Source Identification for a Two-Dimensional Parabolic Equation with an Integral Constraint

Abstract

We consider a two-dimensional parabolic problem subject to both Neumann and Dirichlet boundary conditions, along with an integral constraint. Based on the integral observation, we solve the inverse problem of a recovering time-dependent right-hand side. By exploiting the structure of the boundary conditions, we reduce the original inverse problem to a one-dimensional formulation. We conduct a detailed analysis of the existence and uniqueness of the solution to the resulting one-dimensional loaded initial-boundary value problem. Furthermore, we derive estimates for both the solution and the unknown function. The direct and inverse problems are numerically solved by finite difference schemes. Numerical verification of the theoretical results is provided.

1. Introduction

Parabolic partial differential equations (PDEs) are widely used to model a variety of phenomena in science and engineering. This study focuses on a class of non-classical parabolic initial-boundary value problems that incorporate an integral condition for the solution over the spatial domain. Such problems provide accurate representations of processes such as heat and mass transfer, chemical diffusion, agricultural engineering, population dynamics, biochemistry, thermoelasticity, and atmospheric pollution (see, for example, [1,2,3,4,5,6,7]).

In the mathematical modeling of these processes, it is common to encounter optimization or inverse problems constrained by PDEs, where additional constraints are imposed on the state variables. These may include integral or pointwise measurements (see, for example, [2,3,8,9,10,11]). Integral constraints in particular arise naturally in applications where only spatially averaged or cumulative data can be obtained, such as the total heat, pollutant concentration, or moisture content over a region, making them more practical and relevant in real experimental and industrial contexts.

It should be noted that the integral constraint is called the isoperimetric constraint in optimal control theory (see [12,13,14]). These constraints, which may be either equalities or inequalities, are used when the state or control variables must satisfy integral rather than pointwise conditions, and they arise naturally in many optimal control formulations [12]. Applications appear in various problems involving fractional-order delay and multidimensional PDEs (see [13,14]). These studies introduced Riemann–Liouville isoperimetric constraints in fractional optimal control problems with Caputo derivatives, which were efficiently solved using quadratic or linear programming.

Our interest lies in a two-dimensional parabolic inverse problem that frequently arises in applied contexts, where both the solution and the source term are unknown. We briefly outline the nature of inverse problems and compare them with direct problems. In direct problems, the goal is typically to determine exact or approximate quantities such as temperature distribution, sound propagation, or wave motion, assuming that the media properties, initial state of the process, and its properties on the boundary are fully known. These problems are governed by differential equations with specified parameters. A classical reference for the theory and applications of direct parabolic equations is the monograph [15].

In many practical situations, the properties of the medium are not directly observable, which necessitates the formulation of inverse problems to compensate for this lack of information in mathematical models. Foundational results and techniques for inverse problems in partial differential equations can be found in [7,11,16,17,18,19,20]. In particular, inverse source problems based on final-time observations, boundary measurements, and related data have been extensively studied (see, for instance, [21,22,23]).

In [10], the author investigated the inverse source problem and inverse diffusion coefficient problem for parabolic equations with applications in geology. He developed and analyzed analytical and numerical methods for recovering unknown coefficients or source terms, proved the existence, uniqueness, and stability of the solutions, and validated the results through finite element simulations. The authors of [24] developed a semigroup-based method for inverse source problems in the heat equation, provided a solution representation that revealed non-uniqueness, and offered conditions for uniqueness based on final time or pointwise measurements. The work in [25] addressed inverse moving source problems for parabolic equations and proved the uniqueness in recovering either the source’s trajectory or profile from final time data using Fourier methods and complex analysis. The authors of [26] presented a numerical solution to an inverse source problem under point observations for a time-fractional diffusion-reaction equation on disjoint intervals with different Caputo fractional orders, using a decomposition algorithm on an adaptive time mesh.

The solvability of an inverse problem involving the determination of a time-dependent source term, subject to an additional condition given by a spatial integral observation, was discussed in [27] for a degenerate parabolic equation. A similar problem was considered in [28] for a higher-order parabolic equation defined on a plane. Furthermore, the inverse problem of recovering the source function in a non-uniformly parabolic equation with multiple independent variables in a bounded domain was studied in [29], where an integral observation condition was also imposed. The existence and uniqueness results for the solution of the inverse problem of the simultaneous determination of the right-hand side and the lowest coefficient in multidimensional parabolic equations with integral observation were obtained in [30].

The coefficient inverse problem for the two-dimensional heat equation with integral overdetermination was investigated in [31]. The authors proved the existence and uniqueness of the solution by applying the Schauder fixed-point theorem.

Inverse boundary source problems under integral solution measurements were investigated in [8]. The authors established the existence and uniqueness of the solution to the inverse problem for the two-dimensional heat equation and proposed a numerical approach. The authors of [32] investigated an inverse problem in a 1D magnetohydrodynamic flow system, where a time-dependent convection coefficient and source were determined using two integral observations, transforming the problem into a non-classical direct one analyzed via a Galerkin finite element method, with proofs of existence, uniqueness, and numerical validation.

In [33], the authors studied the inverse problem of determining a time-dependent source in a Dirichlet boundary condition for a two-dimensional parabolic problem, based on an integral solution observation of the solution. The main idea was to use the integral condition to reduce the two-dimensional inverse problem to a one-dimensional direct (forward) problem with a Dirichlet and a non-local condition. A second-order finite difference method was developed in [34] to solve a two-dimensional boundary source identification problem for the heat equation by reducing it to two simpler one-dimensional problems, and its stability and accuracy were analyzed through numerical experiments.

Numerical methods based on reducing the two-dimensional inverse boundary source problem to a one-dimensional formulation for a two-dimensional pseudoparabolic equation and heat equation on disjoint domains were developed in [35,36].

A numerical method for solving non-local problems with an integral condition was developed in [37]. The author presented a second-order finite difference scheme for solving a two-dimensional non-local heat equation problem by transforming it into two simpler problems, followed by analyzing the method’s stability and error and validating it through numerical experiments.

This paper focuses on establishing stability estimates for the determination of a time-dependent source in a two-dimensional parabolic equation. Inverse problems with integral observations of the solution were previously studied in [33,34], where the main approach involves reducing the two-dimensional inverse problem to a one-dimensional direct problem. Our aim is to address the theoretical gap concerning existence, uniqueness, and stability estimates for both the solutions and unknown sources.

The remaining part of this paper is organized as follows. In the next section, we formulate the inverse problems. In Section 3, the reduction of the two-dimensional problem to its equivalent direct (forward) problems is described. The well posedness of the problem (i.e., the existence, uniqueness, and continuous dependence on input data of the solution to the inverse problem) is studied in Section 4. A priory estimates of the solution to the inverse problem are derived in Section 5. A numerical method is presented in Section 6, and validation of the theoretical results is demonstrated by test examples in Section 7. This paper is finalized by some conclusions.

2. Set-Up of the Inverse Problem

We consider the heat condition equation

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

(1)

with the initial condition

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

(2)

and boundary conditions

$${\displaystyle \frac{\partial u(0,y,t)}{\partial x}}=g_0(y,t),0<t\le T,0\le y\le 1,$$

(3)

$${\displaystyle \frac{\partial u(1,y,t)}{\partial x}}=g_1(y,t),0<t\le T,0\le y\le 1,$$

(4)

$$u(x,0,t)={\psi }_0(x,t),0<t\le T,0\le x\le 1,$$

(5)

$$u(x,1,t)={\psi }_1(x,t),0<t\le T,0\le x\le 1,$$

(6)

as well as the integral measurements

$$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}u(x,y,t)dxdy=m\left(t\right),$$

(7)

where f, $p\left(t\right)\le 0$, $u_0$, $g_0$, $g_1$, ${\psi }_0$, ${\psi }_1$, and m are known functions, while the temperature u and the source intensity $q\left(t\right)$ are unknown.

When we use the classical solution, the following compatibility conditions are assumed:

$$\begin{array}{ccc}& & {\psi }_0(x,0)=u_0(x,0),{\psi }_1(x,0)=u_0(x,1),g_0(y,0)={\displaystyle \frac{\partial u_0}{\partial x}}(0,y),g_1(y,0)={\displaystyle \frac{\partial u_0}{\partial x}}(1,y),\hfill \\ & & {\displaystyle \frac{\partial {\psi }_0}{\partial x}(0,t)=g_0(0,t),\frac{\partial {\psi }_0}{\partial x}(1,t)=g_1(0,t),\frac{\partial {\psi }_1}{\partial x}(0,t)=g_0(1,t),\frac{\partial {\psi }_1}{\partial x}(1,t)=g_1(1,t).}\hfill \end{array}$$

In [38], instead of the boundary condition in Equation (4), the authors considered

$${\displaystyle \frac{\partial u(1,y,t)}{\partial x}}=g_1(y,t)p\left(t\right),$$

and instead of Equations (5) and (6), Neumann boundary conditions were imposed. The boundary source function $p\left(t\right)$ is unknown and is determined using the integral specification in Equation (7). The existence and uniqueness of a weak solution to this inverse problem are established via the Rothe method.

3. Reducing the Two-Dimensional Problem to Be One-Dimensional

In this section, we reformulate the inverse two-dimensional problem om Equations (1)–(7) to its equivalent one-dimensional inverse problem.

We introduce the function [33,34]

$$v(y,t)=\underset{0}{\overset{1}{\int }}u(x,y,t)dx,0\le y\le 1,$$

(8)

to obtain from Equations (1)–(7) the following initial boundary value problem:

$${\displaystyle \frac{\partial v}{\partial t}}={\displaystyle \frac{{\partial }^{2}v}{\partial y^2}}+p\left(t\right)v+q\left(t\right)+G(y,t),0<y<1,0<t\le T,$$

(9)

with the initial condition

$$v(y,0)=v_0\left(y\right),0\le y\le 1,$$

(10)

as well as the boundary condition

$$v(0,t)={\phi }_0\left(t\right),0<t\le T,$$

(11)

$$v(1,t)={\phi }_1\left(t\right),0<t\le T,$$

(12)

and the integral constraint, corresponding to Equation (7):

$$\underset{0}{\overset{1}{\int }}v(y,t)dy=m\left(t\right),0<t\le T,$$

(13)

where

$$\begin{array}{cc}\hfill & {\displaystyle G(y,t)=f_0(y,t)+g_1(y,t)-g_0(y,t),v_0\left(y\right)=\underset{0}{\overset{1}{\int }}{u}_0(x,y)dx,}\\ & {\displaystyle f_0(y,t)=\underset{0}{\overset{1}{\int }}f(x,y,t)dx,{\phi }_0\left(t\right)=\underset{0}{\overset{1}{\int }}{\psi }_0(x,t)dx,{\phi }_1\left(t\right)=\underset{0}{\overset{1}{\int }}{\psi }_1(x,t)dx.}\end{array}$$

In the problem in Equations (9)–(13), the functions $v_0$, ${\phi }_0$, ${\phi }_1$, $f_0$, $p\left(t\right)$, and G are known, while the functions $v(y,t)$ and $q\left(t\right)$ are to be determined.

For theoretical investigations, it is more convenient to reformulate Equations (9)–(13) as one with zero Dirichlet boundary conditions and homogeneous measurement data. To this end, we apply the following substitution in the original problem in Equations (9)–(13):

$$v(y,t)=w(y,t)+{\mathsf{\Psi }}_0\left(t\right)y^2+{\mathsf{\Psi }}_1\left(t\right)y+{\mathsf{\Psi }}_2\left(t\right),$$

(14)

where

$$\begin{array}{ccc}& & {\mathsf{\Psi }}_0\left(t\right)=-6m\left(t\right)+3({\phi }_1\left(t\right)+{\phi }_0\left(t\right)),\hfill \\ & & {\mathsf{\Psi }}_1\left(t\right)=6m\left(t\right)-2{\phi }_1\left(t\right)-4{\phi }_0\left(t\right),\hfill \\ & & {\mathsf{\Psi }}_2\left(t\right)={\phi }_0\left(t\right).\hfill \end{array}$$

Then, the unknown functions $w(y,t)$ and $q\left(t\right)$ are the solution to the problem

$$\begin{array}{ccc}& & {\displaystyle \frac{\partial w}{\partial t}}={\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}+p\left(t\right)w+R(y,t)+q\left(t\right),\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & w(y,0)=v_0\left(y\right)-{\mathsf{\Psi }}_0\left(0\right)y^2-{\mathsf{\Psi }}_1\left(0\right)y-{\mathsf{\Psi }}_2\left(0\right),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & w(0,t)=w(1,t)=0,0\le y\le 1,\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& & \underset{0}{\overset{1}{\int }}w(y,t)dy=0,t\in [0,T],\hfill \end{array}$$

(18)

where

$$R(y,t)=G(y,t)+2{\mathsf{\Psi }}_0\left(t\right)-{\mathsf{\Psi }}_0^{\prime }\left(t\right)y^2-{\mathsf{\Psi }}_1^{\prime }\left(t\right)y-{\mathsf{\Psi }}_0^{\prime }\left(t\right)+p\left(t\right)\left({\mathsf{\Psi }}_0\left(t\right)y^2-{\mathsf{\Psi }}_1\left(t\right)y-{\mathsf{\Psi }}_2\left(t\right)\right).$$

We integrate Equation (15) with respect to y from 0 to 1 and use the condition in Equation (18) to obtain

$$q\left(t\right)={\displaystyle \frac{\partial w}{\partial y}}(0,t)-{\displaystyle \frac{\partial w}{\partial y}}(1,t)-\underset{0}{\overset{1}{\int }}R(y,t)dy.$$

(19)

4. Well Posedness of the Loaded Equation’s Direct Problem

In this section, we study the well posedness of the inverse problem in Equations (15)–(19) (i.e., the existence, uniqueness, and continuous dependence of the solution for the input data). The approach involves substituting the expression for $q\left(t\right)$ from Equation (19) into the right-hand side of Equation (15), leading to a modified loaded equation for the unknown function

$${\displaystyle \frac{\partial w}{\partial t}}-{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}-p\left(t\right)w-{\displaystyle \frac{\partial w}{\partial y}}(0,t)+{\displaystyle \frac{\partial w}{\partial y}}(1,t)=R(y,t)-\underset{0}{\overset{1}{\int }}R(y,t)dy.$$

(20)

We define the space $C^k(0,1)$ as the set of functions having continuous derivatives up to the order k on $(0,1)$, and $C^k[0,1]=\{u\in C^k(0,1):{\displaystyle \frac{d^{\alpha }u}{d{y}^{\alpha }}}$ has a continuous extension on [0,1] for $\left.\alpha =1,2,\dots ,k\right\}$.

If H is a real vector space, then a mapping $(·,·):H\times H\to R$ is an inner product on H, with the associated norm defined by $\parallel u\parallel :=\sqrt{(u,u)}$.

Furthermore, it is necessary to replace the classical function space $C^k[0,1]$ with the Sobolev spaces $W^{k,2}(0,1)(=H^2(0,1))$, which consist of all functions $u\in L^2(0,1)$ that possess (weak) derivatives up to the order k. For more details, see, for example, [15,39].

In the estimates below, the $\epsilon$ inequality and the following inequalities are used (see, for example, [15,39]):

$$\begin{array}{ccc}& & ab\le 2a^2+{\displaystyle \frac{b}{4\epsilon }},\epsilon >0,\hfill \\ & & {\varphi }^2\left(0\right)\le 2\epsilon {∥{\displaystyle \frac{d\varphi }{dy}}∥}_0^2+{\displaystyle \frac{2}{\epsilon }}{\parallel \varphi \parallel }_0^2,\epsilon >0,\hfill \\ & & {\varphi }^2\left(1\right)\le 2\epsilon {∥{\displaystyle \frac{d\varphi }{dy}}∥}_0^2+{\displaystyle \frac{2}{\epsilon }}{\parallel \varphi \parallel }_0^2,\epsilon >0\hfill \end{array}$$

where $\varphi =\varphi \left(x\right)$, $x\in [0,1]$, $\varphi \in C^1[0,1]$, ${\parallel \varphi \parallel }_0^2={\int }_0^1{\varphi }^2\left(x\right)dx$.

We begin with the following assertions.

Proposition 1.

If ${\phi }_0\left(t\right)$, $\phi \left(t\right)$, and $m\left(t\right)$ belong to the space $C^1[0,T]$, and $G(y,t)\in C\left((0,1)\times [0,T]\right)$, then the problem in Equations (9)–(13) is equivalent to the problem in Equations (14)–(18), and the function $q\left(t\right)$ is determined (if $w(x,t)$ is already known) from the equality in Equation (19).

Proof. 

Indeed, if the pair $\{v,q\}$ is a solution to the problem in Equations (9)–(13), then by the construction outlined above, the pair $\{w,q\}$ solves the problem in Equations (15)–(18). Conversely, assuming that $\{w,q\}$ is a solution to the problem in Equations (15)–(18), it is straightforward to verify that the function $v(y,t)$, defined by Equation (14), satisfies the problem in Equations (9)–(13). □

We now state the following theorem concerning the well posedness of the inverse problem in Equations (15)–(19).

Theorem 1.

Let the conditions of Proposition 1 be satisfied and $w_0\left(y\right)\in C^1(0,1)$, $p\left(t\right)\in C(0,T)$. Then, there exists a unique global classical solution of the inverse problem in Equations (15)–(19), for which the stability estimate in Equation (31) below holds.

Proof. 

Assume that a solution to Equation (20) exists, subject to the initial condition in Equation (16) and the boundary conditions in Equation (17).

We multiply Equation (20) by ${\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}$ and integrate over the interval $[0,1]$ to obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left({\displaystyle \frac{\partial w}{\partial t}},{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\right)-\left({\displaystyle \frac{{\partial }^{2}w}{\partial y^2}},{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\right)-p\left(t\right)\left(w,{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\right)-{\displaystyle \frac{\partial w}{\partial y}}(0,t)\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy\hfill \\ & +{\displaystyle \frac{\partial w}{\partial y}}(1,t)\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy=\underset{0}{\overset{1}{\int }}R(y,t){\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy-\underset{0}{\overset{1}{\int }}R(y,t)dy\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy.\hfill \end{array}\end{array}$$

(21)

Using the boundary conditions in Equation (17) and the $\epsilon$ inequality, for each term in Equation (21), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left({\displaystyle \frac{\partial w}{\partial t}},{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\right)={\displaystyle \frac{\partial w}{\partial t}}{\displaystyle \frac{\partial w}{\partial y}}{|}_0^1-\underset{0}{\overset{1}{\int }}{\displaystyle \frac{\partial w}{\partial y}}{\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{\partial w}{\partial t}}\right)dy& =-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial }{\partial t}}\underset{0}{\overset{1}{\int }}{\left({\displaystyle \frac{\partial w}{\partial y}}\right)}^{2}dy\hfill \\ & =-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial }{\partial t}}{∥{\displaystyle \frac{\partial w}{\partial y}}∥}_0^2,\hfill \end{array}\end{array}$$

(22)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill p\left(t\right)\underset{0}{\overset{1}{\int }}w{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy=p\left(t\right)\left(w{\displaystyle \frac{\partial w}{\partial y}}{|}_0^1-\underset{0}{\overset{1}{\int }}{\left({\displaystyle \frac{\partial w}{\partial y}}\right)}^{2}dy\right)=-p\left(t\right){∥{\displaystyle \frac{\partial w}{\partial y}}∥}_0^2\end{array}\end{array}$$

(23)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \frac{\partial w}{\partial y}}(0,t)\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy\le {\displaystyle \frac{1}{4{\epsilon }_1}}{\left({\displaystyle \frac{\partial w}{\partial y}}(0,t)\right)}^2+{\epsilon }_1{∥{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}∥}_0^2,\end{array}\end{array}$$

(24)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \frac{\partial w}{\partial y}}(1,t)\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy\le {\displaystyle \frac{1}{4{\epsilon }_2}}{\left({\displaystyle \frac{\partial w}{\partial y}}(1,t)\right)}^2+{\epsilon }_2{∥{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}∥}_0^2,\end{array}\end{array}$$

(25)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \underset{0}{\overset{1}{\int }}R{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy\le {\displaystyle \frac{1}{4{\epsilon }_3}}{\parallel R\parallel }_0^2+{\epsilon }_3{∥{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}∥}_0^2,\end{array}\end{array}$$

(26)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \underset{0}{\overset{1}{\int }}R(y,t)dy\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy\le {\displaystyle \frac{1}{4{\epsilon }_4}}\left|\underset{0}{\overset{1}{\int }}R(y,t)dy\right|+{\epsilon }_4{∥{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}∥}_0^2.\end{array}\end{array}$$

(27)

The application of the trace inequality to Equations (24) and (25) implies

$$\begin{array}{ccc}& & {\displaystyle \frac{\partial w}{\partial y}}(0,t)\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy\le \left({\displaystyle \frac{{\epsilon }_5}{2{\epsilon }_1}}+{\epsilon }_1\right){∥{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}∥}_0^2+{\displaystyle \frac{1}{2{\epsilon }_1{\epsilon }_5}}{∥{\displaystyle \frac{\partial w}{\partial y}}∥}_0^2,\hfill \end{array}$$

(28)

$$\begin{array}{ccc}& & {\displaystyle \frac{\partial w}{\partial y}}(1,t)\underset{0}{\overset{1}{\int }}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}dy\le \left({\displaystyle \frac{{\epsilon }_6}{2{\epsilon }_2}}+{\epsilon }_2\right){∥{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}∥}_0^2+{\displaystyle \frac{1}{2{\epsilon }_2{\epsilon }_6}}{∥{\displaystyle \frac{\partial w}{\partial y}}∥}_0^2.\hfill \end{array}$$

(29)

Now, let us insert all of the estimates in Equations (22), (23) and (26)–(29) into Equation (21) to derive

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial t}}{∥{\displaystyle \frac{\partial w}{\partial y}}∥}_0^2+{∥{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}∥}_0^2\le & \left(p\left(t\right)+{\displaystyle \frac{1}{2{\epsilon }_1{\epsilon }_5}}+{\displaystyle \frac{1}{2{\epsilon }_2{\epsilon }_6}}\right){∥{\displaystyle \frac{\partial w}{\partial y}}∥}_0^2\hfill \\ & +\left({\epsilon }_3+{\epsilon }_4+{\displaystyle \frac{{\epsilon }_5}{2{\epsilon }_1}}+{\epsilon }_1+{\epsilon }_2+{\displaystyle \frac{{\epsilon }_6}{2{\epsilon }_2}}\right){∥{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}∥}_0^2\hfill \\ & +{\displaystyle \frac{1}{4}}\left({\displaystyle \frac{1}{{\epsilon }_3}}{\parallel R\parallel }_0^2+{\displaystyle \frac{1}{{\epsilon }_4}}{\left(\underset{0}{\overset{1}{\int }}R(y,t)dy\right)}^2\right).\hfill \end{array}\end{array}$$

(30)

By applying a Cauchy inequality to the last term in Equation (30), we rewrite the inequality in the form

$${\displaystyle \frac{\partial }{\partial t}}{∥{\displaystyle \frac{\partial w}{\partial y}}∥}_0^2+C{∥{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}∥}_0^2\le \tilde{p}\left(t\right){∥{\displaystyle \frac{\partial w}{\partial y}}∥}_0^2+{\displaystyle \frac{1}{4}}\left({\displaystyle \frac{1}{{\epsilon }_2}}+{\displaystyle \frac{1}{{\epsilon }_4}}\right){\parallel R\parallel }_0^2,$$

where $\tilde{p}\left(t\right)=p\left(t\right)+1/\left(2{\epsilon }_1{\epsilon }_5\right)+1/\left(2{\epsilon }_2{\epsilon }_6\right)$, $C=1-\widehat{\epsilon }$, and $\widehat{\epsilon }$ is the coefficient of ${\displaystyle {∥\frac{{\partial }^{2}w}{\partial y^2}∥}_0^2}$ on the right-hand side in Equation (30). Then, by choosing $C=1-\widehat{\epsilon }>0$, we find the inequality for ${\displaystyle W\left(t\right)={∥\frac{\partial w}{\partial y^2}∥}_0}$:

$${\displaystyle \frac{d}{dt}}W\left(t\right)-\tilde{p}\left(t\right)W\left(t\right)\le {\displaystyle \frac{1}{4}}\left({\displaystyle \frac{1}{{\epsilon }_2}}+{\displaystyle \frac{1}{{\epsilon }_4}}\right){\parallel R_0\left(t\right)\parallel }_0^2,W\left(0\right)={∥{\displaystyle \frac{\partial W_0}{\partial y}}∥}^2.$$

By solving this inequality, one can find explicitly the function $C\left(t\right)$ in Equation (31).

Next, by choosing the constants ${\epsilon }_i$, $i=1,2,\dots ,5$ appropriately, we integrate the inequality in Equation (30) from 0 to $t\le T$ and then apply Gronwall’s inequality to the result, obtaining

$${∥{\displaystyle \frac{\partial w}{\partial y}}∥}_0^2+C{∥{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}∥}_{2,Q_t}^2\le C\left(t\right){\parallel R\parallel }_{2,Q_T}^2,$$

(31)

where for a function $z(y,t)\in L^2\left(Q_T\right)$, we have

$${\parallel z\parallel }_0^2=\underset{0}{\overset{1}{\int }}{z}^2(x,t)dx,{\parallel z\parallel }_{2,Q_T}^2=\underset{Q_T}{\int }{z}^2(x,t)dx.$$

From this, the uniqueness and continuous dependence of the solution to the inverse problem in Equations (15)–(19) on the input data follow.

The local existence of a weak solution to Equation (20), with the initial condition in Equation (16) and boundary conditions in Equation (17), follows from the general theory of parabolic equations (see, for example, [15,39]). The existence of a global weak solution is a direct consequence of the a priori estimate in Equation (31). Furthermore, using the smoothness of the input data, it follows that (see [15,39]) the weak solution is a classical. More detailed estimates for the classical solution and the unknown source term are presented in the next section. □

5. A Priory Estimates of the Solution {w,q} to the Inverse Problem

In this section, we discuss the uniqueness and some properties of the solution to the problem in Equations (15)–(19).

Lemma 1.

The solution to the problem in Equations (15)–(19) is unique.

Proof. 

We multiply Equation (15) by $w(y,t)$ and then integrate the result, using the boundary condition in Equation (17):

$${\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{1}{2}}\underset{0}{\overset{1}{\int }}{w}^2(y,t)dy\right)+\underset{0}{\overset{1}{\int }}{\left({\displaystyle \frac{\partial w}{\partial y}}\right)}^{2}dy=\underset{0}{\overset{1}{\int }}R(y,t)w(y,t)dy.$$

(32)

Now, from the Cauchy inequality, we have

$$\underset{0}{\overset{1}{\int }}R(y,t)w(y,t)dy\le {\left(\underset{0}{\overset{1}{\int }}{R}^2(y,t)dy\right)}^{1/2}{\left(\underset{0}{\overset{1}{\int }}{w}^2(y,t)dy\right)}^{1/2}.$$

Then, using the Friedrichs inequality

$$\underset{0}{\overset{1}{\int }}{w}^2(y,t)dy\le {\displaystyle \frac{1}{{\pi }^2}}\underset{0}{\overset{1}{\int }}{\left({\displaystyle \frac{\partial w}{\partial y}}\right)}^{2}dy,$$

we obtain from Equation (32) the following:

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{dz}{dt}}+{\pi }^{2}z-r\left(t\right)\sqrt{z}\le 0,$$

(33)

where

$$z\left(t\right)=\underset{0}{\overset{1}{\int }}{w}^2(y,t)dy,r\left(t\right)={\left(\underset{0}{\overset{1}{\int }}{R}^2(y,t)dy\right)}^{1/2}.$$

By solving the inequality in Equation (33), we find the estimate:

$$\underset{0}{\overset{1}{\int }}{w}^2(y,t)dy\le {\left[{\left(\underset{0}{\overset{1}{\int }}{w}_0^2\left(y\right)dy\right)}^{1/2}+\underset{0}{\overset{t}{\int }}{e}^{{\displaystyle \frac{{\pi }^2\tau }{2}}}r\left(\tau \right)d\tau \right]}^2{e}^{-{\pi }^{2}t}.$$

(34)

From this inequality, the uniqueness of the solution to the problem in Equations (15)–(18) and the uniqueness of $q\left(t\right)$ from Equation (19) follow. □

Furthermore, we derive an estimate for $\left|w\right(x,t\left)\right|$ which is uniform with respect to $x\in (0,1)$ and $t\in [0,T]$.

We multiply Equation (15) by $w_t$ and then integrate the result over the interval $(0,1)$ to find

$$\underset{0}{\overset{1}{\int }}{w}^2(y,t)dy+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial }{\partial t}}\underset{0}{\overset{1}{\int }}{\left({\displaystyle \frac{\partial w}{\partial y}}\right)}^{2}dy=-\underset{0}{\overset{1}{\int }}R(y,t){\displaystyle \frac{\partial w}{\partial t}}(y,t)dt.$$

From here, we obtain

$$\underset{0}{\overset{1}{\int }}{\left({\displaystyle \frac{\partial w}{\partial y}}\right)}^{2}dy\le \underset{0}{\overset{1}{\int }}{\left({\displaystyle \frac{\partial w_0}{\partial y}}\right)}^{2}dy+\underset{0}{\overset{t}{\int }}{r}^2\left(t\right)dt.$$

(35)

Now, using Equations (34) and (35), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle w^2(y,t)=}& {\displaystyle 2\underset{0}{\overset{y}{\int }}w\frac{\partial w}{\partial y}dy\le 2{\left(\underset{0}{\overset{1}{\int }}{w}^{2}dy\right)}^{1/2}{\left(\underset{0}{\overset{1}{\int }}{\left(\frac{\partial w}{\partial y}\right)}^{2}dy\right)}^{1/2}}\hfill \\ \hfill \le & {\displaystyle 2\left[{\left(-\underset{0}{\overset{1}{\int }}{w}_0^2\left(y\right)dy\right)}^{1/2}+\underset{0}{\overset{t}{\int }}{e}^{\frac{{\pi }^2\tau }{4}}r\left(t\right)d\tau \right)\times }\hfill \\ & {\displaystyle \times \left(\underset{0}{\overset{1}{\int }}{\left(\frac{\partial w_0}{\partial y}\right)}^2+\underset{0}{\overset{t}{\int }}{r}^2\left(\tau \right)d\tau \right)e^{-\frac{{\pi }^2\tau }{4}}={\tilde{r}}^2\left(t\right)e^{-\frac{{\pi }^2\tau }{4}}}\hfill \end{array}\end{array}$$

(36)

Therefore, Equation (36) implies that

$$\left|w(x,t)\right|\le \tilde{r}\left(t\right)e^{-{\displaystyle \frac{{\pi }^2\tau }{8}},}$$

(37)

for each $x\in [0,1]$ and $t\in [0,T]$. Thus, if the integral

$$\underset{0}{\overset{\infty }{\int }}{e}^{{\displaystyle \frac{{\pi }^2\tau }{2}}}r\left(t\right)d\tau ,$$

(38)

is convergent, then the function $\tilde{r}\left(t\right)$ is bounded for each $t\in [0,\infty )$, and $\left|w\right(x,t\left)\right|\to 0$ as $t\to \infty$ uniformly with respect to $x\in [-1,1]$:

$$w^2(y,t)=2\underset{0}{\overset{y}{\int }}w{\displaystyle \frac{\partial w}{\partial y}}dy\le 2{\left(\underset{0}{\overset{1}{\int }}{w}^{2}dy\right)}^{1/2}{\left(\underset{0}{\overset{1}{\int }}{\left({\displaystyle \frac{\partial w}{\partial y}}\right)}^{2}dy\right)}^{1/2}.$$

In order to obtain an estimate for the time derivative of the solution $w(y,t)$, we differentiate the problem in Equations (15)–(19) with respect to t. This results in an analogous problem for ${\displaystyle \frac{\partial w}{\partial t}}$, where $q\left(t\right)$ is replaced by $q^{\prime }\left(t\right)$, $R(y,t)$ by ${\displaystyle \frac{\partial R}{\partial y}}(y,t)$ and the initial data in Equation (16) are replaced by

$${\displaystyle \frac{\partial w}{\partial t}}(x,0)=w_0^{″}\left(y\right)+w_0^{\prime }\left(0\right)-w_0^{\prime }\left(1\right)-\underset{0}{\overset{1}{\int }}R(y,0)dy\equiv w_{10}\left(y\right).$$

(39)

such that the estimate in Equations (34) and (37) holds for ${\displaystyle \frac{\partial v}{\partial t}}(y,t)$, where instead of $r\left(t\right)$, we will stay with the function

$$r_1\left(t\right)={\left(\underset{0}{\overset{1}{\int }}{\displaystyle \frac{\partial R}{\partial y}}(y,t)dy\right)}^{1/2}$$

(40)

and $w_0\left(y\right)$ is the function $w_{10}\left(y\right)$ from Equation (39).

Therefore, we have

$$\left|{\displaystyle \frac{\partial w}{\partial t}}(y,t)\right|\le {\tilde{r}}_1\left(t\right)e^{-{\displaystyle \frac{{\pi }^{2}t}{8}}},$$

(41)

with the bounded function

$${\overline{r}}_1\left(t\right)=\left\{2\left[{\left(\underset{-1}{\overset{1}{\int }}{w}_{10}^2\left(y\right)dy\right)}^{1/2}+\underset{0}{\overset{t}{\int }}{e}^{{\displaystyle \frac{{\pi }^2\tau }{4}}}{r}_1\left(\tau \right)d\tau \right]{\left(\underset{0}{\overset{1}{\int }}{\left(w_{10}^{\prime }\right)}^{2}dy+\underset{0}{\overset{t}{\int }}{r}_1^2\left(\tau \right)\right)}^{1/2}\right\}.$$

(42)

In order to obtain an estimate for the unknown function $q\left(t\right)$, we multiply Equation (15) by $y(y-1)$ and integrate the resulting expression over the interval $[0,1]$.

Thus, using Equations (17) and (18), we find

$$q\left(t\right)=6\left[\underset{0}{\overset{1}{\int }}y(y-1)R(y,t)dy-\underset{0}{\overset{1}{\int }}y(y-1){\displaystyle \frac{\partial w}{\partial t}}(y,t)dy\right].$$

(43)

By applying the Cauchy–Schwarz inequality to the first integral in Equation (41), and in view of Equations (42) and (43) for the second one, we obtain the estimate

$$\left|q\left(t\right)\right|\le \sqrt{{\displaystyle \frac{6}{5}}}r\left(t\right)+{\tilde{r}}_1\left(t\right)e^{-{\displaystyle \frac{{\pi }^{2}t}{8}}}.$$

(44)

Let us note that if we additionally assume the convergence of the integral

$$\underset{0}{\overset{\infty }{\int }}{e}^{{\displaystyle \frac{{\pi }^2\tau }{4}}}{r}_1\left(t\right)d\tau ,$$

(45)

then the function ${\overline{r}}_1\left(t\right)$ is bounded for all $t\ge 0$ and $\left|q\right(t\left)\right|\to 0$ as $t\to \infty$ along the exponential law. Also, from Equation (15) and the estimates in Equations (37), (41) and (44), an analogical result follows for $\left|{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}\right|.$

In addition, from Equation (15) and the estimates in Equations (37), (41) and (44), an analogous result follows.

In summarizing the results above, we have the following assertion.

Theorem 2.

Assume that $w_0\left(y\right)\in C[0,1]\cup C^2(0,1)$ and $R(y,t),{\displaystyle \frac{\partial R}{\partial t}}(y,t)\in C((0,1)\times [0,T])$. Then, there exists a unique solution $q^{\prime }\left(t\right)\in C(0,T)$ to the problem in Equations (15)–(18), and the estimates in Equations (37) and (41) hold uniformly for all $x\in [0,1]$ and $t\in [0,T]$.

Moreover, if the integrals in Equations (38) and (45) are convergent, then the functions $w(y,t)$ and $q\left(t\right)$ converge to zero as $t\to \infty$ with the exponential law.

Furthermore, based on the results for the reduced one-dimensional problem, we discuss the existence and uniqueness of the solution to the problem in Equations (1)–(7).

Theorem 3.

Suppose that the conditions of Theorem 2 are fulfilled. Then, the problem in Equations (1)–(7) has a unique global classical solution.

Proof. 

Suppose that the problem in Equations (1)–(7) admits two solutions $\{u_1(x,y,t),q_1\left(t\right)\}$ and $\{u_2(x,y,t),q_2\left(t\right)\}$. Then, the pair $\{U(x,y,t)=u_2(x,y,t)-u_1(x,y,t),Q\left(t\right)=q_2\left(t\right)-q_1\left(t\right)\}$ satisfies the equation

$${\displaystyle \frac{\partial U}{\partial t}}={\displaystyle \frac{{\partial }^{2}U}{\partial t^2}}+p\left(t\right)U+Q\left(t\right),0\le x,y\le 1,0<t\le T,$$

with the initial condition

$$U(x,y,0)=0,0\le x,y\le 1,$$

and boundary conditions

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

as well as the integral measurement

$$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}U(x,y,t)dxdy=0,0<t\le T,$$

where $p\left(t\right)\le 0$ is a known function and the temperature U and the source intensity $Q\left(t\right)$ are unknown.

Furthermore, we proceed as demonstrated in Section 3 and Section 4 to construct and analyze the corresponding reduced one-dimensional problem. As a result, we deduce that $Q\left(t\right)\equiv 0$. Therefore, the problem above becomes purely homogeneous, which implies $U(x,y,t)=0$ (see, for example, [15,39]). □

6. Numerical Solution

In this section, we present the numerical method used to validate the theoretical results. The method is constructed in alignment with the approach developed in the theoretical analysis. To recover the solution $u(x,y,t)$ and the source term $q\left(t\right)$ in Equations (1)–(7), we solve the corresponding one-dimensional problem derived in Section 3. Since the problems in Equations (9)–(13) and (14)–(19) are proven to be equivalent (see Proposition 1), either formulation can be used for the numerical implementation.

The explicit representation of the unknown source term $q\left(t\right)$, as given in Equation (19), can also be derived for the problem in Equations (9)–(13). Indeed, by integrating Equation (9) with respect to y over the interval $[0,1]$ and applying the measurement condition (13), we obtain

$$q\left(t\right)=m^{\prime }\left(t\right)-{\displaystyle \frac{\partial v}{\partial y}}(1,t)+{\displaystyle \frac{\partial v}{\partial y}}(0,t)-p\left(t\right)m\left(t\right)-\underset{0}{\overset{1}{\int }}G(y,t)dy.$$

(46)

The problem in Equations (9)–(12) and (46) is therefore also a non-local problem, similar to Equations (14)–(19).

We consider the problems in Equations (9)–(12) and (46).

Let us introduce a uniform mesh in both the time and space directions ${\overline{w}}_{h\tau }={\overline{w}}_h\times {\overline{w}}_{\tau }$, where

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

We denote the values of a mesh function $\nu$ at the grid points by ${\nu }_j^n=\nu (y_j,t_n)$. The second derivative with respect to space is approximated using the second-order central difference formula ${\nu }_{\overline{y}y,j}=({\nu }_{j+1}-2{\nu }_j+{\nu }_{j-1})/h^2$.

Equation (46) is approximated as follows:

$$q^{n+1}={\left(m^{\prime }\right)}^{n+1}-v_{\overleftarrow{y},J}^{n+1}+v_{\overrightarrow{y},0}^{n+1}-p^{n+1}{m}^{n+1}-I_G^{n+1},$$

(47)

where

$$\begin{array}{cc}& {\displaystyle v_{\overrightarrow{y},0}^n=\frac{-3v_0^n+4v_1^{n+1}-v_2^n}{2h},v_{\overleftarrow{y},J}^n=\frac{3v_J^n-4v_{J-1}^n+v_{J-2}^n}{2h},}\\ & {\displaystyle I_G^n=\frac{h}{2}\left(G_0\left(t_n\right)+2\sum _{s=1}^{J-1}{G}_s\left(t_n\right)+G_J\left(t_n\right)\right).}\end{array}$$

By substituting Equation (46) into Equation (9), replacing the derivatives with second-order finite differences, and also using Equation (47), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle \frac{v_j^{n+1}-v_j^n}{\tau }}-v_{\overline{y}y,j}^{n+1}-p^{n+1}{v}_j^{n+1}+v_{\overleftarrow{y},J}^{n+1}-v_{\overrightarrow{y},0}^{n+1}=G_j^{n+1}\hfill \\ & +{\left(m^{\prime }\right)}^{n+1}-p^{n+1}{m}^{n+1}-I_G^{n+1},j=1,2,\dots ,J-1,\hfill \\ & v_0^{n+1}={\phi }_0^{n+1},v_J^{n+1}={\phi }_1^{n+1},\hfill \\ & v_j^0=v_0\left(y_j\right),j=0,1,\dots ,J.\hfill \end{array}\end{array}$$

(48)

To avoid the non-locality in Equation (48), we apply the decomposition of the solution in a similar way to that in [40] and adapt it to our problem:

$$v_j^{n+1}=V_j^{n+1}-W_j^{n+1}{\mathcal{S}}^{n+1},{\mathcal{S}}^{n+1}=v_{\overleftarrow{y},J}^{n+1}-v_{\overrightarrow{y},0}^{n+1}j=0,1,\dots ,J.$$

(49)

By replacing $v^{n+1}$ in Equation (48) with Equation (49), we derive the following two problems:

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill & {\displaystyle \frac{V_j^{n+1}-v_j^n}{\tau }}-V_{\overline{y}y,j}^{n+1}-p^{n+1}{V}_j^{n+1}=G_j^{n+1}\hfill \\ & +{\left(m^{\prime }\right)}^{n+1}-p^{n+1}{m}^{n+1}-I_G^{n+1},j=1,2,\dots ,J-1,\hfill \\ & V_0^{n+1}={\phi }_0^{n+1},V_J^{n+1}={\phi }_1^{n+1},\hfill \\ & v_j^0=v_0\left(y_j\right),j=0,1,\dots ,J,\hfill \end{array}\hfill \end{array}$$

(50)

and

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill & {\displaystyle \frac{W_j^{n+1}}{\tau }}-W_{\overline{y}y,j}^{n+1}-p^{n+1}{W}_j^{n+1}=1,j=1,2,\dots ,J-1,\hfill \\ & W_0^{n+1}=0,W_J^{n+1}=0.\hfill \end{array}\hfill \end{array}$$

(51)

Furthermore, by substituting the decomposition in Equation (49) with ${\mathcal{S}}^{n+1}$, we obtain

$$S^{n+1}={\displaystyle \frac{V_{\overleftarrow{y},J}^{n+1}-V_{\overrightarrow{y},0}^{n+1}}{1+W_{\overleftarrow{y},J}-W_{\overrightarrow{y},0}}}.$$

(52)

By solving the systems in Equations (50) and (51), we derive the solutions $V^{n+1}$, $W^{n+1}$, and therefore, from Equations (52) and (49), we determine ${\widehat{v}}_i$, $i=0,1,\dots ,I$. Consequently, from Equation (47), we find $q^{n+1}$.

Once the solution $q^n$, $n=1,2,\dots ,N$ is obtained, we solve the direct problem in Equations (1)–(6) using a standard second-order in space finite difference scheme on uniform spatial meshes with step sizes k and h and with I and J as the number of grid nodes in the x and y directions, respectively. The temporal mesh is ${\overline{\omega }}_{\tau }$.

7. Computational Verification

In this section, we verify the theoretical results. Additionally, we demonstrate the accuracy of the presented numerical approach. For the computations, we deal with exact solution of the problem in Equations (1)–(7).

The errors and order of convergence are given in both the maximum norm and $L_2$ norm:

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

Furthermore, all computations are performed for $I=J$ ($h=k$) and $T=1$.

Example 1.

We consider the inverse problem in Equations (1)–(7), where

$$\begin{array}{cc}& {\displaystyle p=-2,f(x,y,t)=\left(\frac{6+{\pi }^2}{4}\left(sin\frac{\pi x}{2}+cos\frac{\pi y}{2}\right)-1\right)e^{t/2},m\left(t\right)=\frac{4e^{t/2}}{\pi },}\\ & {\displaystyle u_0(x,y)=sin\frac{\pi x}{2}+cos\frac{\pi y}{2},g_0\left(t\right)=\frac{\pi }{2}{e}^{t/2},g_1=0,}\\ & {\displaystyle {\psi }_0(x,t)=e^{t/2}\left(sin\frac{\pi x}{2}+1\right),{\psi }_1(x,t)=e^{t/2}sin\frac{\pi x}{2}.}\end{array}$$

In this case, the exact solution is $u(x,y,t)=e^{t/2}\left(sin{\displaystyle \frac{\pi x}{2}}+cos{\displaystyle \frac{\pi y}{2}}\right)$, $q\left(t\right)=e^{t/2}$.

The errors and convergence orders in the maximum and $L_2$ discrete norms for both the numerically determined source term q and the solution u are presented in

Table 1. Errors and convergence rates of the numerically recovered function q and solution u, where $\tau =h$ (Example 1).

N	${\mathcal{E}}_{\mathit{\infty }}$	${\mathit{CR}}_{\mathit{\infty }}$	${\mathcal{E}}_2$	${\mathit{CR}}_2$	${\mathit{e}}_{\mathit{\infty }}$	${\mathit{cr}}_{\mathit{\infty }}$	${\mathit{e}}_2$	${\mathit{cr}}_2$
20	4.2739 $\times {10}^{-3}$		1.9993 $\times {10}^{-3}$		3.1659 $\times {10}^{-3}$		2.4170 $\times {10}^{-3}$
40	1.7469 $\times {10}^{-3}$	1.2908	8.1957 $\times {10}^{-4}$	1.2866	1.1736 $\times {10}^{-3}$	1.4318	9.0008 $\times {10}^{-4}$	1.4251
80	7.7572 $\times {10}^{-4}$	1.1712	3.6570 $\times {10}^{-4}$	1.1642	4.8376 $\times {10}^{-4}$	1.2785	3.7163 $\times {10}^{-4}$	1.2762
160	3.6328 $\times {10}^{-4}$	1.0944	1.7195 $\times {10}^{-4}$	1.0886	2.1606 $\times {10}^{-4}$	1.1629	1.6606 $\times {10}^{-4}$	1.1621
320	1.7549 $\times {10}^{-4}$	1.0497	8.3269 $\times {10}^{-5}$	1.0462	1.0156 $\times {10}^{-4}$	1.0890	7.8075 $\times {10}^{-5}$	1.0888
640	8.6210 $\times {10}^{-5}$	1.0255	4.0959 $\times {10}^{-5}$	1.0236	4.9165 $\times {10}^{-5}$	1.0467	3.7797 $\times {10}^{-5}$	1.0466

. All runs were carried out with a fixed $\tau =h$. The results show that the numerical solution converged to the exact one. The convergence order of the restored function q was of the first order. Since $\tau =h$, the results presented in Table 1 also reflect the temporal convergence rate of the solution u, which is clearly of the first order.

We demonstrate the spatial convergence order in

Table 2. Errors and convergence rates of the numerically recovered solution u, where $\tau =h^2$ (Example 1).

N	${\mathcal{E}}_{\mathit{\infty }}$	${\mathit{CR}}_{\mathit{\infty }}$	${\mathcal{E}}_2$	${\mathit{CR}}_2$
10	6.2377 $\times {10}^{-3}$		2.7601 $\times {10}^{-3}$
20	1.5829 $\times {10}^{-3}$	1.9785	6.8328 $\times {10}^{-4}$	2.0142
40	3.9739 $\times {10}^{-4}$	1.9940	1.6979 $\times {10}^{-4}$	2.0087
80	9.9586 $\times {10}^{-5}$	1.9965	4.2308 $\times {10}^{-5}$	2.0048
160	2.4917 $\times {10}^{-5}$	1.9988	1.05591 $\times {10}^{-5}$	2.0024
320	6.2311 $\times {10}^{-6}$	1.9996	2.6395 $\times {10}^{-6}$	2.0001

. These computations were performed with a fixed $\tau =h^2$.

We plotted the numerical solution u and the error $E_{i,j}^n$ in Figure 1. The error of the method at interior points was greater than that at the boundary since we had exact boundary conditions, whereas the source function $g\left(t\right)$ was unknown and was determined numerically by the proposed method.

Example 2.

Now, we set

$$\begin{array}{cc}& {\displaystyle p=-2t^2-1,f(x,y,t)=\left(\frac{{\pi }^2}{4}\left(sin\frac{\pi x}{2}+cos\frac{\pi y}{2}\right)-1\right)e^{-2t^3/3-t},m\left(t\right)=\frac{4}{\pi }{e}^{-2t^3/3-t},}\\ & {\displaystyle u^0(x,y)=sin\frac{\pi x}{2}+cos\frac{\pi y}{2},g_0\left(t\right)=\frac{\pi }{2}{e}^{-2t^3/3-t},g_1=0,}\\ & {\displaystyle {\psi }_0(x,t)=e^{-2t^3/3-t}\left(sin\frac{\pi x}{2}+1\right),{\psi }_1(x,t)=e^{-2t^3/3-t}sin\frac{\pi x}{2}.}\end{array}$$

In this case, the exact solution of the inverse problem in Equations (1)–(7) is $q\left(t\right)=e^{-2t^3/3-t}$, $u(x,y,t)=e^{-2t^3/3-t}$$\left(sin{\displaystyle \frac{\pi x}{2}}+cos{\displaystyle \frac{\pi y}{2}}\right)$.

We present the computational results for $\tau =h$ and $\tau =h^2$ in

Table 3. Errors and convergence rates of the numerically recovered function q and solution u, where $\tau =h$ (Example 2).

N	${\mathcal{E}}_{\mathit{\infty }}$	${\mathit{CR}}_{\mathit{\infty }}$	${\mathcal{E}}_2$	${\mathit{CR}}_2$	${\mathit{e}}_{\mathit{\infty }}$	${\mathit{cr}}_{\mathit{\infty }}$	${\mathit{e}}_2$	${\mathit{cr}}_2$
20	6.7844 $\times {10}^{-3}$		4.2826 $\times {10}^{-3}$		5.4018 $\times {10}^{-3}$		2.0772 $\times {10}^{-3}$
40	3.2675 $\times {10}^{-3}$	1.0540	1.9029 $\times {10}^{-3}$	1.1703	2.6324 $\times {10}^{-3}$	1.0371	9.4988 $\times {10}^{-4}$	1.1288
80	1.6021 $\times {10}^{-3}$	1.0282	8.9073 $\times {10}^{-4}$	1.0951	1.2995 $\times {10}^{-3}$	1.0185	4.5307 $\times {10}^{-4}$	1.0680
160	7.9311 $\times {10}^{-4}$	1.0144	4.3018 $\times {10}^{-4}$	1.0501	6.4546 $\times {10}^{-4}$	1.0095	2.2112 $\times {10}^{-4}$	1.0349
320	3.9456 $\times {10}^{-4}$	1.0073	2.1130 $\times {10}^{-4}$	1.0256	3.2167 $\times {10}^{-4}$	1.0047	1.0922 $\times {10}^{-4}$	1.0177
640	1.9678 $\times {10}^{-4}$	1.0037	1.0471 $\times {10}^{-4}$	1.0129	1.6057 $\times {10}^{-4}$	1.0024	5.4273 $\times {10}^{-5}$	1.0089

and

Table 4. Errors and convergence rates of the numerically recovered solution u, where $\tau =h^2$ (Example 2).

N	${\mathcal{E}}_{\mathit{\infty }}$	${\mathit{CR}}_{\mathit{\infty }}$	${\mathcal{E}}_2$	${\mathit{CR}}_2$
10	3.5473 $\times {10}^{-3}$		1.5730 $\times {10}^{-3}$
20	9.1236 $\times {10}^{-4}$	1.9591	3.9116 $\times {10}^{-4}$	2.0077
40	2.2996 $\times {10}^{-4}$	1.9882	9.7260 $\times {10}^{-5}$	2.0078
80	5.7674 $\times {10}^{-5}$	1.9954	2.4232 $\times {10}^{-5}$	2.0049
160	1.4444 $\times {10}^{-5}$	1.9975	6.0471 $\times {10}^{-6}$	2.0026
320	3.6131 $\times {10}^{-6}$	1.9991	1.5117 $\times {10}^{-6}$	2.0001

, respectively. The accuracy of the recovered source q is $O\left(\tau \right)$, and the accuracy of the solution u is $O(\tau +h^2)$.

We plotted the numerical solution u and the error $E_{i,j}^n$ in Figure 2.

Example 3.

We consider the inverse problem in Equations (1)–(7) for identifying the source $q\left(t\right)$ and the solution $u(x,y,t)$ with the following input data:

$$\begin{array}{cc}& {\displaystyle p\left(t\right)=-\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}+2t^2,}\hfill & t\ge \frac{1}{2},\hfill \\ {\displaystyle \frac{3}{2}-2t^2,}\hfill & t\le \frac{1}{2},\hfill \end{array}\right.f(x,y,t)=2sin\left(\pi (x+y)t\right),}\\ & {\displaystyle m\left(t\right)=\frac{1}{3}(cos\left(\pi t\right)-1)sin\left(\pi t\right),u_0(x,y)=0,g_0=0,}\\ & {\displaystyle g_1(y,t)=-\pi xsin\left(\pi t\right)sin\left(\pi yt\right),{\psi }_0=0,{\psi }_1(x,t)=cos\left(\pi tx\right)sin\left(\pi t\right).}\end{array}$$

Now, the exact solution is not known.

We plotted the numerically recovered unique solutions q and u at final time in Figure 3.

8. Conclusions

In this paper, we investigated an inverse source initial-boundary value problem for the linear two-dimensional diffusion equation. The problem was formulated both with Dirichlet and Neumann boundary conditions, and a solution integral constraint governed the determination of the time-dependent source. Unlike many existing studies that focused on estimating the source within either the differential operator or the boundary conditions, we derived a priori estimates of the solution and the unknown source in a strong norm. Furthermore, the existence and uniqueness of a classical solution to the inverse problem were rigorously proven.

Using the structure of the boundary conditions, we reduced the original two-dimensional inverse source problem to an equivalent one-dimensional formulation. We first proved the well posedness of the inverse problem and then derived a series of a priori estimates for the solution and source term. In the final part of this process, the existence and uniqueness of a classical solution to the inverse problem were proven.

To support the theoretical analysis and provide a complete investigation, we presented a numerical method for solving the inverse problem. The proposed scheme is second-order accurate in space and first-order accurate in time. The computational results demonstrated the convergence of the numerical solution to the exact solution.

In our future work, we aim to extend both the theoretical framework and numerical approach to more complex two-dimensional convection-diffusion parabolic problems.