Identification of the unknown heat source terms in a 2D parabolic equation

1 Introduction

In the last few decades, inverse problems for the parabolic equation have received great interest in research (Cannon, 1968; Johansson and Lesnic, 2007a,b; Hazanee et al., 2013; Hussein et al., 2018; Erdem et al., 2013; Hasanov and Pekta, 2014; Trong et al., 2006; Wang et al., 2014; Li and Qian, 2012; Singh et al., 2019). Farcas and Lesnic (2006) used the conditions of the direct problem and over-determination while Johansson and Lesnic (2007a,b) used the standard conditions of the direct and information from one supplementary temperature measurement for investigating a space-dependent heat source term for the parabolic heat equation. Li and Qian (2012) investigated the inverse problem of determining the time-dependent heat source coefficient. Authors of Hazanee et al. (2013) studied the inverse problem of finding the timewise coefficient along with the temperature solution of heat equation from nonlocal and integral conditions while in Hazanee et al. (2015), they investigated the same with a non-classical boundary and an integral over-determination conditions. Hazanee and Lesnic (2013) investigated it with non-local boundary and over-determination conditions. Huntul et al. (2018) studied the inverse problem of identifying the time- and space-dependent terms in the heat equation. Trong et al. (2006) considered the inverse problem of determining a two-dimensional heat source to construct regularized solutions and obtained error estimation explicitly. A method of reproducing kernel Hilbert space was proposed by Wang et al. Wang et al. (2014) for the inverse problem of a two-dimensional heat source. Kulbay et al. (2016) uniquely determined the solution for heat source terms $F\left(x\right)$ and $H\left(t\right)$ , respectively, under some regularity assumptions of inverse problems of the variable coefficients advection–diffusion equation with $F\left(x\right)H\left(t\right)$ type separable sources from additional time-dependent temperature measurement. Recently, Hussein et al. (2018) examined the inverse problem of identifying a multi-dimensional space-dependent heat source term from boundary data. Although, the problem was linear but ill-posed. Chen et al. (2020) established the stability of an inverse source problem. The authors of Kian and Yamamoto, 2019 considered the inverse problem of recovering the time and spacewise source terms for diffusion equation. Mierzwiczak and Kolodziej (2011) investigated an inverse problem for determining unknown right-hand side in the steady two-dimensional parabolic equation while Yang (1998) and Yang et al. (2013) investigated for finding the time- and space-dependent heat sources, respectively, from the heat flux at chosen points on the boundary and final temperature measurements. The authors of Huntul and Lesnic (2020) and Huntul (2020) studied an inverse problem to reconstruct the thermal conductivity from heat flux conditions in a two-dimensional heat equations, respectively, while in Huntul (2021), author studied it for thermal conductivity and free boundary coefficients.

The inverse problem of reconstructing an unknown time or space-dependent heat source term in the heat equation has been the point of attention of many recent studies, e.g. Wang et al. (2020), Damirchi et al. (2021), Ahmadabadi et al. (2009), Yan et al. (2009), Yang and Fu (2010) and Yang et al. (2011). In these studies, in addition to being mostly restricted to one-dimensional computations, the supplementary information required to compensate for the lack of knowledge of the space-dependent heat source is a spacewise internal measurement of the temperature or a time-average of it. In this work, we study the two-dimensional parabolic problem to recover the time-dependent heat source terms numerically, for the first time, in the prescribed domain using the initial and Neumann boundary conditions, and the additional temperature data as over-determination conditions. The problem considered in this paper has already been shown to be uniquely solvable in Pabyrivska and Pabyrivskyy (2018), but the numerical reconstruction has not been studied yet. Therefore, the preeminent goal of the present work is to undertake the numerical aspect of this problem.

The paper is structured as follows. Section 2 formulated the two-dimensional inverse time-dependent source problem for the parabolic equation. Section 3 discretized the direct problem. The minimization technique of the regularized objective function is described in Section 4. Section 5 presents the computational results. Finally, Section 6 highlights the conclusions.

2 Statement of the 2D heat source problem

In the rectangular domain ${\mathrm{\Omega }}_T=\left\{\right(x,y,t):0<x<l_1,0<y<l_2,0<t<T\}$ , consider the inverse problem of identifying the time-dependent heat source terms $f_{\mathit{ij}}\left(t\right)$ for $i,j=\overline{0,1}$ in the two-dimensional parabolic equation

The uniqueness of the solution of the inverse problem (1)–(6) has been established in Pabyrivska and Pabyrivskyy, 2018 and reads as follows.

3 Solution of the direct problem

Now, consider the direct problem (1)–(4). When $f_{\mathit{ij}}\left(t\right),i,j=\overline{0,1},\psi ,{\kappa }_i,i=\overline{1,4}$ are known and $u(x,y,t)$ is to be found. Subdivide the domain ${\mathrm{\Omega }}_T$ into intervals $M_1,M_2$ and N of equal widths $\mathrm{\Delta }x=l_1/M_1,\mathrm{\Delta }y=l_2/M_2$ , and $\mathrm{\Delta }t=T/N$ . We denote $u_{i,j}^n≔u(x_i,y_j,t_n)$ , where $x_i=i\mathrm{\Delta }x,y_j=j\mathrm{\Delta }y,t_n=n\mathrm{\Delta }t,f_{00}^n≔f_{00}\left(t_n\right),f_{01}^n≔f_{01}\left(t_n\right),f_{10}^n≔f_{10}\left(t_n\right)$ , and $f_{11}^n≔f_{11}\left(t_n\right)$ for $i=\overline{0,M_1},j=\overline{0,M_2},n=\overline{0,N}$ .

We apply the forward time central space (FTCS) FDM to solve the Eq. (1) which is conditionally stable, LeVeque, 2007. So we obtain

The initial condition (2) gives

The stability condition of the explicit expression (8) is given as (Morton and Mayers, 2005)

4 Inverse solution of the 2D heat source problem

Our aim is to obtain simultaneously stable reconstructions of the heat source terms $f_{\mathit{ij}}\left(t\right)$ for $i,j=\overline{0,1}$ and the temperature $u(x,y,t)$ , satisfying Eqs. (1)–(6). The inverse problem is formulated as minimizing the regularized function

The MATLAB subroutine lsqnonlin (Mathworks, 2016) is employed to minimize the objective function (14). The inverse problem given by (1)–(6) is solved subject to both exact and noisy measurements (5) and (6). The noisy data is numerically formulated, as follows:

In the case of noisy data (15), we replace $v_{\mathit{ij}}\left(t_n\right)$ by $v_{\mathit{ij}}^{{∊}_{\mathit{ij}}}\left(t_n\right)$ in (14) for $i,j=\overline{0,1}$ .

5 Results and discussion

The solutions for $f_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ and $u(x,y,t)$ are presented for analytical and perturbed (noisy) data (15). The accuracy is measured by

We take $l_1=l_2=T=1$ , for simplicity. The lower and upper bounds for the coefficients $f_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ are taken as $-{10}^3$ and ${10}^3$ , respectively.

5.1

5.1 Example 1

Consider the inverse problem (1)–(6) with unknown $f_{00}\left(t\right),f_{01}\left(t\right),f_{10}\left(t\right),f_{11}\left(t\right)$ , and with the input data $\psi ,{\kappa }_i,i=\overline{1,4},v_{\mathit{ij}},i,j=\overline{0,1}$ , $$\psi (x,y)=-4-{(-2+x)}^2-{(-2+y)}^2,{\kappa }_1(y,t)=4+t^2+t^{2}y,$$

(19)

$${\kappa }_2(y,t)=2+t^2+t^{2}y,{\kappa }_3(x,t)=4+t^2+t^{2}x,{\kappa }_4(x,t)=2+t^2+t^{2}x,$$ $$v_{00}\left(t\right)=-8-{(-2-t)}^2,v_{01}\left(t\right)=-5-{(-2-t)}^2+t^2,$$

(20)

$$v_{10}\left(t\right)=-5-{(-2-t)}^2+t^2,v_{11}\left(t\right)=-2-{(-2-t)}^2+3t^2.$$

It can easily be checked that with this data, the conditions $\left(A1\right)$ – $\left(A6\right)$ of Theorem 1 are fulfilled, hence the uniqueness of the solution is guaranteed. The analytical solution is given by

(21)

$$u(x,y,t)=-{(-2+x)}^2-{(-2+y)}^2-{(-2-t)}^2+\left(x\right(1+y)+y)t^2,$$

(22)

$$f_{00}\left(t\right)=-2t,f_{01}\left(t\right)=2t,f_{10}\left(t\right)=2t,f_{11}\left(t\right)=2t.$$

First of all, the accuracy of the direct problem (1)–(4) is assessed with the data (19), when $f_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ are known and given by (22), using the FTCS-FDM described in Section 3. Table 1 demonstrates that the exact and approximate solutions for (5) and (6), which exactly is given by (20), obtained with $M_1=M_2=5$ and $N\in \{120,140\}$ , are in excellent agreement. The exact (21) and approximate $u(x,y,t)$ are depicted in Fig. 1.

Table 1 The exact (20) and approximated $v_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ , with $M_1=M_2=5$ and various $N\in \{120,140\}$ , for Example 1.

t	0.1	0.2	0.3	…	0.8	0.9	1	N
$v_{00}\left(t\right)$	−12.4096 −12.4096 −12.4100	−12.8395 −12.8396 −12.8400	−13.2896 −13.2897 −13.2900	… … …	−15.8406 −15.8405 −15.8400	−16.4108 −16.4107 −16.4100	-s17.0010 −17.0009 −17.0000	120 140 Exact
$v_{01}\left(t\right)$	−9.4001 −9.4001 −9.4000	−9.8003 −9.8003 −9.8000	−10.2005 −10.2005 −10.2000	… … …	−12.2016 −12.2014 −12.2000	−12.6018 −12.6015 −12.6000	−13.0020 −13.0017 −13.0000	120 140 Exact
$v_{10}\left(t\right)$	−9.4001 −9.4001 −9.4000	−9.8003 −9.8003 −9.8000	−10.2005 −10.2005 −10.2000	… … …	−12.2016 −12.2014 −12.2000	−12.6018 −12.6015 −12.6000	−13.0020 −13.0017 −13.0000	120 140 Exact
$v_{11}\left(t\right)$	−6.3810 −6.3809 −6.3800	−6.7215 −6.7213 −6.7200	−7.0218 −7.0215 −7.0200	… … …	−7.9229 −7.9225 −7.9200	−7.9831 −7.9826 −7.9800	−8.0033 −8.0028 −8.0000	120 140 Exact

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Fig. 1

The $u(x,y,1)$ and absolute errors with $M_1=M_2=5$ for: (a) $N=120$ and (b) $N=140$ .

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In the inverse problem (1)–(6), we take the initial guesses for ${\mathbf{f}}_{\mathbf{00}},{\mathbf{f}}_{\mathbf{01}},{\mathbf{f}}_{\mathbf{10}}$ and ${\mathbf{f}}_{\mathbf{11}}$ as: $$f_{00}^0\left(t_n\right)=f_{00}\left(0\right)=0,f_{01}^0\left(t_n\right)=f_{01}\left(0\right)=0,$$

(23)

$$f_{10}^0\left(t_n\right)=f_{10}\left(0\right)=0,f_{11}^0\left(t_n\right)=f_{11}\left(0\right)=0,n=\overline{1,N}.$$

Next, we examine the inverse problem. We take a mesh size with $M_1=M_2=5$ and $N=120$ satisfying the stability condition (12). We solve the inverse problem (1)–(6) of finding the heat source terms $f_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ and $u(x,y,t)$ with $p=0$ in $v_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ . Although not illustrated, it is reported that a rapid monotonic decreasing convergence of the objective function (14) to a very small minimum value of $O\left({10}^{-28}\right)$ is achieved in about 8 iterations. Fig. 2 depicts the obtained timewise heat source terms with $p=0$ and $\lambda =0$ . An excellent agreement among the analytical (22) and computational heat sources can be noticed with $\mathrm{RMSE}\left(f_{00}\right)$ =3.3E-3, $\mathrm{RMSE}\left(f_{01}\right)$  = 8.3E−3, $\mathrm{RMSE}\left(f_{10}\right)$  = 8.3E−3 and $\mathrm{RMSE}\left(f_{11}\right)$  = 8.3E−3.

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Fig. 2

The exact (22) and numerical solutions for: (a) $f_{00}\left(t\right)$ , (b) $f_{01}\left(t\right)$ , (c) $f_{10}\left(t\right)$ and (d) $f_{11}\left(t\right)$ , with $p=0$ and $\lambda =0$ , for Example 1.

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Now, the stability of the computational solution is examined with respect to perturbed data. We add $p=0.1\%$ noise generated by Eq. (17) to simulate the input noisy data, via Eq. (15) for $v_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ . Fig. 3 shows the objective function (14) versus the number of iterations, with $\lambda \in \{0,{10}^{-6},{10}^{-5},{10}^{-4},{10}^{-3},{10}^{-2}\}$ , where a monotonically decreasing convergence is obtained. The identification of the terms $f_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ is shown in Fig. 4, where the unstable (oscillation) results are obtained, if no regularization, i.e. $\lambda =0$ , is imposed with $\mathrm{RMSE}\left(f_{00}\right)=3.1329,\mathrm{RMSE}\left(f_{01}\right)=3.5603,\mathrm{RMSE}\left(f_{10}\right)=3.6040$ and $\mathrm{RMSE}\left(f_{11}\right)=4.2657$ , respectively. In order to stabilize these coefficients, we employ regularization with $\lambda ={10}^{-3}$ , obtaining $\mathrm{RMSE}(f_{00},f_{01},f_{10},f_{11})\in \{0.4111,0.3222,0.3087,0.1970\}$ . Also, from Table 2 and Fig. 4 it can be noticed that the effect of $\lambda >0$ is decreasing the unbounded behaviour (oscillatery) of the heat source terms. Therefore, the numerical results achieved with $\lambda ={10}^{-3}$ are stable and accurate. The CPU-time is calculated to analyze the performance of the method. One can notice from Table 2 that the CPU-time is very less.

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Fig. 3

The objective function F (14) with $\lambda$ : (a) 0, and (b) ${10}^{-4}$ and ${10}^{-3}$ , for $p=0.1\%$ , for Example 1.

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Fig. 4

The exact (22) and numerical solutions for: $f_{00}\left(t\right),f_{01}\left(t\right),f_{10}\left(t\right)$ and $f_{11}\left(t\right)$ , with $p=0.1\%$ and $\lambda \in \{0,{10}^{-4},{10}^{-3}\}$ , for Example 1.

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Table 2 The RMSE (18) for $p\in \{0,0.1\%\}$ , with different regularizations, for Example 1.

p	$\lambda$	$\mathrm{RMSE}\left(f_{00}\right)$	$\mathrm{RMSE}\left(f_{01}\right)$	$\mathrm{RMSE}\left(f_{10}\right)$	$\mathrm{RMSE}\left(f_{11}\right)$	CPU time (Mins)
0	0	3.3E−3	8.3E−3	8.3E−3	8.3E−3	3.34
$0.1\%$	0 ${10}^{-6}$ ${10}^{-5}$ ${10}^{-4}$ ${10}^{-3}$ ${10}^{-2}$	3.1329 2.8121 1.8183 0.7362 0.4111 0.8798	3.5603 3.0771 1.6971 0.5410 0.3222 0.7182	3.6040 3.0994 1.6573 0.4929 0.3087 0.7146	4.2657 3.5082 1.5006 0.3672 0.1970 0.6320	17.23 7.01 8.11 9.21 9.45 10.49

5.2

5.2 Example 2

In Example 1, we reconstructed the smooth heat source terms $f_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ as given in (22). Now, for compliance, we examine the method to recover a nonlinear test: $$f_{00}\left(t\right)=4+4\pi \cos \left(2\pi t\right),f_{01}\left(t\right)=2\pi \cos \left(2\pi t\right),$$

(24)

$$f_{10}\left(t\right)=2\pi \cos \left(2\pi t\right),f_{11}\left(t\right)=2\pi \cos \left(2\pi t\right),$$ and the input data $\psi ,{\kappa }_i,i=\overline{1,4}$ and $v_{\mathit{ij}},i,j=\overline{0,1}$ :

(25)

$$\begin{array}{cc}\hfill & \psi (x,y)=2-{(-2+x)}^2-{(-2+y)}^2,{\kappa }_1(y,t)=4+(1+y)\sin \left(2\pi t\right),\hfill \\ \hfill & {\kappa }_2(y,t)=2+(1+y)\sin \left(2\pi t\right),{\kappa }_3(x,t)=4+(1+x)\sin \left(2\pi t\right),\hfill \\ \hfill & {\kappa }_4(x,t)=2+(1+x)\sin \left(2\pi t\right),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & v_{00}\left(t\right)=-6+2\sin \left(2\pi t\right),v_{01}\left(t\right)=-3+3\sin \left(2\pi t\right),\hfill \\ \hfill & v_{10}\left(t\right)=-3+3\sin \left(2\pi t\right),v_{11}\left(t\right)=5\sin \left(2\pi t\right).\hfill \end{array}$$

Also, the conditions $\left(A1\right)$ – $\left(A6\right)$ of Theorem 1 are satisfied and therefore, the solvability of the solution is guaranteed. The analytical solution of (1)–(4) is

(27)

$$u(x,y,t)=2-{(-2+x)}^2-{(-2+y)}^2+(2+x+y+\mathit{xy})\sin \left(2\pi t\right).$$

We take the initial guesses for ${\mathbf{f}}_{\mathbf{00}},{\mathbf{f}}_{\mathbf{01}},{\mathbf{f}}_{\mathbf{10}}$ , and ${\mathbf{f}}_{\mathbf{11}}$ as

(28)

$$\begin{array}{cc}\hfill & f_{00}^0\left(t_n\right)=f_{00}\left(0\right)=4+4\pi ,f_{01}^0\left(t_n\right)=f_{01}\left(0\right)=2\pi ,\hfill \\ \hfill & f_{10}^0\left(t_n\right)=f_{10}\left(0\right)=2\pi ,f_{11}^0\left(t_n\right)=f_{11}\left(0\right)=2\pi ,n=\overline{1,N}.\hfill \end{array}$$

We take $M_1=M_2=5$ and $N=140$ , which together with the upper bound ${10}^3$ for the source terms $f_{00},f_{01},f_{10}$ and $f_{11}$ satisfying the stability condition (12).

As in Example 1, first consider the case where we include no noise in $v_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ . Although not illustrated, it is reported that the F (14) decreases fastly to a low tolerance of $O\left({10}^{-29}\right)$ is reported in 17 iterations. The analytical (24) and approximate source terms $f_{00}\left(t\right),f_{01}\left(t\right),f_{10}\left(t\right)$ and $f_{11}\left(t\right)$ are plotted in Fig. 5, where the reconstructed source terms are in excellent agreement with the exact solutions.

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Fig. 5

The exact (24) and numerical solutions for: (a) $f_{00}\left(t\right)$ , (b) $f_{01}\left(t\right)$ , (c) $f_{10}\left(t\right)$ and (d) $f_{11}\left(t\right)$ , with $p=0$ and $\lambda =0$ , for Example 2.

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Next, we include $p\in \{0.1\%,1\%\}$ noise in the input data $v_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ , as in (15). The corresponding exact (24) and numerical solutions for the unknown coefficients are presented in Figs. 6 and 7 for various regularizations, respectively. When $\lambda =0$ , we obtain inaccurate and unstable approximations, with RMSE values (18) of $\{1.6971,1.9709,2.0596,2.3671\}$ , for $p=0.1\%$ , and $\{16.8578,19.6828,20.5685,23.6539\}$ , for $p=1\%$ . We apply the Tikhonov regularization method to overcome this instability. We deduce that $\lambda ={10}^{-4}$ for $p=0.1\%$ , and $\lambda ={10}^{-3}$ for $p=1\%$ provides a stable and accurate numerical solutions for the unknown heat sources having the RMSE values (18) of $\{0.6815,0.3890,0.3645,0.5980\}$ and $\{1.7343,0.8316,0.7640,1.3496\}$ , respectively. Also, from Figs. 6, 7 and Table 3 it is observed that when p decreasing from $1\%$ to $0.1\%$ and then to zero, accuracy and stability of the approximated results increased. Finally, Figs. 6, 7 and Table 3 have the same source terms as Fig. 4 and Table 2, and we can draw the similar conclusions about the stable reconstructions for the heat source terms. It is clear from Table 3 that the CPU-time is very less.

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Fig. 6

The exact (24) and numerical solutions for: $f_{00}\left(t\right),f_{01}\left(t\right),f_{10}\left(t\right)$ and $f_{11}\left(t\right)$ , with $p=0.1\%$ and $\lambda \in \{0,{10}^{-5},{10}^{-4}\}$ , for Example 2.

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Fig. 7

The exact (24) and numerical solutions for: $f_{00}\left(t\right),f_{01}\left(t\right),f_{10}\left(t\right)$ and $f_{11}\left(t\right)$ , with $p=1\%$ and $\lambda \in \{0,{10}^{-4},{10}^{-3}\}$ , for Example 2.

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Table 3 The RMSE (18) for $p\in \{0,0.1\%,1\%\}$ , for Example 2.

p	$\lambda$	$\mathrm{RMSE}\left(f_{00}\right)$	$\mathrm{RMSE}\left(f_{01}\right)$	$\mathrm{RMSE}\left(f_{10}\right)$	$\mathrm{RMSE}\left(f_{11}\right)$	CPU time (Mins)
0	0	8.1E−3	9.1E−3	9.1E−3	9.1E−3	4.94
$0.1\%$	0 ${10}^{-6}$ ${10}^{-5}$ ${10}^{-4}$ ${10}^{-3}$	1.6971 1.4640 0.8894 0.6815 1.6373	1.9709 1.6129 0.8109 0.3890 0.7038	2.0596 1.6946 0.8554 0.3645 0.6933	2.3671 1.8327 0.7671 0.5980 1.3568	18.26 7.81 8.37 9.28 9.54
$1\%$	0 ${10}^{-5}$ ${10}^{-4}$ ${10}^{-3}$ ${10}^{-2}$	16.8578 8.4550 2.9819 1.7343 3.4669	19.6828 7.9392 2.2630 0.8316 1.1545	20.5685 8.4390 2.2954 0.7640 1.1702	23.6539 7.4939 1.8396 1.3496 2.1615	29.34 13.89 14.72 15.48 16.91

6 Conclusions

The inverse problem relating to the reconstruction of the time-dependent heat source terms $f_{\mathit{ij}}\left(t\right),i,j=\overline{0,1}$ and the temperature $u(x,y,t)$ in a two-dimensional parabolic equation from the over-determination conditions has been numerically studied for the first time. The direct problem has been discretized using the FTCS-FDM. The RMSE values for noise with and without regularization for Example 1 and 2 are compared. The numerical results for the inverse problem are presented and discussed. It has been concluded that $\lambda ={10}^{-4}$ for $p=0.1\%$ , and $\lambda ={10}^{-3}$ for $p=1\%$ provides a stable and accurate solution for the unknown heat source terms. Finally, the generalization of the proposed method to reconstruct the heat source coefficients in the three-dimensional parabolic problem is an interesting topic for future research.