Inverse Problems in Science and Engineering ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/gipe20 Inverse source problem of heat conduction equation with time-dependent diffusivity on a spherical symmetric domain Xiaoxiao Geng, Hao Cheng & Mian Liu To cite this article: Xiaoxiao Geng, Hao Cheng & Mian Liu (2021): Inverse source problem of heat conduction equation with time-dependent diffusivity on a spherical symmetric domain, Inverse Problems in Science and Engineering, DOI: 10.1080/17415977.2021.1899172 To link to this article: https://doi.org/10.1080/17415977.2021.1899172 Published online: 13 Mar 2021 Submit your article to this journal Article views: 14 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=gipe20 INVERSE PROBLEMS IN SCIENCE AND ENGINEERING https://doi.org/10.1080/17415977.2021.1899172 Inverse source problem of heat conduction equation with time-dependent diffusivity on a spherical symmetric domain Xiaoxiao Geng, Hao Cheng and Mian Liu School of Science, Jiangnan University, Wuxi, Jiangsu, People’s Republic of China ABSTRACT ARTICLE HISTORY In this paper, we consider the inverse source problem of heat conduction equation with time-dependent diffusivity on a spherical symmetric domain This problem is ill-posed, i.e the solution of the problem does not depend continuously on the measured data To solve this problem, we propose an iterative regularization method and obtain the Hölder type error estimates Numerical examples are presented to demonstrate the effectiveness of the proposed method Received August 2020 Accepted 24 February 2021 KEYWORDS Inverse source problem; heat conduction equation; time-dependent diffusivity; iterative regularization method; error estimates 2010 MATHEMATICS SUBJECT CLASSIFICATIONS 35R25; 47A52 Introduction In recent years, the inverse source problem has played an important role in dealing with inverse problems for partial differential equations Meanwhile, the inverse source problem often occurs in the migration of groundwater, the identification of radiogenic source for helium diffusion, the identification and control of pollution sources and so forth [1,2] The purpose of the inverse source problem is to determine the unknown source term from the final temperature distribution or the boundary temperature distribution In [3], the authors reconstructed the spatially variable heat source from the measured data by applying the dual least squares method Wei and Zhang [4] used a boundary element method combined with a generalized Tikhonov regularization to determine the time-dependent source In [5], the authors proved the existence, uniqueness and continuously dependence on data of heat source depending on time variable by using the generalized Fourier method Ma et al [6] applied a variational method to reconstruct the unknown source term depending on both time and space variables and so on [7–13] However, most of these works were only devoted to the heat conduction equation with constant coefficient In practical engineering, the variable case of diffusivity has more applications than the constant one In [14], the authors considered the model of helium production and diffusion chenghao@jiangnan.edu.cn School of Science, CONTACT Hao Cheng Jiangsu, People’s Republic of China © 2021 Informa UK Limited, trading as Taylor & Francis Group Jiangnan University, Wuxi, 214122 X GENG ET AL in a spherical diffusion geometry: He = 8238 U(t)(eλ238 t − 1) + 7235 U(t)(eλ235 t − 1) + 6232 Th(t)(eλ232 t − 1), D(t) ∂ He(r, t) = ∂t a ∂ He(r, t) ∂ He(r, t) + 8λ238 + 238 U(t) ∂r2 r ∂r 232 + 7λ235 235 U(t) + 6λ232 Th(t), where He(t), U(t) and Th(t) are amounts present at time t, λ’s are the decay constants, a and r are the radius and radial position of spherical diffusion domain, respectively, D(t) is the time-dependent diffusion coefficient obeying an Arrhenius relationship such that: D(t) D0 Ea , = exp − a a RT(t) where D0 is the diffusivity at infinite temperature and Ea the activation energy measured in laboratory experiments, R is the constant, and T(t) is an arbitrary thermal history In this paper, we consider the following heat conduction equation with time-dependent diffusivity on a spherical symmetric domain: ⎧ ⎪ ⎪ ut (r, t) = a(t) urr (r, t) + ur (r, t) + f (r), < r < R0 , < t < T, ⎪ ⎪ r ⎪ ⎪ ⎪ ⎨ u(r, 0) = φ(r), ≤ r ≤ R0 , (1) ⎪ ⎪ ⎪ u(R0 , t) = 0, ≤ t ≤ T, ⎪ ⎪ ⎪ ⎪ ⎩ lim u(r, t) bounded, ≤ t ≤ T, r→0 where a(t) is a time-dependent diffusion coefficient, assuming that it has a lower limit a0 and an upper limit a1 , i.e < a0 ≤ a(t) ≤ a1 (2) Here we want to identify the unknown source term f (r) from the additional final value data u(r, T) = g(r), ≤ r ≤ R0 (3) For the uniqueness results, we refer to the papers [15–18] and the references cited therein Since the measurement is noise-contaminated inevitably, we denote the noise measurement of g(r) as g δ (r) which satisfies g(r) − g δ (r) ≤ δ, (4) where · is the L2 ([0, R0 ]) norm and δ > is the noise level For the diffusion coefficient a(t) = constant, Cheng and Fu [19] and Yang et al [20] used a spectral method and an a posteriori truncation regularization method to solve this problem, respectively For the diffusion coefficient a(t) = function of t, Bao et al [18] and Zhang and Yan [21] used Tikhonov regularization and spectral truncation regularization to solve this problem, respectively Inspired by Wang and Ren [22], we will adopt an iterative regularization method to solve this inverse problem Compared with the above methods, INVERSE PROBLEMS IN SCIENCE AND ENGINEERING the iterative regularization method seems more simple and direct Meanwhile, the iterative regularization method may be used to solve some nonlinear inverse problems The structure of this paper is as follows In Section 2, we provide some preliminary knowledge In Section 3, we analyse the ill-posedness of the inverse problem and conditional stability An iterative regularization method is introduced in Section 4, and the error estimates under two parameter choice rules are obtained In Section 5, three different numerical examples are presented to verify the effectiveness of our method Finally, a brief conclusion is given in Section Preliminaries Lemma 2.1: Assuming λn = ( nπ R0 ) , for any n ≥ 1, there holds C ≤ n2 where C = (1−e−a1 λ1 T )R20 a1 π and C = T e−λn T τ a(s) ds dτ ≤ C , n2 (5) R20 a0 π Proof: For n ≥ 1, λn ≥ · · · ≥ λ1 > and according to (2), we have T e−λn T τ a(s) ds dτ ≥ T e−a1 λn (T−τ ) dτ = − e−a1 λn T − e−a1 λ1 T ≥ a1 λn a1 λn (1 − e−a1 λ1 T )R20 C = = 2, 2 a1 π n n T e−λn T τ a(s) ds dτ ≤ T e−a0 λn (T−τ ) dτ = R20 C − e−a0 λn T ≤ = = 2 a0 λn a0 λn a0 π n n Lemma 2.2 ([23]): For < λ < and k ≥ 1, define pk (λ) = − λpk (λ) = (1 − λ)k Then, pk (λ)λμ ≤ k1−μ , ν ≤ μ ≤ 1, rk (λ)λ ≤ θν (k + 1) −ν , k−1 i i=0 (1 − λ) and rk (λ) = (6) (7) where θν = 1, ≤ ν ≤ 1, ν ν , ν > Ill-posedness and conditional stability Let √ ωn (r) := 2nπ sin(nπ r/R0 ) R3 (nπ r/R0 ) The eigenfunctions ωn (r) are standard orthogonal function systems with weight r2 in the [0, R0 ], and are complete in the class of square integrable functions on [0, R0 ] [24] The X GENG ET AL solution u(r, t) and source term f (r) can be represented in the following form: ∞ R30 Tn (t)ωn (r), √ 2nπ (8) fn ωn (r), (9) u(r, t) = n=1 ∞ f (r) = n=1 where R0 fn = r2 f (r)ωn (r) dr, n = 1, 2, Substituting (8) and (9) into the equation and initial condition in (1), and denoting φn = (φ(r), ωn (r)), we can obtain ⎧ √ 2nπ ⎪ ⎪ ⎪ fn , Tn (t) + λn a(t)Tn (t) = ⎪ ⎪ ⎪ ⎨ R30 √ ⎪ ⎪ 2nπ ⎪ ⎪ φn , Tn (0) = ⎪ ⎪ ⎩ R30 ∗ where λn = ( nπ R0 ) for n ∈ N The solution of this problem is √ Tn (t) = 2nπ R30 φn e−λn t a(s) ds + fn t t τ e−λn a(s) ds dτ , t τ a(s) ds n = 1, 2, (10) Thus, we have ∞ φn e−λn u(r, t) = t a(s) ds + fn n=1 t e−λn dτ ωn (r) (11) Using the final value data u(r, T) = g(r), it can be obtained that gn = g(r), ωn (r) = φn e−λn Denote hn = gn − φn e−λn T a(s) ds ∞ Kf (r) = n=1 T T a(s) ds + fn T e−λn T τ a(s) ds dτ (12) and define the linear operator K : f (·) −→ h(·), e−λn T τ a(s) ds dτ f (r), ωn (r) ωn (r) = h(r) It is easy to show that the operator K is a linear self-adjoint operator with singular values T −λn e T τ a(s) ds dτ and eigenfunctions ωn (r) According to the property of the INVERSE PROBLEMS IN SCIENCE AND ENGINEERING eigenfunctions ωn (r), we have ∞ hn f (r) = T τ T −λn e n=1 a(s) ds dτ ωn (r) (13) From Lemma 2.1, we find T τ T −λn e a(s) ds ≥ dτ n2 → ∞, C n → ∞ Therefore, it can be seen from (13) that small error in the measured data will lead to great changes in the source term f (r), so the inverse problem is ill-posed In order to ensure the stability of the solution, we assume that an a priori bound of the source term f (r) is known as f (r) where the norm f (·) p p ≤ E, p > 0, (14) is defined as ∞ f (·) p p (1 + n2 ) (f (·), ωn (·))ωn (·) = n=1 Theorem 3.1: If f (r) satisfies f (r) p ≤ E, then we have f (r) ≤ C1 E p+2 h(r) where C1 = C p p+2 p > 0, , (15) p − p+2 is a constant Proof: From Hölder inequality and (13), we have f (r) ∞ ∞ fn2 = = n=1 ⎛ n=1 ( T −λn e ∞ ≤⎝ n=1 T τ a(s) ds T −λn e T τ a(s) ds dτ )2 ⎞ p+2 h2n ( p+2 ∞ h2n dτ )p+2 ⎠ 2p p+2 hn = n=1 ( T −λn e T τ a(s) ds dτ )2 hn p p+2 ∞ h2n (16) n=1 Applying Lemma 2.1, we obtain ∞ h2n n=1 ( ≤ ∞ ≤ h2n n2p T p τ a(s) ds dτ )2 C T −λn T a(s) ds T τ dτ )p+2 n=1 ( e−λn e ∞ fn2 (1 + n2 )p C−p = C−p f (r) p ≤ C−p E2 n=1 (17) X GENG ET AL Combining (16) and (17), it is easy to get that f (r) ≤C 2p − p+2 E p+2 h(r) 2p p+2 Remark 3.1: According to the proof of Theorem 3.1, we can also obtain f (r) ≤ C1 f (r) p+2 p p p+2 Kf (r) , (18) which means that we can easily get the bound as the L2 -norm of f (r) by estimating f (r) and the L2 -norm of Kf (r) p An iterative regularization method and error estimates In this section, we will obtain the regularized approximate solution of the source term f (r) by using an iterative regularization method, and give the error estimates between the exact solution and its approximation under the a priori and a posteriori parameter choice rules, respectively We solve the inverse problem by constructing the following direct problem ⎧ ⎪ ⎪ ukt (r, t) = a(t) ukrr (r, t) + ukr (r, t) + f k,δ (r), < r < R0 , < t < T, ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎨ k u (r, 0) = φ(r), ≤ r ≤ R0 , (19) ⎪ ⎪ uk (R , t) = 0, ≤ t ≤ T, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim uk (r, t) bounded, ≤ t ≤ T, r→0 where the source term f k,δ (r) is given by the following iteration f 0,δ (r) = 0, f k,δ (r) = f k−1,δ (r) − s uk−1 (r, T) − g δ (r) , k = 1, 2, 3, , (20) where k is the number of iterations, equivalent to the regularization parameter, and s is an T acceleration factor satisfying < s e−λn τ a(s) ds dτ < for all n ∈ N Denoting fnk,δ = (f k,δ (r), ωn (r)), we can easily obtain that T ∞ uk (r, t) = φn e−λn t a(s) ds + fnk,δ n=1 For simplicity, we denote σn = s becomes ∞ f k,δ (r) = fnk−1,δ − s φn e−λn T −λn e T a(s) ds T τ t = a(s) ds n=1 a(s) ds T dτ ωn (r) dτ , and the iteration scheme (20) + fnk−1,δ (1 − σn ) fnk−1,δ + s gnδ − φn e−λn t τ n=1 ∞ e−λn T e−λn T τ a(s) ds a(s) ds ωn (r) dτ − gnδ ωn (r) INVERSE PROBLEMS IN SCIENCE AND ENGINEERING ∞ = (1 − σn ) fnk−1,δ + shδn ωn (r) n=1 ∞ k−1 (1 − σn )k fn0,δ + = n=1 (1 − σn )i shδn ωn (r) i=0 ∞ k−1 n=1 i=0 = (1 − σn )i shδn ωn (r) (21) 4.1 Error estimate under the a priori regularization parameter choice rule Theorem 4.1: Let f (r) be the exact solution and f k,δ (r) be the regularized approximate solution given by (21) Assuming that the a priori bound (14) and the noise assumption (4) are both true, if we choose k= s p+2 E δ , (22) we can obtain the following error estimate p p f k,δ (r) − f (r) ≤ + θ p C− E p+2 δ p+2 , (23) where [x] represents a maximum integer not exceeding x Proof: Applying the triangle inequality, we have f k,δ (r) − f (r) ≤ f k,δ (r) − f k (r) + f k (r) − f (r) (24) According to (21), (4) and (6) (taking μ = 0), it can be obtained that ∞ f k,δ (r) − f k (r) = pk (σn )s(hδn − hn )ωn (r) = n=1 ∞ pk (σn )s(gnδ − gn )ωn (r) n=1 δ ≤ sup pk (σn )s g (r) − g(r) ≤ δ sup pk (σn )s ≤ skδ n∈N n∈N p From the a priori bound (14), Lemma 2.1 and (7) (letting v = ), there holds ∞ f k (r) − f (r) = ∞ hn pk (σn )shn ωn (r) − n=1 n=1 ∞ = (−rk (σn )) n=1 hn T −λn e ∞ = rk (σn )fn ωn (r) n=1 T −λn e T τ a(s) ds T τ a(s) ds ωn (r) dτ ωn (r) dτ (25) X GENG ET AL ∞ rk2 (σn )fn2 = ∞ = n=1 rk2 (σn )(1 + n2 )−p (1 + n2 )p fn2 n=1 ≤ sup rk (σn )(1 + n2 ) (1 + n2 )p fn2 n∈N n=1 ≤ E sup rk (σn )n −p ⎛ ≤ E sup ⎝rk (σn )C n∈N = E sC ∞ p −2 n∈N p −2 T p −2 e T −λn τ a(s) ds p dτ ⎞ ⎠ p sup rk (σn )σn2 ≤ θ p sC p −2 n∈N p E (k + 1)− (26) Combining (25) and (26) and choosing k = [ 1s ( Eδ ) p+2 ], we obtain (23) 4.2 Error estimate under the a posteriori regularization parameter choice rule In Section 4.1, the regularization parameter k is chosen by (22), but in fact the a priori bound E is usually unknown or cannot be exactly obtained Therefore, we consider the a posteriori rule that does not depend on the a priori information to choose the regularization parameter Applying the discrepancy principle, a posteriori regularization parameter k is chosen to satisfy Kf k,δ (r) − hδ (r) ≤ τ δ < Kf k−1,δ (r) − hδ (r) , (27) where τ > is a constant Theorem 4.2: Let f (r) be the exact solution and f k,δ (r) be the regularized approximate solution given by (21) Assume that the a priori bound (14) and the noise assumption (4) are both valid, and the regularization parameter k is chosen in (27) Then, ⎛ ⎛ p ⎜ f k,δ (r) − f (r) ≤ ⎝C1 (τ + 1) p+2 + ⎝ p C− θ p+2 τ −1 ⎞ ⎞ p+2 p ⎠ ⎟ ⎠ E p+2 δ p+2 (28) Proof: Applying the triangle inequality, it can be obtained that f k,δ (r) − f (r) ≤ f k,δ (r) − f k (r) + f k (r) − f (r) (29) From (25), we know f k,δ (r) − f k (r) ≤ skδ (30) INVERSE PROBLEMS IN SCIENCE AND ENGINEERING According to (27), Lemma 2.1 and (4), there holds ∞ τ δ < Kf k−1,δ (r) − hδ (r) = rk−1 (σn )hδn ωn (r) n=1 ∞ ≤ rk−1 (σn ) hδn − hn ωn (r) + n=1 ∞ rk−1 (σn )hn ωn (r) n=1 ∞ T rk−1 (σn )fn ≤δ+ ∞ T rk−1 (σn ) e−λn T ≤ δ + E sup rk−1 (σn ) ≤ δ + E sup ⎝rk−1 (σn )C (sk) T τ e−λn p dτ n−p (1 + n2 ) fn ωn (r) a(s) ds T p −2 e n∈N p a(s) ds dτ ωn (r) dτ n−p ⎛ ≤ δ + C− θ p+2 E T τ n=1 n∈N a(s) ds n=1 ≤δ+ T τ e−λn p+2 T −λn τ a(s) ds p+2 dτ ⎞ ⎠ (31) From (31), we have ⎛ sk ≤ ⎝ p C− θ p+2 τ −1 ⎞ p+2 E δ ⎠ p+2 (32) Substituting (32) into (30), we obtain ⎛ f k,δ (r) − f k (r) ≤ ⎝ p C− θ p+2 τ −1 ⎞ p+2 ⎠ p E p+2 δ p+2 (33) On the other hand, ∞ k K f (r) − f (r) = rk (σn )hn ωn (r) n=1 ∞ ≤ rk (σn ) hn − hδn ωn (r) + n=1 ∞ rk (σn )hδn ωn (r) n=1 ≤ δ + τ δ = (1 + τ )δ (34) According to the a priori bound of f (r) and the definition of norm f (·) ∞ f k (r) − f (r) p = rk (σn ) n=1 hn T −λn e T τ p a(s) ds p , we know (1 + n2 ) ωn (r) dτ 10 X GENG ET AL ∞ p fn (1 + n2 ) ωn (r) = f (r) ≤ p ≤ E (35) n=1 So from Remark 3.1, we deduce that p p f k (r) − f (r) ≤ C1 (τ + 1) p+2 E p+2 δ p+2 (36) Combining (33) and (36), we can easily obtain the desired result Numerical implementation and numerical examples In this part, we provide the numerical experiments of the above iterative regularization method Considering the difficulty in getting the exact solution of the direct problem, we use the finite difference method to solve the direct problem According to the given functions a(t), φ(r) and f (r), the final value data g(r) can be obtained In the numerical experiments, the numbers of discrete points in time and space are N + and M + 1, respectively, and the corresponding step sizes are τ = NT and h = RM0 Denote tn = nτ (n = 0, 1, , N) and ri = ih (i = 0, 1, , M), and the values of each grid point are uni ≈ u(ri , tn ) We first present the numerical solution of the direct problem The derivatives of time and space terms are approximated by the following schemes, respectively: n u − un−1 , i τ i urr (ri , tn ) ≈ uni+1 − 2uni + uni−1 , h ut (ri , tn ) ≈ ur (ri , tn ) ≈ n u − uni−1 2h i+1 In this way, the discrete format of equation ut (r, t) = a(t)[urr (r, t) + 2r ur (r, t)] + f (r) can be obtained as follows n 1 ui − un−1 un − uni−1 = a(tn ) uni+1 − 2uni + uni−1 + i τ h ri h i+1 + f (ri ), (37) where i = 1, 2, , M − 1, n = 1, 2, , N Therefore, the matrix equation corresponding to the discrete format is AU n = h2 n−1 U + h2 F + B, τ n = 1, 2, , N, (38) where vector F = f (r1 ), f (r2 ), , f (rM−1 ) B = −a(tn ) T , h h − un0 , 0, , 0, a(tn ) + r1 rM−1 U n = (un1 , un2 , , unM−1 )T , n = 1, 2, , N, unM T , INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 11 Matrix A is a tridiagonal matrix ⎤ ⎡ h2 h 0 ··· τ + 2a(tn ) −a(tn )(1 + r1 ) ⎥ ⎢ ⎥ ⎢a(tn )( rh − 1) hτ + 2a(tn ) −a(tn )(1 + rh ) ··· 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ h h2 h ⎥ )( − 1) + 2a(t ) −a(t )(1 + ) · · · a(t n n n ⎣ rM−2 τ rM−2 ⎦ h − 1) hτ + 2a(tn ) ··· 0 a(tn )( rM−1 Next, we give the numerical solution of the inverse problem From (21), we have ∞ pk (σn )shδn ωn (r) = f k,δ (r) = n=1 ∞ ∞ pk (σn )s gnδ − φn e−λn a(s) ds ωn (r) n=1 pk (σn )s (g δ (r), ωn (r)) − (φ(r), ωn (r))e−λn = T T a(s) ds ωn (r) (39) n=1 We write (g δ (r), ωn (r)) and (φ(r), ωn (r)) in the form of numerical integration, and then apply composite Simpson rule R0 δ (g (r), ωn (r)) = T 0 M1 r g (r)ωn (r) dr ≈ ri2 g δ (ri )ωn (ri )yi , i=0 R0 (φ(r), ωn (r)) = δ M1 r2 φ(r)ωn (r) dr ≈ ri2 φ(ri )ωn (ri )yi , i=0 N1 a(s) ds ≈ ⎧ R0 ⎪ ⎪ , ⎪ ⎪ 3M1 ⎪ ⎪ ⎨ 4R , yi = ⎪ 3M1 ⎪ ⎪ ⎪ 2R0 ⎪ ⎪ ⎩ , 3M1 a(tn )qn , n=0 ⎧ T ⎪ ⎪ , ⎪ ⎪ 3N1 ⎪ ⎪ ⎨ 4T qn = , ⎪ 3N1 ⎪ ⎪ ⎪ 2T ⎪ ⎪ ⎩ , 3N1 i = 0, M1 , i = 1, 3, , M1 − 1, i = 2, 4, , M1 − 2, n = 0, N1 , n = 1, 3, , N1 − 1, n = 2, 4, , N1 − In the calculation, taking N2 as the truncation number of accumulated sum, the numerical solution of the regularized approximate solution f k,δ (r) can be written as follows N2 f k,δ (rj ) = M1 pk (σn )s n=1 i=0 ri2 g δ (ri )ωn (ri )yi M1 − ri2 φ(ri )ωn (ri )yi e−λn i=0 N1 n=0 a(tn )qn ωn (rj ) (40) Noisy data is generated by adding a random disturbance, i.e g δ = g + g · (2rand(size(g)) − 1), where > is relative error level, δ = g δ − g (41) 12 X GENG ET AL To illustrate the accuracy of the regularized approximate solution, we calculate the absolute error ea (f , ) = f k,δ (r) − f (r) (42) and the relative error er (f , ) = f k,δ (r) − f (r) f (r) (43) In the following numerical examples, we always take R0 = 3, T = 1, N = 100, M = 50, a(t) = 2−t The regularization parameter k under the a priori rule is given by (22), Figure The exact solution and its regularized approximate solution for Example 5.1 (a) The initial value and source term (a) The final value (a) A priori rule and (a) A posteriori rule Table Numerical results of Example 5.1 for different ea (f , er (f , ea (f , er (f , )priori )priori )posteriori )posteriori 0.05 0.01 0.001 0.1968 0.0595 0.2073 0.0627 0.0943 0.0285 0.0789 0.0238 0.0718 0.0217 0.0685 0.0207 INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 13 and the regularization parameter k under the a posteriori rule is chosen in (27), where τ = 1.1 Example 5.1: Take φ(r) = sin( 2π r), and the smooth source term is f (r) = e−r (44) √ Example 5.2: Take φ(r) = 0.3r + r, and the non-smooth but continuous source term is f (r) = 2r, ≤ r < 1.5, − 2r, 1.5 ≤ r ≤ 3, (45) Figure The exact solution and its regularized approximate solution for Example 5.2 (a) The initial value and source term (b) The final value (c) A priori rule and (d) A posteriori rule Table Numerical results of Example 5.2 for different ea (f , er (f , ea (f , er (f , )priori )priori )posteriori )posteriori 0.05 0.01 0.001 0.9530 0.0778 0.9561 0.0780 0.5781 0.0472 0.5441 0.0444 0.3947 0.0322 0.3142 0.0256 14 X GENG ET AL Example 5.3: Take φ(r) = −r2 + 3r, and the discontinuous source term is ⎧ 0, ⎪ ⎨ f (r) = 3, ⎪ ⎩ 0, ≤ r < 1, ≤ r ≤ 2, (46) < r ≤ 3, Figures 1–3 show the numerical comparisons between the exact solution and its regularized approximate solution with different noise levels for Examples 5.1–5.3 under the a priori and a posteriori regularization parameter choice rules, respectively Tables 1–3 show Figure The exact solution and its regularized approximate solution for Example 5.3 (a) The initial value and source term (b) The final value (c) A priori rule and (d) A posteriori rule Table Numerical results of Example 5.3 for different ea (f , er (f , ea (f , er (f , )priori )priori )posteriori )posteriori 0.05 0.01 0.001 3.3725 0.2726 3.5223 0.2848 2.8147 0.2276 2.8294 0.2287 2.0729 0.1676 2.0433 0.1652 INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 15 the numerical errors for Examples 5.1–5.3 with different noise levels under the two choice rules, respectively From Figures 1–3, we can find that the smaller the error is, the better the approximation effect is What is more, in these three different examples, the numerical results of smooth source term are the best, the numerical results of the non-smooth but continuous source term are slightly worse, and the numerical results of the discontinuous source term are the worst Last but not least, we can also find that the numerical results under the a posteriori regularization parameter choice rule are even comparable to those under the a priori regularization parameter choice rule The numerical results are all consistent with our theoretical analysis Conclusion In this paper, we identify the unknown source term of heat conduction equation with timedependent diffusivity by additional final value data To overcome the difficulties caused by the ill-posedness, an iterative regularization method is applied to solve this problem Based on the conditional stability, we obtain the Hölder type error estimates under the a priori and a posteriori regularization parameter choice rules, respectively Numerical results show the stability and effectiveness of our method in reconstructing smooth and non-smooth source terms Acknowledgments The authors are grateful for the valuable hints and meaningful suggestions of the Professor Daniel Lesnic and the anonymous referee which have significantly improved the paper Disclosure statement No potential conflict of interest was reported by the authors Funding The project is supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province [KYCX20_1921] References [1] Li GS, Tan YJ, Cheng J, et al Determining magnitude of groundwater pollution sources by data compatibility analysis Inverse Probl Sci Eng 2006;14(3):287–300 [2] Wang ZW, Qiu SF Stability and numerical simulation of pollution point source identification in a watershed Chin J Hydrodyn 2008;23(4):364–317 [3] Dou FF, Fu CL Determining an unknown source in the heat equation by a wavelet dual least squares method Appl Math Lett 2009;22(5):661–667 [4] Wei T, Zhang ZQ Reconstruction of a time-dependent source term in a time-fractional diffusion equation Eng Anal Bound Elem 2013;37(1):23–31 [5] Ismailov MI, Kanca F, Lesnic D Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions Appl Math Comput 2011;218(8):4138–4146 [6] Ma YJ, Fu CL, Zhang YX Identification of an unknown source depending on both time and space variables by a variational method Appl Math Model 2012;36:5080–5090 16 X GENG ET AL [7] Ail MN, Aziz S, Malik SA Inverse source problems for a space-time fractional differential equation Inverse Probl Sci Eng 2020;28(1):47–68 [8] Tuan NH, Long LD Fourier truncation method for an inverse source problem for space-time fractional diffusion equation Electron J Differ Equ 2017;2017(122):1–16 [9] Cheng J, Liu JJ An inverse source problem for parabolic equations with local measurements Appl Math Lett 2020;103:106213 [10] Cheng Q, Xiong XT An iterative method for an inverse source problem of a time-fractional diffusion equation Math Numerica Sinica 2017;39(3):295–308 [11] Qiu SF, Wang ZW, Zeng XL, et al The numerical method for reconstructing source term in a time fractional diffusion equation J Jiangxi Normal Univ (Nat Sci) 2018;42(6):610–615 [12] Wen J, Yamamoto M, Wei T Simultaneous determination of a time-dependent heat source and the initial temperature in an inverse heat conduction problem Inverse Probl Sci Eng 2013;21(3):485–499 [13] Yang L, Deng ZC, Yu JN, et al Two regularization strategies for an evolutional type inverse heat source problem Phys A Math Theoret 2009;42(36):365203 [14] Wolf RA, Farley KA, Kass DM Moldeling of the temperature sensitivity of the apatite (U-Th)/He thermochronometer Chem Geol 1998;148:105–114 [15] Ladyženskaja QA, Solonnikov VA, Ural’ceva NN Linear and quasilinear equations of parabolic type Translations of Mathematical Monographs vol.23 Providence (RI): American Mathematical Society; 1967 [16] Cannon J, Pérez-Esteva S Uniqueness and stability of 3D heat sources Inverse Probl 1991;7(1):57–62 [17] Bushuyev I Global uniqueness for inverse parabolic problems with final observation Inverse Probl 1995;11(4):L11–L16 [18] Bao G, Ehlers TA, Li PJ Radiogenic source identification for the helium production-diffusion equation Commun Comput Phys 2013;14(1):1–20 [19] Cheng W, Fu CL Identifying an unknown source term in a spherically symmetric parabolic equation Appl Math Lett 2013;26(4):387–391 [20] Yang F, Zhang M, Li XX, et al A posteriori truncated regularization method for identifying unknown heat source on a spherical symmetric domain Adv Differ Equ 2017;2017(1):1–11 [21] Zhang YX, Yan L The general a posteriori truncation method and its application to radiogenic source identification for the Helium production-diffusion equation Appl Math Molel 2017;43:126–138 [22] Wang JG, Ren RH An iterative method for an inverse source problem of time-fractional diffusion equation Inverse Probl Sci Eng 2018;26(10):1509–1521 [23] Deng YJ, Liu ZH Iteration methods on sideways parabolic equations Inverse Probl 2009;25(9):095004 [24] Liu ZK Orthogonal function and its application Beijing: Natioal Defence Industry Press; 1982 ... [15] Ladyženskaja QA, Solonnikov VA, Ural’ceva NN Linear and quasilinear equations of parabolic type Translations of Mathematical Monographs vol.23 Providence (RI): American Mathematical Society;... of these works were only devoted to the heat conduction equation with constant coefficient In practical engineering, the variable case of diffusivity has more applications than the constant one... with our theoretical analysis Conclusion In this paper, we identify the unknown source term of heat conduction equation with timedependent diffusivity by additional final value data To overcome the