A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem

Wei Cheng; Yi-Liang Liu

doi:10.1515/math-2023-0171

Article Open Access

A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem

Wei Cheng and Yi-Liang Liu

Published/Copyright: December 31, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we investigate a spherically symmetric backward heat conduction problem, starting from the final temperature. This problem is severely ill posed: the solution (if it exists) does not depend continuously on the final data. A conditional stability result of its solution is given. Further, we propose a quasi-boundary value regularization method to solve this ill-posed problem. Two Hölder type error estimates between the approximate solution and its exact solution are obtained under an a priori and an a posteriori regularization parameter choice rule, respectively.

Keywords: ill-posed problem; backward heat equation; regularization; error estimate; Morozov's; discrepancy principle

MSC 2010: 65M30; 35R25; 35R30

1 Introduction

In many physical and engineering applications, it is sometimes necessary to determine a temperature distribution at 0 ≤ t < T from a measured temperature at a fixed time t = T > 0 in a heat conduction body. For example, in practice, people are often faced with such a problem, how to set the temperature of a bar at an earlier time so that it can reach an expected temperature distribution later. This problem is called backward heat conduction problem (BHCP) or final boundary value problem.

The BHCP is a severely ill-posed problem in the sense of Hadamard [1]. That is, in general, the solution does not always exist, and even if a solution exists, it will not be continuously dependent on the final data. The numerical simulation is very difficult because of the instability of small error of measurement data. Therefore, some special regularization methods and stability estimates are needed.

As we know, the BHCPs have been studied in quite sizeable literature [2,3]. For instance, the numerical methods have been investigated in previous studies [4–11]. The conditional stability has been considered in previous studies [12–16]. Many regularization methods with convergence estimates have been proposed. These regularization methods include the quasi-reversibility regularization method [17–20], the mollification method [21], the difference method [22], the optimal filtering method [23], the Fourier regularization [24], the truncation method [25–28], the variational method [29], the wavelet method [30], the quasi-boundary method [31], the homotopy analysis method [32], and a compact filtering method [33]. It is worth mentioning that Cheng and Liu [34] and Liu [35] investigated the 2D BHCP. However, to our knowledge, the results available in the literature on BHCP are mainly devoted to the heat equation in the Cartesian coordinates system. Very few works are developed for spherically symmetric (or axisymmetric) backward heat equation in the spherical coordinates system (or cylindrical coordinates system). Yang et al. [36] investigated a spherically symmetric initial value of heat equation in spherical coordinates system by the truncation regularization method. Cheng and Fu [37] solved a axisymmetric backward heat equation in the cylindrical coordinates system by the spectral regularization method and gave two quite sharp error estimates under an a priori regularization parameter choice rule.

In this article, we investigate a spherically symmetric BHCP, where the spherically symmetric heat equation in a sphere is given by

(1.1) u t = u r r + 2 r u r , 0 < r < a , 0 < t < T ,

which is independent of the zenith angle φ and azimuth angle θ , with the Cauchy final condition at t = T

(1.2) u ( r , T ) = g ( r ) , 0 < r < a ,

and the boundary conditions

(1.3) u ( a , t ) = 0 , lim r → 0 u ( r , t ) is bounded , 0 < t < T ,

where r denotes the radial distance. We want to recover the temperature distribution u ( ⋅ , t ) for 0 < t < T from the final temperature distribution g ( ⋅ ) . The BHCP given by equations (1.1)–(1.3) is ill posed (the details can be seen in Section 2). In this article, we will apply a quasi-boundary value method to solve the BHCP (1.1)–(1.3).

The quasi-boundary value method, also called the nonlocal boundary value problem method in the study by Hào et al. [38], is a regularization technique by replacing the final condition or the boundary condition by a new approximate condition. This regularization method has been used to solve the inverse source identification problem [39,40], the inverse heat conduction problem [41], the Cauchy problem for elliptic equations [42], and the BHCP [43,44]. Zhang [45] applied a modified quasi-boundary value method to solve a Cauchy problems of elliptic equations with variable coefficients. Jiang and Liu [46] used two new quasi-boundary value methods for regularizing the BHCP.

We note that most of the articles on BHCP regularization methods focus on giving convergence analysis with an a priori regularization parameter choice rule. In this article, we use the quasi-boundary value method to deal with the BHCP (1.1)–(1.3) and provide convergence analysis by using an a posteriori and an a priori regularization parameter choice rule, respectively.

This article is organized as follows: Section 2 provides the ill-posedness and the conditional stability for BHCP (1.1)–(1.3). In Section 3, we propose a quasi-boundary value method to formulate regularized solution. The convergence estimates between the approximate solution and its exact solution are obtained under an a priori and an a posteriori regularization parameter choice rule, respectively.

2 Ill-posedness and conditional stability

In this section, we provide the ill-posedness and the conditional stabilities for problems (1.1)–(1.3). First, we present the ill-posedness.

In this article, L 2 [ 0 , a ; r 2 ] denotes the Hilbert space of Lebesgue measurable functions v with weight r 2 on [ 0 , a ] . Throughout this article, we denote by ( ⋅ , ⋅ ) and ‖ ⋅ ‖ the inner and norm on L 2 [ 0 , a ; r 2 ] , respectively, with the norm

‖ v ‖ = ∫ 0 a r 2 ∣ v ( r ) ∣ 2 d r 1 ∕ 2 .

If a solution of problems (1.1)–(1.3) exists, then it must be unique, which is given in the following lemma:

Lemma 1

If the given fixed value g ( r ) ∈ L 2 [ 0 , a ; r 2 ] and ∑ n = 1 ∞ e 2 n π a 2 ( T − t ) ∣ ( g , ψ n ) ∣ 2 < ∞ , then a solution of problems (1.1)–(1.3) exists and is given by

(2.1) u ( r , t ) = ∑ n = 1 ∞ b n e n π a 2 ( T − t ) j 0 n π a r ,

where

(2.2) b n = 2 ( n π ) 2 a 3 ∫ 0 a r 2 g ( r ) j 0 n π a r d r

and j 0 ( x ) denotes the spherical Bessel function of the first kind [47]

j 0 ( x ) = sin x x .

Proof

Applying separation of variable, a solution of problems (1.1)–(1.3) with the following form is sought.

(2.3) u ( r , t ) = w ( t ) h ( r ) .

By substituting (2.3) into equation (1.1), we discover that w ( t ) satisfies the following equation:

(2.4) w ′ ( t ) + λ w ( t ) = 0 , 0 < t < T ,

and function h ( r ) satisfies

(2.5) h ″ ( r ) + 2 r h ′ ( r ) + λ h ( r ) = 0 , 0 < r < a ,

where λ is an unknown constant. By substituting (2.3) into conditions (1.3), we have

u ( a , t ) = w ( t ) h ( r ) = 0 , lim r → 0 u ( r , t ) = lim r → 0 w ( t ) h ( r ) is bounded , 0 < t < T ,

and then we obtain the boundary conditions

(2.6) h ( a ) = 0 , ∣ h ( 0 ) ∣ < + ∞ .

According to the study by Abramowitz and Stegun [47], we have the general solution of equation (2.5)

(2.7) h ( r ) = A j 0 ( r λ ) + B y 0 ( r λ ) , 0 < r < a ,

where y 0 ( x ) denotes the spherical Bessel function of the second kind

y 0 ( x ) = − cos x x .

From conditions (2.6) and lim x → 0 y 0 ( x ) = − ∞ , we obtain B = 0 , i.e.,

h ( r ) = A ( sin ( r λ ) ) ∕ ( r λ ) ,

and

(2.8) sin ( a λ ) = 0 .

So, the eigenvalues of problems (2.5) and (2.6) are

λ n = n π a 2 , n = 1 , 2 , … ,

and the corresponding eigenfunctions are

(2.9) h n ( r ) = j 0 n π a r , n = 1 , 2 , … .

Then, the corresponding solutions of equation (2.4) are

w n ( t ) = a n e − n π a 2 t , n = 1 , 2 , … .

We have the usual “forward representation” of problems (1.1)–(1.3)

u ( r , t ) = ∑ n = 1 ∞ a n e − n π a 2 t j 0 n π a r .

Let t = T , we obtain the expansions

(2.10) u ( r , T ) = ∑ n = 1 ∞ b n j 0 n π a r with b n = e − n π a 2 T a n ,

then we obtain formula (2.1). According to the properties of function j 0 ( x ) , the system of eigenfunctions { j 0 n π a r } n = 1 + ∞ are complete, and weighted orthogonal with weight r 2 in L 2 [ 0 , a ; r 2 ] [47]. Thus, we have formula (2.2).□

Let

(2.11) ψ n ( r ) = 2 n π a 3 j 0 n π a r ,

then the system of eigenfunctions { ψ n ( r ) } n = 1 + ∞ are complete, and weighted orthonormal with weight r 2 in L 2 [ 0 , a ; r 2 ] . Combining with (2.2) and (2.11), we can rewrite formula (2.1) as follows:

(2.12) u ( r , t ) = ∑ n = 1 ∞ e n π a 2 ( T − t ) ( g ( r ) , ψ n ( r ) ) ψ n ( r ) .

Since measurement errors exist in g , the solution has to be reconstructed from noisy data g δ , which is assumed to satisfy

(2.13) ‖ g ( ⋅ ) − g δ ( ⋅ ) ‖ ≤ δ ,

where the constant δ > 0 is a noise level.

We also assume that there exists an a priori condition for problems (1.1)–(1.3)

(2.14) ‖ u ( ⋅ , 0 ) ‖ ≤ E ,

where E > 0 is constant.

We introduce an operator K ( t ) : u ( ⋅ , t ) → g ( ⋅ ) , and then we can rewrite problems (1.1)–(1.3) as the following operator equation:

(2.15) K ( t ) u ( r , t ) = g ( r ) , 0 < t < T .

From formula (2.12), we have

( g ( r ) , ψ n ( r ) ) = e − n π a 2 ( T − t ) ( u ( r , t ) , ψ n ( r ) ) ,

then there holds

(2.16) K ( t ) u ( r , t ) = ∑ n = 1 ∞ e − n π a 2 ( T − t ) ( u ( r , t ) , ψ n ( r ) ) ψ n ( r ) .

The operator K ( t ) : L 2 [ 0 , a ; r 2 ] → L 2 [ 0 , a ; r 2 ] is a linear self-adjoint compact operator with eigenelements ψ n and eigenvalues

(2.17) k n ( t ) = e − n π a 2 ( T − t ) .

We can see that the eigenvalues k n ( t ) of the operator K ( t ) decay exponentially fast. Therefore, problems (1.1)–(1.3) are a severely ill-posed problem.

Next, we provide the conditional stabilities.

Theorem 1

Let u ( r , t ) , given by (2.12), be the exact solution of BHCP (1.1)–(1.3) with the exact data g ( r ) , and the a-priori condition (2.14) be valid, then there holds the following estimate for a fixed t ∈ ( 0 , T )

(2.18) ‖ u ( ⋅ , t ) ‖ ≤ E 1 − t T ‖ g ‖ t T .

Proof

Using (2.12), (2.14), and the Hölder inequality, there holds

‖ u ( ⋅ , t ) ‖ 2 = ∑ n = 1 ∞ e n π a 2 ( T − t ) ( g , ψ n ) ψ n 2 = ∑ n = 1 ∞ e n π a 2 ( T − t ) 2 ∣ ( g , ψ n ) ∣ 2 = ∑ n = 1 ∞ e 2 n π a 2 ( T − t ) ∣ ( g , ψ n ) ∣ 2 ( 1 − t T ) ∣ ( g , ψ n ) ∣ 2 t T ≤ ∑ n = 1 ∞ e 2 n π a 2 ( T − t ) ∣ ( g , ψ n ) ∣ 2 ( 1 − t T ) 1 1 − t T 1 − t T ∑ n = 1 ∞ ∣ ( g , ψ n ) ∣ 2 t T T t t T = ∑ n = 1 ∞ e 2 n π a 2 T ∣ ( g , ψ n ) ∣ 2 1 − t T ∑ n = 1 ∞ ∣ ( g , ψ n ) ∣ 2 t T = ∑ n = 1 ∞ ∣ ( u ( r , o ) , ψ n ) ∣ 2 1 − t T ∑ n = 1 ∞ ∣ ( g , ψ n ) ∣ 2 t T ≤ E 2 ( 1 − t T ) ‖ g ‖ 2 t T .

This ends the proof.□

Remark 1

Assume u 1 ( r , t ) and u 2 ( r , t ) are the solutions of problems (1.1)–(1.3) with the exact data g 1 ( r ) and g 2 ( r ) , respectively, then there holds for a fixed t ∈ ( 0 , T ) ,

(2.19) ‖ u 1 ( ⋅ , t ) − u 2 ( ⋅ , t ) ‖ ≤ E 1 − t T ‖ g 1 ( ⋅ ) − g 1 ( ⋅ ) ‖ t T .

Note that if ‖ g 1 ( ⋅ ) − g 2 ( ⋅ ) ‖ → 0 , then ‖ u 1 ( ⋅ , t ) − u 2 ( ⋅ , t ) ‖ → 0 for 0 < t < T . Now we construct an example with the conditional stability.

Example 1

It is easy to verify that for 0 < r < a , 0 < t < T , the functions u 0 ( r , t ) ≡ 0 and u n ( r , t ) = e − n π a 2 t j 0 n π a r , n = 1 , 2 , … are the solutions of problems (1.1)–(1.3) with the exact data g 0 ( r ) ≡ 0 and g n ( r ) = e − n π a 2 T j 0 n π a r , respectively. Then we have

‖ g n ( r ) − g 0 ( r ) ‖ = ∫ 0 a r 2 e − n π a 2 T j 0 n π a r 2 d r 1 ∕ 2 = e − n π a 2 T ∫ 0 a r 2 sin n π r a n π r a 2 d r 1 ∕ 2 = a n π e − n π a 2 T ∫ 0 a 1 − cos n π r a 2 d r 1 ∕ 2 = a 3 2 2 π e − n π a 2 T 1 n , n = 1 , 2 , … .

Similarly, we obtain

‖ u n ( r , t ) − u 0 ( r , t ) ‖ = a 3 2 2 π e − n π a 2 t 1 n , n = 1 , 2 , … , 0 < t < T .

It is obviously that there holds simultaneously ‖ g n − g 0 ‖ → 0 , and ‖ u n ( ⋅ , t ) − u 0 ( ⋅ , t ) ‖ → 0 , for n → + ∞ .

This result of conditional stability does not guarantee the stability of numerical calculations with noisy data. Therefore, an effective regularization method is needed to solve problems (1.1)–(1.3).

3 Regularization method and error estimates

In this section, we first use a quasi-boundary value method to derive the approximate solution of the ill-posed problems (1.1)–(1.3) in interval 0 < t < T . Then provide two error estimates by choosing an a priori and an a posteriori regularization parameter, respectively.

The application of quasi-boundary value method on BHCP replaces the final condition with a new approximate condition. So we add a perturbation term in the final condition (1.2) and consider the following final condition:

(3.1) u ( r , T ) + β u ( r , 0 ) = g δ ( r ) , 0 < r < a ,

and here, β plays a role of regularization parameter.

Let u β δ ( r , t ) be the solution of the following regularized problem:

(3.2) u t = u r r + 2 r u r , 0 < r < a , 0 < t < T , u ( r , T ) + β u ( r , 0 ) = g δ ( r ) , 0 < r < a , u ( a , t ) = 0 , 0 < t < T , lim r → 0 u ( r , t ) is bounded , 0 < t < T .

Similar to the derivation of the formula (2.12), we obtain the formal solution of problem (3.2)

(3.3) u β δ ( r , t ) = ∑ n = 1 ∞ e n π a 2 ( T − t ) 1 + β e n π a 2 T ( g δ ( r ) , ψ n ( r ) ) ψ n ( r ) .

We call u β δ ( r , t ) given by (3.3) the quasi-boundary value approximation of the exact solution u ( r , t ) for problems (1.1)–(1.3).

To obtain stability estimate for the regularized solution, we need the following lemma.

Lemma 2

Let 0 < t < T , α > 0 , then there holds

(3.4) sup x ≥ 0 e x ( T − t ) 1 + β e x T ≤ α t − T T .

The proof of Lemma 2 is similar to that of Lemma 4 in [48].

Next, we provide two error estimates by choosing an a priori and an a posteriori regularization parameter.

3.1 Error estimate under an a priori regularization parameter choice rule

Theorem 2

Let u ( r , t ) , given by (2.12), be the exact temperature history and, u β δ ( r , t ) , given by (3.3), be the regularized approximation solution to u ( r , t ) . Suppose that the a priori condition (2.14) is valid and the measured data at t = T , g δ ( r ) , satisfy the condition (2.13). If we take β = δ E , then there holds error estimate for t ∈ ( 0 , T )

(3.5) ‖ u ( ⋅ , t ) − u β δ ( ⋅ , t ) ‖ ≤ 2 E 1 − t T δ t T .

Proof

Due to (2.12), (3.3), and the triangle inequality, we have

‖ u ( ⋅ , t ) − u β δ ( ⋅ , t ) ‖ = ∑ n = 1 ∞ e n π a 2 ( T − t ) ( g , ψ n ) ψ n − ∑ n = 1 ∞ e n π a 2 ( T − t ) 1 + β e n π a 2 T ( g δ , ψ n ) ψ n ≤ ∑ n = 1 ∞ β e n π a 2 ( 2 T − t ) 1 + β e n π a 2 T ( g , ψ n ) ψ n + ∑ n = 1 ∞ e n π a 2 ( T − t ) 1 + β e n π a 2 T ( g − g δ , ψ n ) ψ n ≤ sup n ∈ N + β e n π a 2 ( T − t ) 1 + β e n π a 2 T ∑ n = 1 ∞ ( u ( ⋅ , 0 ) , ψ n ) ψ n + sup n ∈ N + e n π a 2 ( T − t ) 1 + β e n π a 2 T ∑ n = 1 ∞ ( g − g δ , ψ n ) ψ n .

Let x = n π a 2 , with inequality (3.4), conditions (2.13), (2.14), and the choice of β = δ E , we obtain

‖ u ( ⋅ , t ) − u β δ ( ⋅ , t ) ‖ ≤ sup x > 0 β e x ( T − t ) 1 + β e x T ∥ u ( ⋅ , 0 ) ∥ + sup x > 0 e x ( T − t ) 1 + β e x T ∥ g − g δ ∥ ≤ β t T E + β t − T T δ = 2 E 1 − t T δ t T .

This proves estimate (3.5).□

From Theorem 2, we can see that estimate (3.5) is a Hölder stability estimate, and the error bound in (3.5) is consistent with the error bound in (3.4) in the study by Cheng and Fu [37].

3.2 Error estimate under an a posteriori regularization parameter choice rule

In this section, we apply the Morozov's discrepancy principle as a posteriori regularization parameter choice rule. Recalling the definition of Morozov's discrepancy principle, the classical Morozov's discrepancy principle chooses the regularization parameter β > 0 such that [49,50]

(3.6) ‖ K ( t ) u β δ − g δ ‖ = δ .

Scherze [51] extended Morozov's discrepancy principle, which chooses the regularization parameter β > 0 such that

(3.7) ‖ K ( t ) u β δ − g δ ‖ = τ δ ,

where 0 < t < T , τ > 1 is a constant. In this article, we select the regularization parameter β > 0 satisfied equation (3.7), because equation (3.6) will not fit in our framework.

To establish the existence and uniqueness of the solution to equation (3.7), we need the following lemma:

Lemma 3

Assume q ( β ) = ‖ K ( t ) u β δ − g δ ‖ , 0 < t < T . If ‖ g δ ‖ > δ > 0 , then there holds

q ( β ) is a continuous function;
lim β → 0 q ( β ) = 0 ;
lim β → + ∞ q ( β ) = ‖ g δ ‖ ;
q ( β ) is a strictly increasing function over ( 0 , ∞ ) .

Proof

From (2.16) and (3.3), there holds

(3.8) q ( β ) = ∑ n = 1 ∞ 1 1 + β e n π a 2 T ( g δ , ψ n ) ψ n − ∑ n = 1 ∞ ( g δ , ψ n ) ψ n = ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T 2 ∣ ( g δ , ψ n ) ∣ 2 1 2 .

According to (3.8), the results of Lemma 3 are straightforward.□

Theorem 3

Let u ( r , t ) , given by (2.12), be the exact temperature history and u β δ ( r , t ) , given by (3.3), be the regularized approximation solution to u ( r , t ) . Suppose conditions (2.13) and (2.14) are valid, and there exists τ > 1 such that ‖ g δ ‖ > δ > 0 . If the regularization parameter β > 0 is chosen by the Morozov’s discrepancy principle (3.7), then we have error estimate for a fixed t ∈ ( 0 , T )

(3.9) ‖ u ( ⋅ , t ) − u β δ ( ⋅ , t ) ‖ ≤ ( τ + 1 ) t T + ( τ − 1 ) t T − 1 E 1 − t T δ t T .

Proof

Using the triangle inequality, there holds

(3.10) ‖ u ( ⋅ , t ) − u β δ ( ⋅ , t ) ‖ ≤ ‖ u ( ⋅ , t ) − u β ( ⋅ , t ) ‖ + ‖ u β ( ⋅ , t ) − u β δ ( ⋅ , t ) ‖ ≔ I 1 + I 2 .

We give the estimates for I 1 and I 2 , respectively.

From (2.12), (3.3), and the Hölder inequality, we obtain

I 1 2 = ∑ n = 1 ∞ β e n π a 2 ( 2 T − t ) 1 + β e n π a 2 T ( g , ψ n ) ψ n 2 = ∑ n = 1 ∞ β e n π a 2 ( 2 T − t ) 1 + β e n π a 2 T 2 ∣ ( g , ψ n ) ∣ 2 = ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T 2 e n π a 2 ( T − t ) ( g , ψ n ) 2 ≤ ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ∣ ( g , ψ n ) ∣ 2 t T e n π a 2 T ( g , ψ n ) 2 ( T − t ) T ≤ ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ∣ ( g , ψ n ) ∣ 2 t T ∑ n = 1 ∞ e n π a 2 T ( g , ψ n ) 2 T − t T ≤ ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ( g , ψ n ) 2 t T ∥ u ( ⋅ , 0 ) ∥ 2 ( T − t ) T ,

combining with the a priori condition (2.14), we obtain

(3.11) I 1 ≤ ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ( g , ψ n ) t T E 1 − t T .

From (3.7) and (3.8), there holds

(3.12) τ δ = ‖ K ( t ) u β δ − g δ ‖ = ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ( g δ , ψ n ) ψ n ,

then we have

∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ( g , ψ n ) ψ n ≤ ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ( g − g δ , ψ n ) ψ n + ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ( g δ , ψ n ) ψ n ≤ ‖ g − g δ ‖ + τ δ = ( τ + 1 ) δ .

By combining with (3.11), we obtain

(3.13) I 1 ≤ [ ( τ + 1 ) δ ] t T E 1 − t T .

Next, we give the bound for the I 2 . With inequality (3.4), we have

(3.14) I 2 = ∑ n = 1 ∞ e n π a 2 ( T − t ) 1 + β e n π a 2 T ( g − g δ , ψ n ) ψ n ≤ sup n ∈ N + e n π a 2 ( T − t ) 1 + β e n π a 2 T ∑ n = 1 ∞ ( g − g δ , ψ n ) ψ n ≤ sup x > 0 e x ( T − t ) 1 + β e x T ∥ g − g δ ∥ ≤ β t − T T δ .

On the other hand,

τ δ = ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ( g δ , ψ n ) ψ n ≤ ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ( g δ − g , ψ n ) ψ n + ≤ ∑ n = 1 ∞ β e n π a 2 T 1 + β e n π a 2 T ( g , ψ n ) ψ n ≤ ‖ g − g δ ‖ + ∑ n = 1 ∞ β 1 + β e n π a 2 T ( u ( ⋅ , 0 ) , ψ n ) ψ n ≤ ‖ g − g δ ‖ + β ∥ ( u ( ⋅ , 0 ) ) ∥ ≤ δ + β E ,

this yields

(3.15) β ≥ δ E ( τ − 1 ) .

Substituting (3.15) into (3.14), we obtain

(3.16) I 2 ≤ δ E ( τ − 1 ) t − T T δ = ( τ − 1 ) t − T T E 1 − t T δ t T .

Combining (3.13) and (3.16) with (3.10), the error estimate (3.9) is obtained.□

Remark 2

(i) Estimate (3.9) is a Hölder stability estimate, and the error bound in (3.9) is consistent with the error bound in (3.7) in the study by Zhang et al. [28].

(ii) From Theorems 2 and 3, we know that the a priori parameter choice rule has the same convergence rate as the a posteriori parameter choice. However, such a priori information is rarely available in practice. Therefore, we prefer to use the a posteriori choice rule to obtain the regularization parameter.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions on this article.

Funding information: The work was supported by the National Natural Science Foundation of China (Grant Nos 11961044 and 11561045).
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflicts of interest.

References

[1] J. Hadamard, Lectures on the Cauchy Problems in Linear Partial Differential Equations, Yale University Press, New Haven, 1923. Search in Google Scholar

[2] U. Tautenhahn and T. Schröter, On optimal regularization methods for the backward heat equation, Z. Anal. Anwend. 15 (1996), 475–493. 10.4171/ZAA/711Search in Google Scholar

[3] F. Mostajeran and R. Mokhtari, DeepBHCP: Deep neural network algorithm for solving backward heat conduction problems, Comput. Phys. Commun. 272 (2022), 108236. 10.1016/j.cpc.2021.108236Search in Google Scholar

[4] H. Han, D. B. Ingham, and Y. Yuan, The boundary element method for the solution of the backward heat conduction equation, J. Comput. Phys. 116 (1995), 292–299. 10.1006/jcph.1995.1028Search in Google Scholar

[5] N. S. Mera, L. Elliott, D. B. Ingham, and D. Lesnic, An iterative boundary element method for solving the one dimensional backward heat conduction problem, Int. J. Heat Mass Transf. 44 (2001), 1937–1946. 10.1016/S0017-9310(00)00235-0Search in Google Scholar

[6] J. M. Marban and C. Palencia, A new numerical method for backward parabolic problems in the maximum-norm setting, SIAM J. Numer. Anal. 40 (2002), 1405–1420. 10.1137/S0036142901386422Search in Google Scholar

[7] N. S. Mera, The method of fundamental solutions for the backward heat conduction problem, Inverse Probl. Sci. Eng. 13 (2005), 65–78. 10.1080/10682760410001710141Search in Google Scholar

[8] X. T. Xiong, C. L. Fu, and Z. Qian, Two numerical methods for solving a backward heat conduction problem, Appl. Math. Comput. 179 (2006), 370–377. 10.1016/j.amc.2005.11.114Search in Google Scholar

[9] M. Li, T. S. Jiang, and Y. C. Hon, A meshless method based on RBFs method for nonhomogeneous backward heat conduction problem, Eng. Anal. Bound. Elem. 34 (2010), 785–792. 10.1016/j.enganabound.2010.03.010Search in Google Scholar

[10] Y. C. Hon and T. Takeuchi, Discretized Tikhonov regularization by reproducing kernel Hilbert space for backward heat conduction problem, Adv. Comput. Math. 34 (2011), 167–183. 10.1007/s10444-010-9148-1Search in Google Scholar

[11] L. D. Su, A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem, Appl. Math. Comput. 354 (2019), 232–247. 10.1016/j.amc.2019.02.035Search in Google Scholar

[12] J. R. Cannon and J. Douglas jr, The Cauchy problem for the heat equation, SIAM J. Numer. Anal. 4 (1967), 317–327. 10.1137/0704028Search in Google Scholar

[13] S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equation in Banach spaces, Commun. Pure Appl. Math. 16 (1963), 121–239. 10.1002/cpa.3160160204Search in Google Scholar

[14] W. L. Miranker, A well posed problem for the backward heat equation, Proc. Amer. Math. Soc. 12 (1961), 243–254. 10.1090/S0002-9939-1961-0120462-2Search in Google Scholar

[15] D. N. Hào and N. V. Duc, Stability results for backward parabolic equations with time-dependent coefficients, Inverse Probl. 27 (2011), 025003. 10.1088/0266-5611/27/2/025003Search in Google Scholar

[16] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Probl. 25 (2009), 123013. 10.1088/0266-5611/25/12/123013Search in Google Scholar

[17] R. Lattés and J. L. Lions, Methode de Quasi-Reversibility et Applications, Dunod, Paris, 1967. Search in Google Scholar

[18] R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl. 47 (1974), 563–572. 10.1016/0022-247X(74)90008-0Search in Google Scholar

[19] K. A. Ames, G. W. Clark, J. F. Epperson, and S. F. Oppenthermer, A comparison of regularizations for an ill-posed problem, Math. Comp. 67 (1998), 1451–1471. 10.1090/S0025-5718-98-01014-XSearch in Google Scholar

[20] Y. Huang and Q. Zheng, Regularization for ill-posed Cauchy problems associated with generators of analytic semigroups, J. Differential Equations 203 (2004), 38–54. 10.1016/j.jde.2004.03.035Search in Google Scholar

[21] D. N. Hào, A mollification method for ill-posed problems, Numer. Math. 68 (1994), 469–506. 10.1007/s002110050073Search in Google Scholar

[22] X. T. Xiong, C. L. Fu, Z. Qian, and X. Guo, Error estimates of a difference approximation method for a backward heat conduction problem, Int. J. Math. Math. Sci. 2006 (2006), 45489. 10.1155/IJMMS/2006/45489Search in Google Scholar

[23] T. I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal. 33 (1996), 162–170. 10.1137/0733010Search in Google Scholar

[24] C. L. Fu, X. T. Xiong, and Z. Qian, Fourier regularization for a backward heat equation, J. Math. Anal. Appl. 331 (2007), 472–480. 10.1016/j.jmaa.2006.08.040Search in Google Scholar

[25] P. T. Nam, An approximate solution for nonlinear backward parabolic equations, J. Math. Anal. Appl. 367 (2010), 337–349. 10.1016/j.jmaa.2010.01.020Search in Google Scholar

[26] X. L. Feng, L. Eldén, and C. L. Fu, Stability and regularization of a backward parabolic PDE with variable coefficients, Inverse Ill-posed Probl. Ser. 18 (2010), 217–243. 10.1515/jiip.2010.008Search in Google Scholar

[27] N. H. Tuan and D. D. Trong, Two regularization methods for backward heat problems with new error estimates, Nonlinear Anal. Real World Appl. 12 (2011), 1720–1732. 10.1016/j.nonrwa.2010.11.004Search in Google Scholar

[28] Y. X. Zhang, C. L. Fu, and Y. J. Ma, An a posteriori parameter choice rule for the truncation regularization method for solving backward parabolic problems, J. Comput. Appl. Math. 255 (2014), 150–160. 10.1016/j.cam.2013.04.046Search in Google Scholar

[29] Y. J. Ma, C. L. Fu, and Y. X. Zhang, Solving a backward heat conduction problem by variational method, Appl. Math. Comput. 219 (2012), 624–634. 10.1016/j.amc.2012.06.052Search in Google Scholar

[30] J. R. Wang, Shannon wavelet regularization methods for a backward heat equation, J. Comput. Appl. Math. 235 (2011), 3079–3086. 10.1016/j.cam.2011.01.001Search in Google Scholar

[31] J. R. Chang, C. S. Liu, and C. W. Chang, A new shooting method for quasi-boundary regularization of backward heat conduction problems, Int. J. Heat Mass Transfer 50 (2007), 2325–2332. 10.1016/j.ijheatmasstransfer.2006.10.050Search in Google Scholar

[32] J. J. Liu and B. X. Wang, Solving the backward heat conduction problem by homotopy analysis method, Appl. Numer. Math. 128 (2018), 84–97. 10.1016/j.apnum.2018.02.002Search in Google Scholar

[33] A. Shukla and M. Mehra, Compact filtering as a regularization technique for a backward heat conduction problem, Appl. Numer. Math. 153 (2020), 82–97. 10.1016/j.apnum.2020.02.003Search in Google Scholar

[34] J. Cheng and J. J. Liu, A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution, Inverse Probl. 24 (2008), 065012. 10.1088/0266-5611/24/6/065012Search in Google Scholar

[35] J. J. Liu, Numerical solution of forward and backward problem for 2-D heat conduction problem, J. Comput. Appl. Math. 145 (2002), 459–482. 10.1016/S0377-0427(01)00595-7Search in Google Scholar

[36] F. Yang, Y. R. Sun, X. X. Li, and C. Y. Ma, The truncation regularization method for identifying the initial value of heat equation on a spherical symmetric domain, Bound. Value Probl. 2018 (2018), 13. 10.1186/s13661-018-0934-xSearch in Google Scholar

[37] W. Cheng and C. L. Fu, A spectral method for an axisymmetric backward heat equation, Inverse Probl. Sci. Eng. 17 (2009), 1085–1093. 10.1080/17415970903063193Search in Google Scholar

[38] D. N. Hào, N. V. Duc, and H. Sahli, A non-local boundary value problem method for parabolic equations backward in time, J. Math. Anal. Appl. 345 (2008), 805–815. 10.1016/j.jmaa.2008.04.064Search in Google Scholar

[39] T. Wei and J. G. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math. 78 (2014), 95–111. 10.1016/j.apnum.2013.12.002Search in Google Scholar

[40] F. Yang, M. Zhang, and X. X. Li, A quasi-boundary value regularization method for identifying an unknown source in the Poisson equation, J. Inequal. Appl. 2014 (2014), 1–11. 10.1186/1029-242X-2014-117Search in Google Scholar

[41] W. Cheng and Q. Zhao, A modified quasi-boundary value method for a two-dimensional inverse heat conduction problem, Comput. Math. Appl. 79 (2020), 293–302. 10.1016/j.camwa.2019.06.031Search in Google Scholar

[42] X. L. Feng and L. Eldén, Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method, Inverse Probl. 30 (2013), 015005.10.1088/0266-5611/30/1/015005Search in Google Scholar

[43] D. N. Hào, N. V. Duc, and D. Lesnic, Regularization of parabolic equations backward in time by a non-local boundary value problem method, IMA J. Appl. Math. 75 (2010), 291–315. 10.1093/imamat/hxp026Search in Google Scholar

[44] F. Yang, F. Zhang, X. X. Li, and C. Y. Huang, The quasi-boundary value regularization method for identifying the initial value with discrete random noise, Bound. Value Probl. 2018 (2018), 108. 10.1186/s13661-018-1030-ySearch in Google Scholar

[45] H. W. Zhang, Modified quasi-boundary value method for Cauchy problems of elliptic equations with variable coefficients, Electron. J. Differential Equations 2011 (2011), 106. Search in Google Scholar

[46] Y. Jiang and J. Liu, Fast parallel-in-time quasi-boundary value methods for backward heat conduction problems, Appl. Numer. Math. 184 (2023), 325–339. 10.1016/j.apnum.2022.10.006Search in Google Scholar

[47] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1972. Search in Google Scholar

[48] A. Carasso, Determining surface temperature from interior observations, SIAM J. Appl. Math. 42 (1982), 558–574. 10.1137/0142040Search in Google Scholar

[49] V. A. Morozov, On the solution of functional equations by the method of regularization, Dokl. Akad. Nauk 7 (1966), 414–417. Search in Google Scholar

[50] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, 1996. 10.1007/978-1-4612-5338-9Search in Google Scholar

[51] O. Scherzer, The use of Morozovas discrepancy principle for Tikhonov regularization for solving nonlinear ill-posed problems, Computing 51 (1993), 45–60. 10.1007/BF02243828Search in Google Scholar

Received: 2023-08-07

Revised: 2023-12-07

Accepted: 2023-12-12

Published Online: 2023-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2023-0171

Keywords for this article

ill-posed problem; backward heat equation; regularization; error estimate; Morozov's; discrepancy principle

Creative Commons

BY 4.0