in heat conduction are systematically addressed in [32].

Numerical techniques for inverse problems concerned with the identification of outer boundary conditions in two-dimensional parabolic and time-fractional parabolic equations posed on disjoint domains have been developed in our previous works [33,34]. In [33], an averaged left- and right-sided Saulyev scheme is applied, using available information from the internal boundaries, while in [34], a decomposition technique is employed.

Inverse problems for interface parabolic equations have been less extensively studied. In [35], the numerical treatment of an inverse problem related to the identification of outer boundary conditions in a one-dimensional multilayer diffusion model employing pointwise measurements is addressed by combining exponential finite difference schemes combined with a decomposition technique. A similar problem, but with integral observations in the last layer, is numerically investigated in [36]. Other contributions include the determination of interfaces in a parabolic-elliptic model from thermal surface data [37], where a variational formulation and uniqueness results are established, and a time-dependent adaptation of the factorization method for high-conductivity inclusions is developed. Reference [38] treats a coefficient inverse problem for semilinear parabolic interface equations with an additional integral condition in an optimization framework. Existence results for the recovery of diffusion and potential coefficients in a parabolic interface setting are presented in [39], and stability results for the simultaneous recovery of two discontinuous diffusion coefficients in a one-dimensional interface parabolic problem with minimal measurement data are obtained in [40].

In this paper, we develop a fast numerical method for solving inverse problems of boundary condition identification in a time-dependent diffusion equation with multiple interfaces, using integral observations in the final layer, namely the subdomain adjacent to the boundary with the unknown condition. The approach is based on solving the inverse problems sequentially and independently in each layer, including the reconstruction of the solution at the interfaces, instead of tackling the global problem over the entire domain. Moreover, the method employs only explicit techniques, utilizing an unconditionally stable Saulyev-type discretization and a specific strategy for determining the solution at the interfaces.

In our previous work [36], we developed a numerical method for identifying a Dirichlet boundary condition for the same interface problem, but with the integral measurement taken in the layer most distant from the boundary with the unknown condition. This results in a distinct inverse problem, and the numerical method proposed there is entirely different.

The structure of this paper is as follows. The direct and inverse problems are formulated in Section 2. In Section 3, we show that the forward problem is well-posed. A numerical method for identifying the Dirichlet boundary condition is presented in Section 4. Numerical simulation results are reported in Section 5, and the paper ends with concluding remarks.

2. The Direct and Inverse Problems

We consider a parabolic interface problem in the composite domain: $\overline{\mathrm{\Omega }}={\cup }_{m=1}^M{\overline{\mathrm{\Omega }}}_m$, ${\overline{\mathrm{\Omega }}}_m=[l_{m-1},l_m]\times [0,T]$, $m=1,2,\dots ,M$, $a=l_0$, $b=l_M$ (see Figure 1)

$${\displaystyle \frac{\partial u_m}{\partial t}}=p_m{\displaystyle \frac{{\partial }^2{u}_m}{\partial x^2}}+R_m(x,t),l_{m-1}<x<l_m,0<t\le T,$$

(1)

with initial conditions

$$u_m(x,0)=u_{m0}\left(x\right),l_{m-1}\le x\le l_m,$$

(2)

boundary conditions

$$\begin{array}{ccc}& & u_1(a,t)=\mu \left(t\right),0<t\le T,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & u_M(b,t)=\phi \left(t\right),0<t\le T,\hfill \end{array}$$

(4)

and interface relations

$$\begin{array}{ccc}& & {\left[u_m\right]}_{l_m}=u_{m+1}(l_m,t)-u_m(l_m,t)=0,m=1,2,\dots ,M-1,0\le t\le T,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & {\left[{\displaystyle \frac{\partial u_m}{\partial x}}\right]}_{l_m}=q_{m+1}{\displaystyle \frac{\partial u_{m+1}}{\partial x}}(l_m,t)-q_m{\displaystyle \frac{\partial u_m}{\partial x}}(l_m,t)=g_m\left(t\right),m=1,\dots ,M-1,0\le t\le T.\hfill \end{array}$$

(6)

If all coefficients and functions $p_m\ge {\overline{p}}_m>0$, $q_m$, $R_m(x,t)$, $\phi \left(t\right)$, $m\left(t\right)$, $m=1,2,\dots ,M$ in (1)–(6) are given, we call Problems (1)–(6) direct problems.

In this work, we consider boundary condition inverse problems (1)–(6). Namely, both the function $\phi \left(t\right)$ and the solution $u_m(x,t)$, $m=1,2,\dots ,M$ are unknown and must be determined using the following integral observation representing heat energy or mass specification:

$$\underset{l_{M-1}}{\overset{l_M}{\int }}{u}_M(x,t)dx=\psi \left(t\right),0<t\le T.$$

(7)

3. Well-Posedness of the Direct Problem

In this section, we present the results on the existence and uniqueness of a weak solution to the direct problems introduced in Section 2.

For the sake of clarity, but without loss of generality, we restrict our attention to the case $M=3$ with interface conditions (5) and (6). Namely, we consider the following initial boundary value problem:

$$\begin{array}{ccc}& & {\displaystyle \frac{\partial u}{\partial t}}-{\displaystyle \frac{\partial }{\partial x}}\left(p\left(x\right){\displaystyle \frac{\partial u}{\partial x}}\right)=g(x,t)+g_1\left(t\right)\delta \left(l_1\right)+g_2\left(t\right)\delta \left(l_2\right),(x,t)\in \overline{\mathrm{\Omega }},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & u(0,x)=u_0\left(x\right),u(a,t)=\mu \left(t\right),u(b,t)=\phi \left(t\right),0<t\le T.\hfill \end{array}$$

(9)

In (8), $\delta (·)$ is the Dirac-delta function.

In order to obtain the weak formulation of Problem (8), (9), we multiply (8) by a test function $v\in L_2(0,T;H^1\left(D\right))$, $D=(a,l_1)\cup [l_1,l_2]\cup (l_2,b)$, integrate over D, then integrate the resulting equation by parts.

Therefore, we seek $u\in L_2(0,T;H^1\left(D\right))$ with ${\displaystyle \frac{\partial u}{\partial t}\in L_2(0,T;H^{-1}\left(D\right))}$ and $g\in L_2\left(D\right)$, satisfying the identity

$$\underset{0}{\overset{T}{\int }}\left[-\underset{a}{\overset{b}{\int }}u{\displaystyle \frac{\partial v}{\partial t}}dx+\underset{a}{\overset{b}{\int }}p{\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{\partial v}{\partial x}}dx-\underset{a}{\overset{b}{\int }}gvdx+g_1\left(t\right)v(l_1,t)+g_2\left(t\right)v(l_2,t)\right]dt=\underset{a}{\overset{b}{\int }}{u}_0\left(x\right)vdx.$$

for every $v\in L_2(0,T;H^1\left(D\right)).$

Assume that such a weak solution exists, and ${\displaystyle \frac{\partial u}{\partial t}\in L_2\left((0,T);D\right)}$. Let us chose $v(x,t)=w\left(x\right)\psi \left(t\right)$, where $w\left(x\right)\in H^1\left(D\right)$, $w\left(a\right)=\mu \left(0\right)$, $w\left(b\right)=\phi \left(0\right)$, ${\displaystyle \frac{\partial \psi }{\partial t}\in L_2(0,T)}$, $\psi \left(T\right)=0$. We then perform integration by parts in the above identity to obtain the following:

$$\begin{array}{ccc}& & \underset{0}{\overset{T}{\int }}\left[-\underset{a}{\overset{b}{\int }}{\displaystyle \frac{\partial u}{\partial t}}wdx+\underset{a}{\overset{b}{\int }}p{\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{\partial w}{\partial t}}dx-\left(g_1\left(t\right)w\left(l_1\right)+g_2\left(t\right)w\left(l_2\right)\right)\right]dt\hfill \\ & & =\psi \left(0\right)\underset{a}{\overset{b}{\int }}\left(u(x,0)-u_0\left(x\right)\right)wdx.\hfill \end{array}$$

Now, in view of the arbitrariness of the function $\psi \left(t\right)$, we deduce that

$$\underset{a}{\overset{b}{\int }}{\displaystyle \frac{\partial u}{\partial t}}wdx+\underset{a}{\overset{b}{\int }}p{\displaystyle \frac{du}{dx}}{\displaystyle \frac{dw}{dx}}dx=g_1\left(t\right)w\left(l_1\right)+g_2\left(t\right)w\left(l_2\right),$$

for almost all $t\in (0,T)$, and

$$\underset{a}{\overset{b}{\int }}u(x,0)w\left(x\right)dx=\underset{a}{\overset{b}{\int }}{u}_0\left(x\right)w\left(x\right).$$

Based on these preparatory results, we will prove the existence and uniqueness of the weak solution to Problems (8), (9). We will also discuss the classical solution.

The following assertion holds.

Theorem 1.

Suppose that

$$\overline{p}\ge p\ge \underline{p}>0,$$

(10)

and let $g\in L_2\left(\mathrm{\Omega }\right)$, $u_0\in H^1\left(D\right)$, $g_i\in L_2(0,T)$, $i=1,2$. Then, Problems (8), (9) have a weak solution in $H^1\left(\mathrm{\Omega }\right)$.

Proof. 

We apply the Rothe method in the form described in [41], Section 3.2.

In the first stage, we analyze the time-semidiscrete approximation of the solutions as follows.

Let $w^{k,j}(x,jt)$, $j=0,1,\dots ,k$, $\tau =T/k$ be an approximation of the arbitrary function $w(x,t)$. Consider the approximation of (8):

$${\displaystyle \frac{u^{k,j}-u^{k,j-1}}{\tau }}-{\displaystyle \frac{\partial }{\partial x}}\left(p\left(x\right){\displaystyle \frac{\partial u^{k,j}}{\partial x}}\right)=g^{k,j}\left(x\right)+g_1^{k,j}\delta \left(l_1\right)+g_2^{k,j}\delta \left(l_2\right).$$

(11)

Since $u_0^{k,0}=u_0\in H^1\left(D\right)$, and due to the symmetry of the elliptic part in (11), we can conclude that $u^{k,j-1}$ is well defined; see Lemma 1 below. Therefore, we can construct, by induction, the finite sequence $u^{k,1},u^{k,2},\dots ,u^{k,k}$ in $H^1\left(D\right)$, for which uniform estimates hold.

Next, applying the standard procedure for progressing to the subsequent time layer and carrying out estimates for the approximate solutions in the manner described in ([41], Section 3.2), we complete this part of the proof.

In the second stage, we finalize the argument by investigating the stationary Equation (11) in the following form:

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left(p\left(x\right)u^{\prime }\right)}^{\prime }+q\left(x\right)u=g\left(x\right)+Q_1\delta \left(l_1\right)+Q_2\delta \left(l_2\right),\hfill \\ & u\left(a\right)=u_a,u\left(b\right)=u_b.\hfill \end{array}\end{array}$$

(12)

Here, ${(·)}^{\prime }={\displaystyle \frac{d}{dx}}$, $Q_1$, $Q_2$, $u_a$, and $u_b$ are constants. We define the scalar product in $H^1\left(D\right)$ as follows:

$${(u,v)}_{H_{+}^1\left(D\right)}=\underset{D}{\int }(p{u}^{\prime }{v}^{\prime }+quv)dx+Q_{1}u\left(l_1\right)v\left(l_1\right)+Q_{2}u\left(l_2\right)v\left(l_2\right).$$

It is equivalent to the usual symmetric scalar product $(u,v)=\underset{D}{\int }(p{u}^{\prime }{v}^{\prime }+quv)dx$. □

Next, we refine the above results in the direction of obtaining a classical solution of our basic problem (8), (9). We have the next statement.

Lemma 1.

Assume that condition (10) holds, and that $q\left(x\right)\ge q>0$. Then, for every $g\in L_2\left(D\right)$, Problem (12) admits a unique weak solution $u\in H^1\left(D\right)$, and the following uniform estimate is satisfied:

$${\parallel u\parallel }_{H_{+}^1\left(D\right)}\le C\left({\parallel g\parallel }_{L_2\left(D\right)}+|u_a|+|u_b|\right),$$

(13)

where the constant C does not depend on g, $u_a$, $u_b$, or $Q_1$, $Q_2$. The interface jump conditions are satisfied:

$${\left[u\right]}_{x=l_i}=u(l_i+0)-u(l_i-0)=0,{\left[u^{\prime }\right]}_{x=l_i}=Q_i,i=1,2.$$

(14)

Assume that $p,q\in L_{\infty }\left(D\right)\cap C^1\left(D_1\right)\cap C^1\left(D_2\right)\cap C^1\left(D_3\right)$ and $g\in L_{\infty }\left(D\right)\cap C^2\left(D_1\right)\cap C^2\left(D_2\right)\cap C^2\left(D_3\right)$. Then, $u\in L_{\infty }\left(D\right)\cap C^2\left(D_1\right)\cap C^2\left(D_2\right)\cap C^2\left(D_3\right)$; that is, u is a classical solution of Problem (12), and the following estimate is valid:

$${\parallel u\parallel }_{\infty }\le \left(|u_a|+|u_b|+{\displaystyle \frac{1}{q_0}}{\parallel g\parallel }_{\infty }\right).$$

(15)

Proof. 

The estimate (13) follows from the application of the Lax theorem, see, e.g., [41,42,43,44].

The interface jump relations (14), as well as the estimate (15), are well studied in [45], Section 3.3.

□

Remark 1.

Note that, to prove the existence and uniqueness of the solution to the inverse problem (1)–(7), it can be treated in a manner similar to the direct problem, using the auxiliary problem (12), where the right boundary condition is replaced by the nonlocal condition ${\int }_{l_2}^{b}u\left(x\right)dx={\psi }_b$ in view of the observation (7). The existence and uniqueness of such a modified problem (12) is investigated in [9].

4. Numerical Method

This section introduces the numerical method for the inverse problem (1)–(7). The main idea is to transform the boundary condition identification inverse multiple interface problem in the domain $\overline{\mathrm{\Omega }}$ into M separate boundary condition identification inverse problems, each posed in a smaller region; namely, within each layer ${\overline{\mathrm{\Omega }}}_m$, $m=1,2,\dots ,M$.

We introduce a uniform partition of the time interval $t_n=n\tau$, $\tau =T/N$, $n=0,1,\dots ,N$, and in each spatial subdomain ${\overline{\mathrm{\Omega }}}_m$, we define a uniform mesh:

$$\begin{array}{ccc}& & x_{1,i_1}=a+i_1{h}_1,i_1=0,1,\dots ,I_1,l_1=a+I_1{h}_1,\hfill \\ & & x_{2,i_2}=l_1+i_2{h}_2,i_2=0,1,\dots ,I_2,l_2=l_1+I_2{h}_2,\hfill \\ & & \dots \dots \dots \hfill \\ & & x_{m,i_m}=l_{m-1}+i_m{h}_m,i_m=0,1,\dots ,I_m,l_m=l_{m-1}+I_m{h}_m,\hfill \\ & & \dots \dots \dots \hfill \\ & & x_{M,i_M}=l_{M-1}+i_M{h}_M,i=0,1,\dots ,I_M,l_M=b=l_{M-1}+I_M{h}_M.\hfill \end{array}$$

Note that $x_{m,I_m}=x_{m+1,0}$, $m=1,2,\dots ,M-1$. We denote by $U_{m,i_m}^n$ the numerical approximation of the function $u_m$ at grid node $(x_{m,i},t_n)$.

First, we solve the problem in the region ${\overline{\mathrm{\Omega }}}_1$. Taking into account (3), we consider unconditionally stable Saulyev-type approximation of (1) for $m=1$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & U_{1,0}^{n+1}=\mu \left(t_{n+1}\right),\hfill \\ & {\displaystyle \frac{U_{1,i_1}^{n+1}-U_{1,i_1}^n}{\tau }}=p_1{\displaystyle \frac{U_{1,i_1-1}^{n+1}-U_{1,i_1}^{n+1}-U_{1,i_1}^n+U_{1,i_1+1}^n}{h_1^2}}+R_1(x_{1,i_1},t_{n+1}),i_1=1,2,\dots I_1-1.\hfill \end{array}\end{array}$$

(16)

The scheme in (21) can be realized in an explicit manner. We consequently determine the solution $U_{1,i_1}^{n+1}$ for known $U_{1,i_1-1}^{n+1}$, $i_1=1,2,\dots ,I_1-1$.

At spatial grid node $x_{1,I_1-1}=l_1-h_1$ (see Figure 1), we have the following:

$${\displaystyle \frac{U_{1,I_1-1}^{n+1}-U_{1,I_1-1}^n}{\tau }}=p_1{\displaystyle \frac{U_{1,I_1-2}^{n+1}-U_{1,I_1-1}^{n+1}-U_{1,I_1-1}^n+U_{1,I_1}^n}{h_1^2}}+R_1(x_{1,I_1-1},t_{n+1}).$$

(17)

Further, we denote ${\lambda }_m=\tau /h_m^2$, $m=1,2,\dots ,M$ and express $U_{1,I_1-1}^{n+1}$ from (17):

$$U_{1,I_1-1}^{n+1}={\displaystyle \frac{p_1{\lambda }_1}{1+p_1{\lambda }_1}}(U_{1,I_1-2}^{n+1}+U_{1,I_1}^n)+{\displaystyle \frac{1-p_1{\lambda }_1}{1+p_1{\lambda }_1}}{U}_{1,I_1-1}^n+{\displaystyle \frac{\tau }{1+p_1{\lambda }_1}}{R}_1(x_{1,I_1-1},t_{n+1})$$

From the interface conditions (5) and (6) for $m=1$, we obtain the following:

$$\begin{array}{ccc}\hfill U_1^{n+1}(l_1,t^{n+1})=U_2(l_1,t^{j+1})& \Rightarrow & U_{1,I_1}^{n+1}=U_{2,0}^{n+1}=:V_1^{n+1},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill q_2{\displaystyle \frac{\partial U_2}{\partial x}}(l_1,t)=q_1{\displaystyle \frac{\partial U_1}{\partial x}}(l_1,t)+g_1\left(t\right)& \Rightarrow & q_2{\displaystyle \frac{U_{2,1}^{n+1}-U_{2,0}^{n+1}}{h_2}}=q_1{\displaystyle \frac{U_{1,I_1}^{n+1}-U_{1,I_1-1}^{n+1}}{h_1}}+g_1^{n+1}.\hfill \end{array}$$

(19)

Similarly, at first spatial node $x_{2,1}$ after the interface $l_1$, in view of (18), we have the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill U_{2,1}^{n+1}& ={\displaystyle \frac{p_2{\lambda }_2}{1+p_2{\lambda }_2}}(U_{2,0}^{n+1}+U_{2,2}^n)+{\displaystyle \frac{1-p_2{\lambda }_2}{1+p_2{\lambda }_2}}{U}_{2,1}^n+{\displaystyle \frac{\tau }{1+p_2{\lambda }_2}}{R}_2(x_{2,1},t_{n+1})\hfill \\ & =\underset{A_2}{\underbrace{{\displaystyle \frac{p_2{\lambda }_2}{1+p_2{\lambda }_2}}}}{V}_1^{n+1}+\underset{B_2}{\underbrace{{\displaystyle \frac{p_2{\lambda }_2}{1+p_2{\lambda }_2}}{U}_{2,2}^n+{\displaystyle \frac{1-p_2{\lambda }_2}{1+p_2{\lambda }_2}}{U}_{2,1}^n+{\displaystyle \frac{\tau }{1+p_2{\lambda }_2}}{R}_2(x_{2,1},t_{n+1})}}\hfill \\ & =:A_2{V}_1^{n+1}+B_2\hfill \end{array}\end{array}$$

(20)

Then, from (18)–(20), we obtain

$$q_2{\displaystyle \frac{A_2{V}_1^{n+1}+B_2-V_1^{n+1}}{h_2}}=q_1{\displaystyle \frac{V_1^{n+1}-U_{1I_1-1}^{n+1}}{h_1}}+g_1^{n+1}$$

and find an explicit representation for $V_1^{n+1}$

$$V_1^{n+1}={\displaystyle \frac{h_1{q}_2{B}_2+h_2{q}_1{U}_{1I_1-1}^{n+1}-h_1{h}_2{g}_1^{n+1}}{h_1{q}_2(1-A_2)+h_2{q}_1}}.$$

Once $V_1^{n+1}$ is computed, we determine $U_{2,1}^{n+1}$ from (20).

We apply the same procedure for ${\overline{\mathrm{\Omega }}}_2$ to find $V_2^{n+1}:=U_{2,I_2}^{n+1}=U_{3,0}^{n+1}$ and $U_{3,1}^{n+1}$. Then, we repeat this procedure in the domains ${\overline{\mathrm{\Omega }}}_3,{\overline{\mathrm{\Omega }}}_4,\dots ,{\overline{\mathrm{\Omega }}}_{M-1}$ (see Figure 1).

At layer ${\overline{\mathrm{\Omega }}}_{M-1}$, we solve the following:

$$\begin{array}{ccc}\hfill {\displaystyle U_{M-1,0}^{n+1}}& =& U_{M-2,I_{M-2}}^{n+1},\hfill \\ \hfill {\displaystyle U_{M-1,i_{M-1}}^{n+1}}& =& {\displaystyle \frac{1-p_{M-1}{\lambda }_{M-1}}{1+p_{M-1}{\lambda }_{M-1}}}{U}_{M-1,i_{M-1}}^n\hfill \\ & & +{\displaystyle \frac{p_{M-1}{\lambda }_{M-1}}{1+p_{M-1}{\lambda }_{M-1}}}(U_{M-1,i_{M-1}-1}^{n+1}+U_{M-1,i_{M-1}+1}^n)\hfill \\ & & {\displaystyle +\frac{\tau }{1+p_{M-1}{\lambda }_{M-1}}{R}_{M-1}(x_{M-1,i_{M-1}},t_{n+1}),i_{M-1}=1,2,\dots I_{M-1}-1.}\hfill \end{array}$$

Further, we determine $V_{M-1}^{n+1}=U_{M-1,I_M}^{n+1}=U_{M,0}^{n+1}$ from

$$V_{M-1}^{n+1}={\displaystyle \frac{h_{M-1}{q}_M{B}_M+h_M{q}_{M-1}{U}_{M-1I_{M-1}-1}^{n+1}-h_{M-1}{h}_M{g}_{M-1}^{n+1}}{h_{M-1}{q}_M(1-A_M)+h_M{q}_{M-1}}},$$

where

$$A_M={\displaystyle \frac{p_M{\lambda }_M}{1+p_M{\lambda }_M}},B_M={\displaystyle \frac{p_M{\lambda }_M}{1+p_M{\lambda }_M}}{U}_{M,2}^n+{\displaystyle \frac{1-p_M{\lambda }_M}{1+p_M{\lambda }_M}}{U}_{M,1}^n+{\displaystyle \frac{\tau }{1+p_M{\lambda }_M}}{R}_M(x_{M,1},t_{n+1})$$

Then, we find $U_{M,1}^{n+1}=A_M{V}_{M-1}^{n+1}+B_M$.

For the last domain, ${\mathrm{\Omega }}_M$, taking into account that $V_{M-1}^{n+1}:=U_{M-1,I_{M-1}}^{n+1}=U_{M,0}^{n+1}$ is known from the previous computations, we compute the solution $U_{M,i_M}$, $i_M=1,2,\dots ,I_M-1$, using Saulyev-type approximation of (1) for $m=M$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill U_{M,0}^{n+1}=& V_{M-1}^{n+1},\hfill \\ \hfill U_{M,i_M}^{n+1}=& {\displaystyle \frac{p_M{\lambda }_M}{1+p_M{\lambda }_M}}(U_{2,i_M-1}^{n+1}+U_{2,i_M+1}^n)+{\displaystyle \frac{1-p_M{\lambda }_M}{1+p_M{\lambda }_M}}{U}_{2,i_M}^n\hfill \\ & +{\displaystyle \frac{\tau }{1+p_M{\lambda }_M}}{R}_M(x_{M,i_M},t^{n+1}),i_M=1,2,\dots I_M-1.\hfill \end{array}\end{array}$$

(21)

Next, we approximate the integral measurement (7) via the trapezoidal quadrature rule and represent the unknown value of the solution $U_{M,I_M}^{n+1}$ at the boundary $x=l_M=b$:

$$U_{M,I_M}^{n+1}\approx \phi \left(t_{n+1}\right)={\displaystyle \frac{2}{h_M}}\psi \left(t^{n+1}\right)-2\sum _{i_M=1}^{I_M-1}{U}_{M,i_M}^{n+1}-U_{M,0}^{n+1}.$$

Therefore, the problem is solved sequentially, progressing layer by layer, independently within each subdomain ${\overline{\mathrm{\Omega }}}_m$. During the computational process, the solutions at the interfaces are determined and used as Dirichlet boundary conditions for each layer ${\overline{\mathrm{\Omega }}}_m$. In this way, the original problem is decomposed into independent subproblems on each ${\overline{\mathrm{\Omega }}}_m$, which are solved separately but in sequence.

Moreover, the inverse problem in each layer is solved in an explicit fashion so that no matrix inversion is required.

The expected accuracy of the proposed approach is ${\displaystyle O\left(\tau +\left|h\right|+\tau /h\right)}$, where $h=\underset{m}{min}{h}_m$, $\left|h\right|=\underset{m}{max}{h}_m$, which corresponds to the accuracy of the one-sided Saulyev discretization.

5. Numerical Simulations

This section illustrates the effectiveness of the developed numerical method for addressing the inverse problem (1)–(7), and it provides computational results in the case of exact and perturbed measurements $\psi \left(t\right)$.

Let us define $I={\displaystyle \sum _{m=1}^M}{I}_m-M-1$. Further, all computations are carried out on a uniform mesh $h=h_m$ and $T=1$.

Example 1 (Exact data, $M=3$). We consider the inverse problem (1)–(7), setting

$$\begin{array}{cc}& {\displaystyle M=3,a=-2,l_1=0,l_2=0.5,b=1.5,p_1=4,p_2=2,p_3=0.25,}\\ & {\displaystyle q_1=2,q_2=1,q_3=3,g_1\left(t\right)=-2e^{-t/2},g_2\left(t\right)=-1.75e^{-t/2},}\\ & {\displaystyle R_1(x,t)=e^{-t/2}\left(4{\pi }^2-\frac{1}{2}\right)cos\pi x,R_2(x,t)=-e^{-t/2}\left(\frac{x^3}{2}+11x+\frac{1}{2}\right),}\\ & {\displaystyle R_3(x,t)=\frac{1}{32}{e}^{-t/2}\left({\pi }^2-2\right)sin\pi x,\mu \left(t\right)=e^{-t/2},}\\ & {\displaystyle u_{m0}\left(x\right)=\left\{\begin{array}{cc}{\displaystyle cos\pi x,}\hfill & {\displaystyle a\le x\le l_1,}\hfill \\ {\displaystyle (x^3-2x+1),}\hfill & l_1\le x\le l_2,\hfill \\ {\displaystyle \frac{1}{8}sin\pi x,}\hfill & l_2\le x\le b.\hfill \end{array}\right.}\end{array}$$

In this case, the exact solution is ${\displaystyle u_1(x,t)=e^{-t/2}cos\pi x}$, ${\displaystyle u_2(x,t)=e^{-t/2}(x^3-2x+1)}$, ${\displaystyle u_3(x,t)=\frac{1}{8}{e}^{-t/2}sin\pi x}$, ${\displaystyle \phi \left(t\right)=-\frac{1}{8}{e}^{-t/2}}$. The measurements (7) are generated from the exact solution $u_3(x,t)$.

We estimate the error in maximum and $L_2$ discrete norms at the final time in each subdomain ${\overline{\mathrm{\Omega }}}_m$, $m=1,2,\dots ,M$:

$$\begin{array}{cc}& {\displaystyle E_m=E_m^{I_m}=\underset{0\le i_m\le I_m}{max}|u_m(x_{i_m},T)-U_{m,i_m}^N|,{CR}_m={log}_2\frac{E_m^{I_m}}{E_{m,\infty }^{2I_m}}}\\ & {\displaystyle {\mathcal{E}}_m={\mathcal{E}}_m^{I_m}=\sqrt{\sum _{i_m=0}^{I_m}h{\left(u_m(x_{i_m},T)-U_{m,i_m}^N\right)}^2},{\mathcal{CR}}_m={log}_2\frac{{\mathcal{E}}_m^{I_m}}{{\mathcal{E}}_m^{2I_m}},}\end{array}$$

Let $\tau =h^2$. In

Table 1. Errors and convergence rates of solutions $U_m$, $m=1,2,3$, Example 1.

I	h	${\mathit{E}}_1$	${\mathbf{CR}}_1$	${\mathcal{E}}_1$	${\mathcal{CR}}_1$	${\mathit{E}}_2$	${\mathbf{CR}}_2$	${\mathcal{E}}_2$	${\mathcal{CR}}_2$	${\mathit{E}}_3$	${\mathbf{CR}}_3$	${\mathcal{E}}_3$	${\mathcal{CR}}_3$
70	0.05	1.188 $\times {10}^{-1}$		1.006 $\times {10}^{-1}$		1.188 $\times {10}^{-1}$		6.019 $\times {10}^{-2}$		3.363 $\times {10}^{-2}$		2.067 $\times {10}^{-2}$
140	0.025	5.947 $\times {10}^{-2}$	0.998	4.953 $\times {10}^{-2}$	1.023	5.947 $\times {10}^{-2}$	0.998	2.930 $\times {10}^{-2}$	1.039	1.674 $\times {10}^{-2}$	1.007	9.903 $\times {10}^{-3}$	1.061
280	0.0125	2.975 $\times {10}^{-2}$	1.999	2.457 $\times {10}^{-2}$	1.012	2.975 $\times {10}^{-2}$	0.999	1.444 $\times {10}^{-2}$	1.020	8.342 $\times {10}^{-3}$	1.005	4.840 $\times {10}^{-3}$	1.033
560	0.00625	1.488 $\times {10}^{-2}$	1.000	1.223 $\times {10}^{-2}$	1.006	1.488 $\times {10}^{-2}$	1.000	7.170 $\times {10}^{-3}$	1.010	4.163 $\times {10}^{-3}$	1.003	2.392 $\times {10}^{-3}$	1.017
1120	0.003125	7.440 $\times {10}^{-3}$	1.000	6.105 $\times {10}^{-3}$	1.003	7.440 $\times {10}^{-3}$	1.000	3.572 $\times {10}^{-3}$	1.005	2.079 $\times {10}^{-3}$	1.001	1.189 $\times {10}^{-3}$	1.009
2240	0.0015625	3.720 $\times {10}^{-3}$	1.000	3.050 $\times {10}^{-3}$	1.001	3.720 $\times {10}^{-3}$	1.000	1.786 $\times {10}^{-3}$	1.000	1.039 $\times {10}^{-3}$	1.000	5.941 $\times {10}^{-4}$	1.001

, we present errors and orders of convergence of the numerically recovered solutions $U_m$. Note that the function $\phi$ is a part of the solution in ${\overline{\mathrm{\Omega }}}_3$, i.e., ${\phi }^n=U_{3,I_3}^n$. We observe that the convergence rate in each subdomain is first-order in both the maximum and $L_2$ norms, since for $\tau =h^2$, the term $O(\tau /h)$ reduces to $O\left(h\right)$. Note that the solution at interface node $l_1$ and $l_2$ belongs to both subdomains ${\overline{\mathrm{\Omega }}}_1$ and ${\overline{\mathrm{\Omega }}}_2$, and the maximum errors of the solutions $U_1$ and $U_2$ are equal. This implies that the largest error occurs at the interfaces or at the boundary $x=b$. In the present example, the maximum of the solution occurs at the interface $x=l_1$, which explains the largest error at that point.

Computations with step size $\tau =h$ yield almost the same (slightly increasing) accuracy of the solution for different numbers of spatial grid nodes I, which confirms that the order of convergence is $O\left({\displaystyle \frac{\tau }{h}}+\tau +h\right)$, reducing in this case to $O\left(1\right)$. When the spatial step size is halved while keeping $\tau$ fixed with $\tau <h^2$, the error increases approximately linearly with $\tau /h$, which indicates that h cannot be refined independently of $\tau$.

Furthermore, all computations are performed with $\tau =h^2$.

Example 2 (Exact data, $M=5$). Now, we consider the inverse problem (1)–(7) for $M=5$. Let

$$\begin{array}{cc}& {\displaystyle M=5,a=-2,l_1=0,l_2=0.5,l_3=1.5,l_4=2,b=4,}\\ & {\displaystyle p_1=4,p_2=0.5,p_3=0.5,p_4=3,p_5=1,}\\ & {\displaystyle q_1=2,q_2=1,q_3=-2,q_4=-1,q_5=5,\mu \left(t\right)=t^3+t,u_{m0}\left(x\right)=0,}\\ & {\displaystyle g_1\left(t\right)=-\pi (t^3+t),g_2\left(t\right)=t^3+t,g_3\left(t\right)=15(t^3+t),g_4\left(t\right)=\left(\frac{29}{4}-5e^{-3/2}\right)(t^3+t),}\end{array}$$

and

$$\begin{array}{ccc}& & R_1(x,t)=({\pi }^2{t}^3+3t^2+{\pi }^{2}t+1)sin{\displaystyle \frac{\pi x}{2}}+t^3+t,\hfill \\ & & R_2(x,t)=\left({\displaystyle \frac{{\pi }^2}{2}}{t}^3+3t^2+{\displaystyle \frac{{\pi }^2}{2}}t+1\right)cos\pi x,\hfill \\ & & R_3(x,t)=2{\left(x-{\displaystyle \frac{1}{2}}\right)}^2\left({\left(x-{\displaystyle \frac{1}{2}}\right)}^2(3t^2+1)-6(t^3+t)\right),R_4(x,t)=(3t^2+1)\left(x+{\displaystyle \frac{1}{2}}\right),\hfill \\ & & R_5(x,t)=e^{-x}(-5t^3+3t^2-5t+1)+(3t^2+1)\left({\displaystyle \frac{5x}{4}}-e^{-2}\right).\hfill \end{array}$$

Thus, the exact solution of the inverse problem (1)–(7) is ${\displaystyle u_1(x,t)=(t^3+t)\left(sin\frac{\pi x}{2}+1\right)}$, $u_2(x,t)=(t^3+t)cos\pi x$, ${\displaystyle u_3(x,t)=2(t^3+t){\left(x-\frac{1}{2}\right)}^4}$, ${\displaystyle u_4(x,t)=(t^3+t)\left(x+\frac{1}{2}\right)}$, ${\displaystyle u_5(x,t)=(t^3+t)\left(e^{-x}-e^{-2}+\frac{5x}{4}\right)}$, ${\displaystyle \phi \left(t\right)=(t^3+t)\left(e^{-4}-e^{-2}+5\right)}$. The measurements (7) are obtained from the exact solution $u_5(x,t)$.

We estimate the errors and the convergence rate of the solution in each layer (as in Example 1) and also present the error in the $L_2$-norm for the entire domain $\overline{\mathrm{\Omega }}$:

$$\begin{array}{c}\hfill {\displaystyle \mathcal{E}={\mathcal{E}}^I=\sqrt{\sum _{i=0}^{I}h{\left(u(x_i,T)-U_i^N\right)}^2},\mathcal{CR}={log}_2\frac{{\mathcal{E}}^I}{{\mathcal{E}}^{2I}},}\end{array}$$

where $x=[x_{1,0},x_{1,1},\dots ,x_{1,I_1-1},\dots ,x_{m,0},\dots ,x_{m,I_m-1},\dots ,x_{M,0},x_{M,1},\dots ,x_{M,I_M-1},x_{M,I_M}]$ and $u=(u_1,u_2,\dots ,u_M)$, $U=(U_1,U_2,\dots ,U_M)$, constructed similarly by excluding the duplicate solution quantities at interface nodes.

In

Table 2. Errors and convergence rates of solutions $U_m$, $m=1,2,\dots ,5$ in maximum norm, Example 2.

I	h	${\mathit{E}}_1$	${\mathbf{CR}}_1$	${\mathit{E}}_2$	${\mathbf{CR}}_2$	${\mathit{E}}_3$	${\mathbf{CR}}_3$	${\mathit{E}}_4$	${\mathbf{CR}}_4$	${\mathit{E}}_5$	${\mathbf{CR}}_5$
120	0.05	1.353 $\times {10}^{-1}$		1.336 $\times {10}^{-1}$		2.272 $\times {10}^{-1}$		2.272 $\times {10}^{-1}$		3.041 $\times {10}^{-1}$
240	0.025	7.062 $\times {10}^{-2}$	0.939	7.011 $\times {10}^{-2}$	0.930	1.114 $\times {10}^{-1}$	1.028	1.114 $\times {10}^{-1}$	1.028	1.517 $\times {10}^{-1}$	1.003
480	0.0125	3.602 $\times {10}^{-2}$	0.971	3.584 $\times {10}^{-2}$	0.968	5.521 $\times {10}^{-2}$	1.013	5.521 $\times {10}^{-2}$	1.013	7.582 $\times {10}^{-2}$	1.001
960	0.00625	1.819 $\times {10}^{-2}$	0.986	1.811 $\times {10}^{-2}$	0.985	2.748 $\times {10}^{-2}$	1.007	2.748 $\times {10}^{-2}$	1.007	3.790 $\times {10}^{-2}$	1.000
1920	0.003125	9.137 $\times {10}^{-3}$	0.993	9.102 $\times {10}^{-3}$	0.993	1.371 $\times {10}^{-2}$	1.003	1.371 $\times {10}^{-2}$	1.003	1.895 $\times {10}^{-2}$	1.000
3840	0.0015625	4.575 $\times {10}^{-3}$	0.998	4.557 $\times {10}^{-3}$	0.998	6.855 $\times {10}^{-3}$	1.000	6.855 $\times {10}^{-3}$	1.000	9.475 $\times {10}^{-3}$	1.000

, we give the errors and the convergence rate in the maximum norm, while in

Table 3. Errors and convergence rates of solutions $U_m$, $m=1,2,\dots ,5$ and U in $L_2$ norm, Example 2.

I	h	${\mathit{E}}_1$	${\mathbf{CR}}_1$	${\mathcal{E}}_1$	${\mathcal{CR}}_1$	${\mathit{E}}_2$	${\mathbf{CR}}_{2,}$	${\mathcal{E}}_2$	${\mathcal{CR}}_2$	${\mathit{E}}_3$	${\mathbf{CR}}_{3,}$	${\mathcal{E}}_3$	${\mathcal{CR}}_3$
120	0.05	2.540 $\times {10}^{-1}$		1.169 $\times {10}^{-1}$		5.552 $\times {10}^{-2}$		1.054 $\times {10}^{-1}$		9.398 $\times {10}^{-2}$		1.822 $\times {10}^{-1}$
240	0.025	1.250 $\times {10}^{-1}$	1.023	5.991 $\times {10}^{-2}$	1.023	2.587 $\times {10}^{-2}$	1.102	4.798 $\times {10}^{-2}$	1.135	4.313 $\times {10}^{-2}$	1.124	8.858 $\times {10}^{-2}$	1.040
480	0.0125	6.211 $\times {10}^{-2}$	1.009	3.029 $\times {10}^{-2}$	1.009	1.259 $\times {10}^{-2}$	1.039	2.293 $\times {10}^{-2}$	1.065	2.063 $\times {10}^{-2}$	1.064	4.369 $\times {10}^{-2}$	1.020
960	0.00625	3.097 $\times {10}^{-2}$	1.004	1.523 $\times {10}^{-2}$	1.004	6.226 $\times {10}^{-3}$	1.016	1.121 $\times {10}^{-2}$	1.032	1.008 $\times {10}^{-2}$	1.033	2.170 $\times {10}^{-2}$	1.010
1920	0.003125	1.546 $\times {10}^{-2}$	1.002	7.634 $\times {10}^{-3}$	1.002	3.097 $\times {10}^{-3}$	1.008	5.543 $\times {10}^{-3}$	1.016	4.984 $\times {10}^{-3}$	1.017	1.081 $\times {10}^{-3}$	1.004
3840	0.0015625	7.725 $\times {10}^{-3}$	1.002	3.814 $\times {10}^{-3}$	1.001	1.544 $\times {10}^{-3}$	1.004	2.756 $\times {10}^{-3}$	1.008	2.477 $\times {10}^{-3}$	1.009	5.401 $\times {10}^{-3}$	1.001

, we show the results computed in the $L_2$-norm. The order of convergence of the restored solution in all domains ${\overline{\mathrm{\Omega }}}_m$, $m=1,2,3$ is observed to be first. As in the Example 1, since the solution at interface node $l_m$ belongs to both subdomains ${\overline{\mathrm{\Omega }}}_m$ and ${\overline{\mathrm{\Omega }}}_{m+1}$, the maximum errors of the solutions $U_m$ and $U_{m+1}$, $m=1,2,3$ are equal, and the largest error occurs at the interfaces or at the boundary $x=b$. In the present example, the maximum of the solution occurs at the boundary $x=b$, which explains the largest error at that point. This is also illustrated in Figure 2, where the numerical solution U, together with the corresponding final-time error, are shown.

On the base of the computational results, we conclude that the accuracy of the proposed numerical method does not depend on the number of layers M.

Example 3 (Noisy data, $M=5$). We evaluate the performance of the methods for perturbed observations:

$${\psi }^{\rho }\left(t_n\right)=\psi \left(t_n\right)+2\rho (\sigma \left(t_n\right)-0,5),$$

where ρ is the noise level, $\sigma \left(t_n\right)$ is a random function uniformly distributed on the interval $[0,1]$, and $\psi \left(t_n\right)$ is the exact observation, obtained from the exact solution. Degree-5 polynomial curve fitting is applied to smooth the measured data.

The test problem is the one from Example 2. In

Table 4. Errors of the solution U for different noise levels, Example 3.

$\mathit{\rho }$	E	$\mathcal{E}$
0.002	3.341 $\times {10}^{-1}$	2.744 $\times {10}^{-1}$
0.01	3.526 $\times {10}^{-1}$	3.942 $\times {10}^{-1}$
0.05	3.872 $\times {10}^{-1}$	4.395 $\times {10}^{-1}$
0.1	4.611 $\times {10}^{-1}$	5.073 $\times {10}^{-1}$

, we present the errors of the numerical solution in the whole spatial domain for $I=120$ at the final time and for different noise levels. The errors are estimated in the $L_2$-norm and maximal discrete norm $E=\underset{0\le i\le I}{max}|u(x_i,T)-U_i^N|$. We observe that for $\rho =0.002$, the results are close to those obtained with exact measurements (Table 2 and Table 3). Note that the relative maximum error is approximately $0.0341$, which is satisfactory for a spatial step size of $0.05$. When the noise level increases, the error also increases but remains within an optimal range.

In Figure 3, the exact and numerically reconstructed function $\phi$ are displayed for noise levels $\rho =0.03$ and $\rho =0.1$. It can be seen that the function $\phi$, recovered through the inverse problem, closely matches the exact function. For $\rho =0.1$, a noticeably larger error also appears at grid nodes near the initial time $t=0$ compared to the case with $\rho =0.03$.

Example 4. (Noisy data, nonsmooth $\phi \left(t\right)$, $M=5$). We use the same model parameters in problem (1)–(7) as in Example 2 but define the following:

$$\phi \left(t\right)=\left\{\begin{array}{cc}{\displaystyle (t^3+t)(e^{-4}-e^{-2}+5),}\hfill & t\le 0.5,\hfill \\ {\displaystyle \frac{5}{8}(0.5+t)(e^{-4}-e^{-2}+5),}\hfill & t>0.5,\hfill \end{array}\right.$$

In this case, on the time interval $[0,0.5]$, the exact solution coincides with that in Example 2. For $t\in [0.5,1]$, the exact solution is constructed analogously, replacing the function $t^3+t$ (and its derivative) with $(0.5+t)(t^3+t)$. The initial condition, the boundary conditions, and the right-hand side in (1)–(7) are then derived accordingly, based on the exact solution.

In Figure 4, we illustrate that the recovered function $\phi$ fits the exact function well. Computations are performed for $\rho =0.3$, $\rho =0.1$, and $I=120$. In Figure 5, we plot the numerical solution U and the error $|u-U|$ over the entire computational domain. It can be seen that the recovery is achieved with satisfactory accuracy.

6. Conclusions

In this work, we constructed an unconditionally stable and fast numerical scheme for addressing the inverse problem of determining Dirichlet boundary conditions in a parabolic equation with multiple interfaces. By determining the solution at the interfaces, the problem is decomposed and solved numerically in each layer sequentially and independently. Moreover, the discretization is implemented in an explicit manner, requiring no matrix inversion.

By determining the solution at the interfaces, the problem is decomposed and solved numerically in each layer sequentially and independently. Moreover, the discretization is implemented explicitly, requiring no matrix inversion.

Numerical results with exact measurements showed that the order of convergence of the inverse problem solution matches that of the underlying one-sided Saulyev discretization used for the direct problem, namely $O(\tau +h+\tau /h)$. Results with noisy observations demonstrated that the boundary condition can be recovered with satisfactory precision, allowing the numerical solution to achieve optimal accuracy.

The method is efficient for multilayer problems, and its accuracy does not depend on the number of layers, provided that the parameter ratios are moderate.

A natural extension of this work is the study of two-dimensional direct and inverse problems to develop numerical methods that identify the interfaces. It is also interesting to consider problems with Neumann or mixed boundary conditions.