A spacetime collocation Trefftz method for solving the inverse heat conduction problem

Abstract

In this article, a novel spacetime collocation Trefftz method for solving the inverse heat conduction problem is presented. This pioneering work is based on the spacetime collocation Trefftz method; the method operates by collocating the boundary points in the spacetime coordinate system. In the spacetime domain, the initial and boundary conditions are both regarded as boundary conditions on the spacetime domain boundary. We may therefore rewrite an initial value problem (such as a heat conduction problem) as a boundary value problem. Hence, the spacetime collocation Trefftz method is adopted to solve the inverse heat conduction problem by approximating numerical solutions using Trefftz base functions satisfying the governing equation. The validity of the proposed method is established for a number of test problems. We compared the accuracy of the proposed method with that of the Trefftz method based on exponential basis functions. Results demonstrate that the proposed method obtains highly accurate numerical solutions and that the boundary data on the inaccessible boundary can be recovered even if the accessible data are specified at only one-fourth of the overall spacetime boundary.

Keywords

Spacetime collocation Trefftz method inverse heat conduction problem inverse boundary value problem numerical solutions

Introduction

The inverse heat conduction problem (IHCP) is a crucial issue in various physical, precision mechanical, and industrial mechatronic applications.^1–3 Among IHCPs, the absence of the initial temperature of the heat conduction problem is also referred to as a final boundary value problem or the backward heat conduction problem (BHCP). The main task in solving the BHCP, which has received increasing attention in recent years, is therefore to find the initial temperature. Although boundary conditions for such a system are given, the BHCP is difficult to solve because of the absence of the initial condition.⁴ The IHCP is a typical example of an ill-posed problem that cannot be solved through numerical methods and requires special regularization methods.⁵ In the context of the approximation method for this problem, many approaches have been investigated, such as the operator-splitting method,⁶ the three spectral regularization methods,⁷ and the Fourier regularization method.⁸

Some meshless methods have evoked considerable interest for finding solutions of initial value problems and boundary value problems such as heat conduction problems. A significant number of meshless methods have been proposed, such as the Trefftz method,^9–12 the method of fundamental solution (MFS),^13,14 the radial basis functions method,^15,16 the modified polynomial expansion algorithm,¹⁷ and the Local Petrov–Galerkin Method.¹⁸ The collocation Trefftz method is one of the boundary-type meshless methods for solving boundary value problems where approximate solutions are expressed as a linear combination of basis functions.^19,20 The collocation Trefftz method had originally been developed to deal with the boundary value problems in Euclidean space. In the past, several applications of the Trefftz method for solving time-dependent heat conduction problems have been developed.^21,22 Different types of Trefftz functions for stationary and non-stationary linear differential equations have been developed.²³ The advantage is that it can deal with various types of direct and inverse problems. In addition, the Trefftz functions for heat conduction equation in cylindrical dimensionless coordinates have also been proposed. The advantage is that it can identify the heat transfer coefficient during the cooling process.²⁴ However, solving the IHCP still presents a great challenge because the IHCP is a highly ill-posed system, which is one of the difficult issues of many inverse problems.

Recently, a novel collocation meshless method based on Minkowski spacetime for transient modeling of subsurface flow problems has been developed by Ku et al.²⁵ The spacetime collocation Trefftz method (SCTM) is highly accurate and computationally efficient for solving boundary value problems. Accordingly, we proposed a pioneering work using the SCTM for solving the IHCP in this study. Unlike the conventional collocation method based on a set of unstructured points in space, a novel concept in this study was using the SCTM by collocating the boundary points in the spacetime coordinate system. In the spacetime domain, the initial and boundary conditions are both regarded as boundary conditions on the spacetime domain boundary. We may therefore rewrite an initial value problem such as the heat conduction problem as a boundary value problem. Similarly, the IHCP can be transformed into an inverse boundary value problem. Hence, the SCTM is adopted to solve the IHCP by approximating numerical solution using Trefftz base functions satisfying the governing equation. The validity of the proposed method is established for a number of test problems. The formulation of the proposed method is described as follows.

Formulation

The IHCP

Considering the one-dimensional IHCP, the spacetime domain, $Ω^{t}$ , enclosed by a spacetime boundary, $Γ^{t}$ , is a two-dimensional rectangular shape. The governing equation of the IHCP is described as follows

\frac{\partial u}{\partial t} = α^{2} \frac{\partial^{2} u}{\partial x^{2}}, 0 < x < L, 0 < t < T

(1)

where u denotes temperature, t denotes time, x denotes a spatial coordinate, $α^{2}$ denotes the heat conductivity, L denotes the length of the space domain, and T denotes final time. Regarding the time dimension, the temperature is a time-dependent variable. The final time condition is expressed as

u (x, t = T) = u_{f} (x, t = T)

(2)

where $u_{f} (x, t = T)$ denotes the distribution of the temperature in the spacetime domain, $Ω^{t}$ , at the final time. To solve equation (1), the boundary conditions on $Γ^{t}$ must be given as

u (x, t) = b (x, t)

(3)

where $b (x, t)$ denotes the Dirichlet boundary condition in the spacetime domain.

The SCTM

A one-dimensional IHCP is one dimensional in both space and time in the spacetime coordinate system. The spacetime domain is therefore a rectangular shape, as shown in Figure 1. We transform the one-dimensional final boundary value problem into a two-dimensional inverse boundary value problem. The final and boundary conditions are both applied on the spacetime boundary. In addition, it becomes an inverse boundary value problem because the final time boundary values may be not assigned. Because the collocation Trefftz method begins with the consideration of T-complete functions, it is necessary to derive the general solutions. We may adopt the method of the separation of variables by assuming the solution in the form^25,26

u (x, t) = X (x) Φ (t)

(4)

Figure 1.

Collocation points of one-dimensional IHCP in the spacetime domain.

For simplicity, we let

X' = \frac{d X (x)}{dx}, X^{″} = \frac{d^{2} X (x)}{d x^{2}}, and Φ' = \frac{d Φ (t)}{dt}

(5)

Inserting equation (4) into equation (1), by taking into account notation equation (5), we obtain

X Φ' = α^{2} X^{″} Φ

(6)

We divide $α^{2} X (x) Φ (t)$ on both sides in the aforementioned equation, and the equation can be rewritten as follows

\frac{X^{″}}{X} = \frac{1}{α^{2}} \frac{Φ'}{Φ}

(7)

Each term in the aforementioned equation must be a constant for a nonzero solution. The aforementioned equations can therefore be rewritten as two differential equations as follows

\frac{X^{″}}{X} = λ, \frac{1}{α^{2}} \frac{Φ'}{Φ} = λ

(8)

Using the constant $υ$ to ensure positive or negative constants, we have $λ = 0$ , $λ = υ^{2}$ , and $λ = - υ^{2}$ . Considering the first case $λ = 0$ , we obtain the solutions as follows

X (x) = c_{1} x + c_{2}, Φ (t) = c_{3}

(9)

where $c_{1}$ , $c_{2}$ , and $c_{3}$ are constants. Substituting equation (9) into equation (4), we have

u (x, t) = {\bar{c}}_{1} x + {\bar{c}}_{2}

(10)

where ${\bar{c}}_{1}$ and ${\bar{c}}_{2}$ are unknown coefficients to be evaluated. Considering the second case $λ = υ^{2}$ , we may obtain the solutions as follows

X (x) = c_{4} e^{υ x} + c_{5} e^{- υ x}, Φ (t) = c_{6} e^{α^{2} υ^{2} t}

(11)

where $c_{4}$ , $c_{5}$ , and $c_{6}$ are constants. Substituting equation (11) into equation (4), we have

u (x, t) = {\bar{c}}_{3} e^{υ x} e^{α^{2} υ^{2} t} + {\bar{c}}_{4} e^{- υ x} e^{α^{2} υ^{2} t}

(12)

where ${\bar{c}}_{3}$ and ${\bar{c}}_{4}$ are unknown coefficients to be evaluated. Finally, we may consider the last case $λ = - υ^{2}$ . The solutions may be obtained as follows

X (x) = c_{7} \cos (υ x) + c_{8} \sin (υ x), Φ (t) = c_{9} e^{- α^{2} υ^{2} t}

(13)

where $c_{7}$ , $c_{8}$ , and $c_{9}$ are constants. Substituting equation (13) into equation (4), we have

u (x, t) = {\bar{c}}_{5} e^{- α^{2} υ^{2} t} \cos (υ x) + {\bar{c}}_{6} e^{- α^{2} υ^{2} t} \sin (υ x)

(14)

where ${\bar{c}}_{5}$ and ${\bar{c}}_{6}$ are unknown coefficients to be evaluated. Collecting all the solutions from the aforementioned results, the linearly independent solutions to the heat equation for a simply connected domain can be obtained. Using the addition theorem, we find infinitely many solutions to equation (1). It is shown that any solution to equation (1) can be written as a series expansion in terms of separable solutions. Using a finite number of $ω$ , we may approach the general solution in the form of infinite series as follows

\begin{matrix} u (x, t) = a_{0} + a_{1} x + \sum_{k = 1}^{ω} [b_{1 k} e^{- α^{2} υ_{k}^{2} t} \cos (υ_{k} x) + b_{2 k} e^{- α^{2} υ_{k}^{2} t} \sin (υ_{k} x) + b_{3 k} e^{υ_{k} x} e^{α^{2} υ_{k}^{2} t} + b_{4 k} e^{- υ_{k} x} e^{α^{2} υ_{k}^{2} t}] \end{matrix}

(15)

where $ω$ is the order of the Trefftz basis functions for approximating the solution; $a_{0}, a_{1}, b_{1 k}, b_{2 k}, b_{3 k}, b_{4 k}$ denote unknown coefficients; $υ_{k} = π k / L_{c}$ ; and $L_{c}$ denotes characteristic length. Collocating the numerical expansion from equation (15) at boundary collocation points to match the given boundary conditions, one obtains the following equation

A κ = B

(16)

where $A = [\begin{matrix} 1 & x_{1} & e^{- α^{2} υ_{1}^{2} t_{1}} \cos (υ_{1} x_{1}) & e^{- α^{2} υ_{1}^{2} t_{1}} \sin (υ_{1} x_{1}) & \dots & e^{- υ_{ω} x_{1}} e^{α^{2} υ_{ω}^{2} t_{1}} \\ 1 & x_{2} & e^{- α^{2} υ_{1}^{2} t_{2}} \cos (υ_{1} x_{2}) & e^{- α^{2} υ_{1}^{2} t_{2}} \sin (υ_{1} x_{2}) & \dots & e^{- υ_{ω} x_{2}} e^{α^{2} υ_{ω}^{2} t_{2}} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & x_{I} & e^{- α^{2} υ_{1}^{2} t_{I}} \cos (υ_{1} x_{I}) & e^{- α^{2} υ_{1}^{2} t_{I}} \sin (υ_{1} x_{I}) & \dots & e^{- υ_{ω} x_{I}} e^{α^{2} υ_{ω}^{2} t_{I}} \end{matrix}],$ $κ = [\begin{matrix} a_{0} & a_{1} & b_{11} & \dots & b_{4 ω} \end{matrix}]^{T}$ , and $B = [\begin{matrix} d_{1} & d_{2} & d_{3} & \dots & d_{I} \end{matrix}]^{T}$ ; $A$ is a matrix of the Trefftz basis functions with the size of $I \times O$ , $κ$ with the size of $O \times 1$ is a vector of unknown coefficients, $B$ with the size of $I \times 1$ is a vector of given values from Dirichlet boundary condition at collocation points; and $I$ is the number of collocation points, $O$ is the number of the terms related to the order of the basis function as depicted in equation (15), which can be defined as $O = 2 + 4 ω$ , $a_{0}, a_{1}, b_{11}, \dots, b_{4 ω}$ are unknown coefficients to be evaluated, and $d_{1}, d_{2}, d_{3}, \dots, d_{I}$ are the values of the Dirichlet boundary condition. Solving the aforementioned simultaneous linear equations, we may obtain the set of unknown coefficients for the spacetime domain. To obtain the distribution of temperature for the spacetime domain, the inner collocation points in the spacetime domain must be placed. The temperature at inner collocation points can be found by equation (15) using $A$ and $κ$ for the spacetime domain.

Numerical implementation

To investigate the effectiveness and accuracy of the proposed SCTM, four numerical examples including two one-dimensional heat conduction problems, a BHCP, and an IHCP are introduced in this section.

Example 1—one-dimensional heat conduction problem

The first example is to examine the accuracy of the proposed method. The transient heat conduction in a homogeneous isotropic space is described as equation (1). The boundary conditions at right and left of the space domain are expressed as follows

u (x = 0, t) = 12 t^{2}

(17)

u (x = L, t) = L^{4} + 12 L^{2} t + 12 t^{2}

(18)

The initial condition is given by

u (x, t = 0) = x^{4}

(19)

The exact solution is obtained as follows⁴

u (x, t) = x^{4} + 12 x^{2} t + 12 t^{2}

(20)

The length of the space domain (L) is 3 m. The heat conductivity $(α^{2})$ is 1. The elapsed time (T) is 10 h. The initial condition is applied on the bottom side of the spacetime domain and the boundary conditions are applied on both the left and right sides of the domain.

The proposed method is similar to the collocation Trefftz method. For a simply connected domain, it is necessary to locate the source point as the reference point inside the domain and the number of source point is only one for the proposed method. We adopt 123 boundary points uniformly distributed on the boundary and one source point. The Dirichlet boundary values obtained from the exact solution are given on boundary collocation points which collocate on three sides of the domain. The order of the basis function and the characteristic length for the analysis are 30 and 180, respectively. To obtain the computed results of the heat distribution, we collocate 10,000 inner points, which are uniformly placed inside the rectangular spacetime domain.

The computed results with the exact solution are illustrated in Figure 2. The accuracy for the SCTM is illustrated in Figure 3. With the SCTM, the maximum absolute error can reach the order of $1 0^{- 9}$ . It is significant that the SCTM can obtain accurate results in this numerical example. The relative error versus the elapsed time is illustrated in Figure 4. It is interesting to note that the SCTM may avoid the error propagation. Results demonstrate that the magnitude of the error for the SCTM is in the order of $1 0^{- 8}$ to $1 0^{- 13}$ .

Figure 2.

Comparison of solutions for the one-dimensional heat conduction problem.

Figure 3.

Comparison of the accuracy at final time.

Figure 4.

Relative error at observation point through time marching.

Example 2—one-dimensional heat conduction problem

To indicate the accuracy of the computed result from the proposed SCTM and that from other meshless methods, we conduct a comparison example with the direct heat conduction problem. In this example, the length of the space domain, the heat conductivity, and the final elapsed time are all assumed to be 1. The boundary conditions at right and left sides of the domain are expressed as follows

u (x = 0, t) = 2 t

(21)

u (x = L, t) = 2 t + 1

(22)

The initial condition is given by

u (x, T = 0) = x^{2}

(23)

The exact solution is obtained as

u (x, t) = x^{2} + 2 t

(24)

In the spacetime domain, the initial condition is applied on the bottom side of the spacetime domain and the boundary conditions are applied on both left and right sides of the domain. The Dirichlet boundary values obtained from the exact solution are given on boundary collocation points, which collocate on three sides of the domain.

Since the proposed method is rooted from the collocation Trefftz method, the accuracy of the solution for the collocation Trefftz method depends on the order of the Trefftz basis functions and the number of boundary collocation points. Accordingly, we conducted a sensitivity analysis of the order of the Trefftz basis functions as well as the number of boundary collocation points in this example. Parameters including the number of the source point and the characteristic length in this example are considered to be 1 and 180, respectively.

Figure 5(a) demonstrates the maximum absolute error versus the order of the basis function. It is found that promising results may be obtained when the order of the basis function ranges from 30 to 65. Figure 5(b) shows the maximum absolute error versus the number of boundary collocation points. It is found that promising results may be obtained when the number of boundary collocation points ranges from 30 to 270.

Figure 5.

Maximum absolute error versus the order of the basis function and the number of collocation points for the sensitivity example: (a) accuracy for the maximum absolute error versus the order of the basis function and (b) accuracy for the maximum absolute error versus the number of boundary collocation points.

Accordingly, the number of boundary collocation points and the order of the basis function adopted in this example can be determined to be 135 and 30, respectively. The number of boundary collocation points in space domain is considered to be 45, which uniformly placed on bottom side of the spacetime boundary. The number of boundary collocation points in time domain is considered to be 90, which uniformly placed on both left and right sides of the spacetime boundary.

To view the results clearly, the profiles of the numerical solution at different times are selected for comparison with the exact solution. We collocate 10,000 inner points (uniformly placed inside the two-dimensional rectangular spacetime) to obtain the computed results of the heat distribution at different times.

Figure 6 depicts the computed results from the SCTM, the MFS, the Trefftz method based on exponential basis functions (EBFs), and the exact solution. It is found that the computed results using the SCTM agree very closely with the exact solution. We compare the maximum relative error of the proposed method and those of the MFS and Trefftz method based on EBFs.^11,14 The maximum relative errors of the MFS and Trefftz method based on EBFs are only on the order of 10⁻⁴ and 10⁻⁶. However, the maximum absolute error of the proposed method can reach the order of 10⁻¹³, as depicted in Figure 7. It is significant that excellent agreement is achieved and highly accurate numerical solutions can be obtained by the SCTM.

Figure 6.

Comparison of solutions for the one-dimensional heat conduction problem: (a) comparison of the results at $t = 0.5$ and (b) comparison of the results at $x = 0.5$ .

Figure 7.

Absolute error of the computed results for one-dimensional heat conduction problem.

The maximum relative error of the computed heat distribution with the consideration of different number of boundary collocation points from the SCTM are compared with those reported in reference,¹¹ as depicted in Table 1. It is found that the maximum relative error of the SCTM remains in the order of 10⁻¹³. The SCTM is able to yield high accuracy performance and is superior to that of Trefftz method based on EBFs.

Table 1.

Maximum relative errors of the SCTM and the Trefftz method based on EBFs.

	Trefftz method based on EBFs¹¹	SCTM
Time step	Maximum relative error	Maximum relative error
0.2	1.73 × 10⁻⁴	2.17 × 10⁻¹³
0.15	6.05 × 10⁻⁵	1.78 × 10⁻¹³
0.10	2.31 × 10⁻⁵	1.58 × 10⁻¹³
0.05	9.75 × 10⁻⁶	1.18 × 10⁻¹³
0.025	5.92 × 10⁻⁷	1.38 × 10⁻¹³
0.01	7.93 × 10⁻⁷	1.63 × 10⁻¹³

SCTM: spacetime collocation Trefftz method; EBF: exponential basis functions.

Example 3—one-dimensional BHCP

The third example under investigation is the modeling of one-dimensional BHCP. The BHCP is considered as a final boundary value problem. It is one of the inverse problems in heat conduction engineering practice for recovery of some past history of heat distribution. Given the ill-posedness of the BHCP, the absent initial condition renders BHCP difficult to be solved. In addition, a small random noise in the measured data may cause arbitrarily large error in the numerical solutions.⁴ To solve the BHCP, it is crucial to examine the effectiveness and stability of the proposed numerical method. In this example, the boundary and final time data contaminated by random noise are adopted. The noised data on accessible boundary and final time data are given by

\hat{b} (x, t) = b (x, t) \times (1 + s \times rand)

(25)

{\hat{u}}_{f} (x, t = T) = u_{f} (x, t = T) \times (1 + s \times rand)

(26)

where $b (x, t)$ is the exact boundary data in equation (25), $u_{f} (x, t = T)$ is the exact final time data in equation (26), $\hat{b} (x, t)$ is the noise data on accessible boundary, ${\hat{u}}_{f} (x, t = T)$ is the noise data on final time data, s is the noise level, and rand is the random number generated by the uniform distribution in the range of $[- 1, 1]$ . The noisy data as depicted in equations (25) and (26) are used in the calculation of BHCP. In this example, the length of the space domain, the heat conductivity, and the final elapsed time are all assumed to be 1. The governing equation is given as equation (1). The boundary conditions are expressed as equations (25) and (26). The final time condition is given by

u (x, t = T) = x^{2} + 2

(27)

The exact solution is given as equation (24). In this example, we attempt to recover the absent initial condition. There is one-dimensional space and one-dimensional time. The spacetime domain is therefore a two-dimensional rectangular shape. The final time condition is applied on the top side of the spacetime domain and the boundary conditions are applied on both left and right sides of the domain, as depicted in Figure 1. Parameters including the number of the boundary collocation points, source point, the order of the basis function, and the characteristic length in this example are the same as Example 2. We consider that the specified data are contaminated by random noise. The noise levels are selected to be $s = 0.01$ and $s = 0.1$ .

To obtain the computed results, we adopt 10,000 inner points, placed uniformly inside the rectangular spacetime domain. Figure 8 shows the maximum absolute error of the numerical solutions with the consideration of noise level $s = 0$ , $s = 0.01$ , and $s = 0.1$ . It is found that the maximum absolute error with the consideration of noise level $s = 0$ , $s = 0.01$ , and $s = 0.1$ can reach the order of $1 0^{- 9}$ , $1 0^{- 5}$ , and $1 0^{- 4}$ , respectively. It is found that the absent initial value can be recovered with very high accuracy using the SCTM.

Figure 8.

Absolute error distribution of one-dimensional BHCP: (a) $s = 0$ , (b) $s = 0.01$ , and (c) $s = 0.1$ .

Example 4—one-dimensional IHCP

The last example is to investigate the effectiveness and stability of the proposed method to solve the IHCP. In this example, the boundary and final time data contaminated by random noise are adopted.

Three cases with different combinations of missing boundary, initial, and final conditions are considered. In Case A, we consider the absence of the initial and right spacetime boundary conditions but the data of the left spacetime boundary and final time conditions are specified. In Case B, we consider the absence of the initial and right spacetime boundary conditions. In addition, partial data of the final time and left spacetime boundary conditions are also absent where the accessible data are provided only at one-fourth of the overall length of the spacetime boundary. In Case C, we consider the absence of both initial and final time conditions given one-fourth of the overall length of the spacetime boundary.

Case A

The transient heat conduction in a homogeneous isotropic space is described as equation (1). The spacetime boundary data using the Dirichlet boundary condition are applied from the following exact solution

u (x, t) = x^{2} + 2 t

(28)

The IHCP in this study is defined such that the initial conditions are absent, and the partial data of the final time and the spacetime boundary conditions are also not specified, as depicted in Figure 9. In this example, we consider that the length of the space domain is 10, the heat conductivity is 1, and the final elapsed time is 3. The scenario with missing boundary data is considered as presented in Case A. In Case A, we consider the absence of the initial and right spacetime boundary conditions but the data of the left spacetime boundary and final time conditions are specified. There are 124 boundary collocation points and a source point. The order of the basis function and the characteristic length for the analysis are 30 and 180, respectively.

Figure 9.

Schematic of one-dimensional IHCP in spacetime domain.

Using the proposed method, the maximum absolute error of Case A is $2.47 \times 1 0^{- 7}$ . Usually, a necessary condition for numerically dealing with the IHCP, especially the BHCP, is that the complete data given on the whole accessible boundary are required. In this particular example, we demonstrate that highly accurate recovered boundary data with the accuracy at the order of $10^{- 7}$ on inaccessible boundary can be obtained even when the initial and one of the spacetime boundary conditions are absent, as depicted in Figure 10(a).

Figure 10.

Absolute error of the computed results with exact solution for Case A: (a) $s = 0$ and (b) $s = 0.1$ .

To further illustrate the capability of the proposed method, we consider that the measurements of the specified data are polluted by random noise where the noise level under consideration in this example is $s = 0.1$ . Results obtained indicate that the maximum absolute error using the proposed method for Case A can reach $1.14 \times 1 0^{- 4}$ with $s = 0.1$ , as depicted in Figure 10(b). Both initial and right boundary data can be recovered with very high accuracy.

Case B and Case C

To further demonstrate the advantages of the proposed method for analyzing the one-dimensional IHCP, we consider two IHCPs which are most challenging and difficult examples, as shown in Figure 11. In Case B, we consider the absence of the initial and right spacetime boundary conditions. In addition, partial data of the final time and left spacetime boundary conditions are also absent where the accessible data are provided only at one-fourth of the overall length of the spacetime boundary, as depicted in Figure 11(a). In Case C, we consider the absence of both initial and final time conditions but given one-fourth of the overall length of the spacetime boundary, as depicted in Figure 11(b). The transient heat conduction in a homogeneous isotropic space is described as equation (1). The spacetime boundary data using the Dirichlet boundary condition are applied from the following exact solution

u (x, t) = x^{2} + 2 t

(29)

Figure 11.

Schematic of one-dimensional ill-posed IHCP in spacetime domain: (a) Case B and (b) Case C.

In this example, the length of the space domain, the heat conductivity, and the final elapsed time are assumed to be 1. Parameters including the number of the boundary collocation points, the order of the basis function, and the characteristic length in this example are the same as previous example.

The maximum absolute error values for Case B and Case C are illustrated in Figure 12. Results obtained show that the maximum absolute error values for Case B and Case C are $3.67 \times 1 0^{- 5}$ and $2.43 \times 1 0^{- 8}$ , respectively. The absent boundary data can be recovered with very high accuracy using the proposed method. To show the accuracy of the proposed method, we consider that the measurements of the specified data are polluted by random noise where the noise level is $s = 0.1$ . The maximum absolute error values using the proposed method for Case B and Case C can reach $2.05 \times 1 0^{- 2}$ and $1.83 \times 1 0^{- 3}$ with $s = 0.1$ , as shown in Figure 13. Recovered boundary data with the accuracy at the order of $1 0^{- 2}$ on inaccessible boundary can be obtained even when three-fourth of the overall length of the spacetime boundary are absent.

Figure 12.

Absolute error of the computed results with exact solutions for Case B and Case C with $s = 0$ : (a) Case B and (b) Case C.

Figure 13.

Absolute error of the computed results with exact solutions for Case B and Case C with $s = 0.1$ : (a) Case B and (b) Case C.

Conclusion

A novel spacetime collocation meshless method for solving the IHCP has been developed. This pioneering study is based on the SCTM and provides a promising solution for the IHCP. The validity of the model is established for a number of numerical problems. The fundamental concepts and the workings of the proposed method are addressed in detail. The findings are addressed as follows.

For the modeling of the IHCP, we propose an innovative concept that one may collocate boundary points in the spacetime coordinate system and both the initial and boundary conditions can be regarded as boundary conditions on the spacetime domain boundary. The IHCP can then be rewritten as an inverse boundary value problem. Therefore, the one-dimensional IHCP can be solved without using the traditional time-marching scheme.

In the conventional time-marching scheme, the local truncation error is proportional to the step sizes and the round-off error may propagate through the time-stepping direction in the solution of the heat equation by finite differences. In this study, we demonstrate that the SCTM may avoid the error propagation. In addition, we demonstrate that recovered boundary data with the accuracy at the order of $1 0^{- 2}$ on inaccessible boundaries can be obtained with noise level $s = 0.1$ even if the accessible data are only specified at one-fourth of the overall length of the spacetime boundary.

Footnotes

Handling Editor: Jan Awrejcewicz

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was partially supported by the Ministry of Science and Technology, Taiwan, the Republic of China (MOST 106-2621-M-019-004-MY2). The authors thank the Ministry of Science and Technology for the generous financial support.

ORCID iDs

Cheng-Yu Ku

Chih-Yu Liu

References

Fan

Tang

Hua

et al . Analytical method for heat conduction problem with internal heat source in irregular domains. Adv Mech Eng 2013; 5: 173237.

Zhu

Fan

Hua

et al . A green’s function solution to the transient heat transfer through the building wall. Adv Mech Eng 2016; 8: 1–12.

Karageorghis

Lesnic

. Reconstruction of an elliptical inclusion in the inverse conductivity problem. Int J Mech Sci 2018; 142–143: 603–609.

Liu

Kuo

Jhao

. The multiple-scale polynomial Trefftz method for solving inverse heat conduction problems. Int J Heat Mass Tran 2016; 95: 936–943.

Muniz

Ramos

de Campos Velho

. Entropy- and Tikhonov-based regularization techniques applied to the backward heat equation. Comput Math Appl 2000; 40: 1071–1084.

Kirkup

Wadsworth

. Solution of inverse diffusion problems by operator-splitting methods. Appl Math Model 2002; 26: 1003–1018.

Xiong

Qian

. On three spectral regularization methods for a backward heat conduction problem. J Korean Math Soc 2007; 44: 1281–1290.

Xiong

Qian

. Fourier regularization for a backward heat equation. J Math Anal Appl 2007; 331: 472–480.

Kita

Kamiya

. Trefftz method: an overview. Adv Eng Softw 1995; 24: 3–12.

10.

et al . Trefftz and collocation methods. Southampton; Boston, MA: WIT Press, 2008.

11.

Movahedian

Boroomand

Soghrati

. A Trefftz method in space and time using exponential basis functions: application to direct and inverse heat conduction problems. Eng Anal Bound Elem 2013; 37: 868–883.

12.

Liu

Zhang

. Numerically solving twofold ill-posed inverse problems of heat equation by the adjoint Trefftz method. Numer Heat Tr B: Fund 2018; 73: 48–61.

13.

Hon

. A discrepancy principle for the source points location in using the MFS for solving the BHCP. Int J Comput Meth 2009; 6: 181–197.

14.

Johansson

Lesnic

. A method of fundamental solutions for transient heat conduction. Eng Anal Bound Elem 2008; 32: 697–703.

15.

Mavrič

Sarler

. Application of the RBF collocation method to transient coupled thermoelasticity. Int J Numer Meth Heat Fluid Flow 2017; 27: 1064–1077.

16.

Jiang

Hon

. A meshless method based on RBFs method for nonhomogeneous backward heat conduction problem. Eng Anal Bound Elem 2010; 34: 785–792.

17.

Chang

Liu

Wang

. A modified polynomial expansion algorithm for solving the steady-state Allen-Cahn equation for heat transfer in thin films. Appl Sci 2018; 8: 983.

18.

Mahmoodabadi

Abedzadeh Maafi

Bagheri

et al . Meshless local Petrov-Galerkin method for 3D steady-state heat conduction problems. Adv Mech Eng 2011; 3: 251546.

19.

Xiao

Liu

et al . On the accuracy of the collocation Trefftz method for solving two and three dimensional heat equations. Numer Heat Tr B: Fund 2016; 69: 334–350.

20.

Grysa

Maciejewska

. Trefftz functions for the non-stationary problems. J Theor Appl Mech 2013; 51: 251–264.

21.

Liu

. A multiple/scale/direction polynomial Trefftz method for solving the BHCP in high dimensional arbitrary simply-connected domains. Int J Heat Mass Tran 2016; 92: 970–978.

22.

Maciejewska

Piasecka

. Trefftz function-based thermal solution of inverse problem in unsteady-state flow boiling heat transfer in a minichannel. Int J Heat Mass Tran 2017; 107: 925–933.

23.

Ciałkowki

Grysa

. Trefftz method in solving the inverse problems. J Inverse Ill Pose P 2010; 18: 595–616.

24.

Grysa

Maciag

Cebo-Rudnicka

et al . Identification of the heat transfer coefficient during cooling process by means of Trefftz method. Eng Anal Bound Elem 2018; 95: 33–39.

25.

Liu

et al . Modeling of transient flow in unsaturated geomaterials for rainfall-induced landslides using a novel spacetime collocation method. Geofluids 2018; 2018: 7892789.

26.

Jeng

Barry

. Analytical solution for tidal propagation in a coupled semi-confined/phreatic coastal aquifer. Adv Water Res 2002; 25: 577–584.