1. Introduction
In practical engineering and physical problems, the physical quantity to be analyzed is affected by more than one variable; thus, it is generally expressed in partial differential equations. For the needs of practice and application, Fourier elucidated the phenomenon of heat conduction and presented a series of laws related to heat conduction, i.e., the classical heat conduction equations in 1822; his research greatly impacted the development of partial differential equations [
1]. Heat conduction appears in many mathematical models. The difference equation in the famous Black Scholes model can be transformed into heat conduction and a simple solution can be derived, but many extension models have no analytical solution. Thus, numerical methods are used for the calculation. For example, the Crank–Nicolson method can be used to effectively determine the numerical solution of heat conduction, and this method can also be used in many models without an analytical solution [
2]. Fayz et al. expressed a numerical study of flow features and heat transport inside an enclosure. Governing equations were discretized by a finite-element process with a collected variable arrangement. Streamlines and isotherm lines were utilized to show the corresponding flow and thermal field inside a cavity. Velocity and temperature profiles were displayed for some selected positions inside an enclosure for a better perception of the flow and thermal field [
3]. Moreover, Navier–Stokes equations are motion equations that describe the momentum conservation of viscous incompressible fluids. They reflect the basic mechanical laws of viscous fluid flow and are of great significance in fluid mechanics, but they are nonlinear partial differential equations, which are very difficult and complex to solve. Moreover, they can be simplified to obtain an approximate solution in some cases. Since the rapid development of computers, the numerical solution of Navier–Stokes equations have made great progress. However, in terms of analytical solutions, the exact solutions can only be obtained on some very simple special flow problems, and further development and novel ideas or technologies are required to obtain the analytical solutions.
Therefore, the heat-conduction equation is an important development equation in partial differential equations and an important theoretical equation in the fields of fluid mechanics [
4], materials [
5], and bioengineering [
6]. In the classical one-dimensional heat-conduction model [
7,
8], the boundary temperature
φ(
t) is a known constant ∆
T0 (i.e., the boundary temperature increases ∆
T0 at the initial instantaneous change and then remains unchanged) when the temperature field follows the first boundary control condition. Here, the analytical solution of the model can be directly obtained using integral transformation methods, such as Laplace transform or Fourier transform [
8,
9,
10]. Through Newton’s law of cooling and Fourier’s law, combined with the relevant physical parameters, Tan et al. established the heat-conduction model of the fire suit and the model could improve the fitting accuracy, which is of great significance for the design and of and research into specialist fire clothing and equipment, and also had a certain reference value for solving similar problems [
11]. It could also be directly solved using integral transformation when the boundary condition had a relatively simple function form, such as a time linear function [
9,
12,
13,
14]. However, it is difficult to solve when
φ(
t) is a definite function form due to its complexity, and the solution can be expressed or an approximate solution can be obtained [
15,
16] by using the special functions [
17,
18].
The heat-conduction model has always been an important research component of mathematical and physical methods [
7,
8,
9]. For instance, Dominic et al. [
17] studied the analytical solution to the unsteady one-dimensional conduction problem with two time-varying boundary conditions, Weigand et al. [
19] studied the analytical methods for heat transfer and fluid flow problems, and Burggraf et al. [
20] studied an exact solution of the inverse problem in heat conduction. Nevertheless, in practical engineering applications, some common problems still exist. According to Newton’s law of cooling [
21], objects transfer heat to the surrounding medium when their temperature is higher than that of the environment, which is one of the fundamental laws of heat transfer. For example, in the ground source heat-pump system, the circulating water temperature in the heat exchanger is naturally cooled, and the geothermal heat conduction around the heat exchanger can be regarded as a heat conduction problem under the boundary conditions of Newton’s law of cooling. To prevent the impact of high temperature on the performance and safety of a power battery, Ma Yan et al. applied Newton’s cooling law, established the battery resistance of the battery with temperature change, determined the convection heat transfer coefficient changes with the coolant flow rate of the cell concentrated-mass heat model, and proposed a fuzzy proportional-integral-derivative direct liquid-cooling strategy for a battery pack [
22]. Rosales et al. used a generalized conformable differential operator and then a simulation of the well-known Newton’s law of cooling was made, which had an advantage with respect to ordinary derivatives [
23]. Melo et al. developed an active thermography algorithm capable of detecting defects in materials, based on the techniques of thermographic signal reconstruction, thermal contrast, and the physical principles of heat transfer. Newton’s law of cooling was used to store the normalized temperature data pixel-by-pixel over time and a compression ratio of 99% was obtained [
24]. Konovalenko et al. proposed a novel method that extends the applicability of Newton’s law of cooling to changeable ambient temperatures based on a set of temperature stability conditions and a sensor measurement error. In this method, an optimal number of measurements that characterize stable ambient temperatures and improve prediction reliability are selected [
25]. Calvo-Schwarzwlder et al. simulated the growth of a one-dimensional solid by considering a modified Fourier law with a size-dependent effective thermal conductivity and a Newton cooling condition at the interface between the solid and the cold environment [
26]. Herrera-Sánchez et al. used Newton’s Law of Cooling for heat transfer, which states the rate of heat exchange between an object and its environment, to solve the problem of the packaging process when handling canned food [
27].
In practice, even if the immediate increase in the boundary temperature and subsequent decline are consistent with Newton’s law of cooling, the heat transfer issue is challenging to solve directly by integral transformation when the boundary conditions are the exponential decay function ∆
T0 e−λt. Moreover, for the above one-dimensional heat-conduction model, from the perspective of mathematical physical models, many problems in nature have similar physical laws, such as diffusion, cooling, charge and discharge, particle spin polarization degree, and other system state evolutions over time. Moreover, the solving of partial differential equations is equivalent to calculating a particular solution under a specific boundary condition [
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46]. The physical law for describing the temporal temperature decrease has been dominated by Newton’s law of cooling (NLC), which assumes that natural cooling occurs by following an exact exponential trend. However, several studies have questioned the broad validity of this law by arguing that cooling occurs following an approximate rather than an exact exponential trend. Silva introduced a new formulation of NLC based on generalized statistics that outperforms the classical NLC, and so demonstrates a new path to cooling analyses [
47]. Yan et al. studied a discrete variable topology optimization method to solve the simplified convective heat transfer (SCHT) design optimization modeled by Newton’s law of cooling. The discrete variable topology optimization was based on the proposed sequential approximate integer programming with trust-region, which could identify the convective boundary and carry out the optimization design [
48]. Thus, the heat-conduction model with the boundary conditions of Newton’s law of cooling needs to be studied.
The goal of this work was to analyze the mathematical and physical implications of a one-dimensional heat-conduction model in a domain with a semi-infinite border using Newton’s law of cooling as a boundary condition. More significantly, the analytical solution approaches that are suggested were examined from the standpoint of fusing mathematical significance with real-world application requirements. For specific research content, the operator of
φ(
t) was used in the model transformation and inverse transformation processes to establish a general theoretical solution. Furthermore, the boundary condition function does not directly participate in the transformation based on the inverse Fourier transformation and the differential properties of the convolution method [
8,
9]. Then,
φ(
t) = ∆
T0 e−λt was substituted into the general theoretical solution, and the analytical solution of this problem is obtained. Note that although the formal transformation of
φ(
t) was not performed, the boundary function must meet the Fourier transformation requirements. Additionally, the specific solutions and corresponding mathematical meanings are discussed. Numerical verification and sensitivity analysis are performed for the proposed model. Finally, an analytical solution is applied for parameter calculation and verification in the case study. The proposed solution method is relatively simple and convenient and does not have the complicated transformation and inverse transformation operation processes of
φ(
t).
The proposed method aims to guide the actual process of solving the reverse heat conduction reverse problem. This could provide a reference basis for numerical calculation under specific complex boundaries, especially for similar physical laws. Finally, the measurement point arrangement and measurement effect inspection of heat sensitive sensors depend on the distribution characteristics of the temperature field in the detection parts. Thus, the analytical solution of the temperature field under the influence of different temperature boundary conditions will provide a convenient and reliable theoretical method for analyzing the temperature field distribution characteristics in different detection parts.
2. Basic Model and Its Solution
In engineering practice, most heat exchange holes are arranged in a dense line in ground coupling heat-pump system, and the heat exchange holes are regarded as a single hole. The water temperature of the linearly arranged heat exchange hole is generalized as the first temperature boundary (Dirichlet boundary). Here, the geothermal field change of the heat-pump system can be generalized into a one-dimensional heat conduction problem in the semi-infinite domain.
Based on the above a thin-layer material with a heat source at one boundary was taken as an example, as shown in
Figure 1. The heat transfer characteristics of materials can be summarized as follows: (1) The experimental material is homogeneous and isotropic and extends infinitely in the
x-direction. (2) One boundary of the material is provided with a heat source under the Dirichlet boundary condition, whereas the outer surface of the other boundary is a heat insulation surface. (3) The initial temperature of the material and boundary is
T(
x,0) = 0, and the temperature field is recorded as
T(
x,
t). (4) The time variation function of the boundary temperature is denoted as
φ(
t). (5) The heat transfer in the thin layer material is one-dimensional heat conduction, and the temperature field near the boundary is shown in
Figure 2.
The above heat conduction problem can be depicted as a mathematical model (I) [
49]:
where
t (s) is the time,
x (m) is the distance from the calculation point to the boundary,
a (m
2/s) is the thermal diffusivity or thermal conductivity of the solid material,
T(
x,
t) (°C) is the temperature, and
φ(
t) is a boundary function.
Model (I) is the basic model. In porous media seepage mechanics,
T(
x,
t) is generally denoted as
H(
x,
t) to represent the water level and
a (m
2/d) is the permeability coefficient of the porous media [
12,
13,
14]. In environmental hydraulics,
T(
x,
t) is generally denoted as
C(
x,
t) to represent the water quality concentration, wherein parameter
a is mostly written as
D (m
2/d), which is the hydrodynamic diffusion coefficient [
50].
In model (I), the boundary condition (3) is
x = 0:
T(
x,
t) =
T(0,0) +
φ(
t). So that the converted model does not rely on the initial time value of the model, the following Fourier transform is applied. Set
u(
x,
t) =
T(
x,
t) −
T(
x,0), and then, in model (II), the boundary condition (7) is converted to
u(
x,
t) =
φ(
x,0). Model (II) is as follows [
7]:
where
u(
x,
t) is the temperature relative to the initial temperature field.
When a definite function of φ(t) exists such that ∆T0 is constant in the classical model, model (II) can be solved via Fourier transform and Laplace transform.
For the above problem, in particular, the PDE is linear and a simple mathematical model. There is a very hot field in mathematics and physics that has been studying nonlinear PDE for a long time. For example, Md and Cemil examined the modified (G′/G)-expansion process for generating closed-form wave answers of the conformable fractional ZK equation, including power law nonlinearity [
51]. So, in the future research, the related nonlinear PDE equations could be studied further.
3. General Theoretical Solution
The heat conduction equation can be solved using methods such as analytical, approximate analytical, and numerical methods. Numerical methods are usually used to deal with this kind of heat conduction problem. Calculation tools and analysis techniques are fairly advanced, and numerical methods have become the main means for solving the complex heat conduction problems [
52]. However, strict analytical solutions can only be obtained under certain specific conditions. In most cases, especially under transient conditions, the strict solution is either too cumbersome or does not exist at all. In existing research, the heat-conduction model and its analytical solution for a ground source heat-pump system are given, and the heat conduction outside a borehole can be generalized as an infinite or a finite length linear heat source releasing heat to the surrounding soil [
53,
54]. The problem can also be regarded as an unsteady heat-conduction process from the column heat source to the surrounding infinite area [
55]. A Kelvin line or infinite line heat source analytical model is established based on Fourier heat conduction law. The theory usually assumes that a borehole is an infinite linear heat source, and the Earth is an infinite homogeneous medium with a specific initial temperature [
53,
56,
57]. Moreover, the column heat source model is based on the Fourier heat conduction law. Assuming that the heat transfer rate is a constant value, Carslaw and Jaeger established the transient heat conduction control equation under the given boundary conditions and initial conditions [
7]. Eskilson assumed that a borehole is a limited linear heat source and considered the heat flux along the borehole axis; thus, their model is suitable for the long-term operation of the ground source heat-pump system [
58]. Based on the Eskilson theory, Zeng et al. considered the influence of finite borehole length with the surface as the boundary, and they derived the analytical solution of a transient finite-line heat source mode [
54]. To solve complex mathematical problems, analytical models usually have some restrictive assumptions and simplifications, and the accuracy of analytical results is reduced [
59]. However, analytical models have high calculation efficiency and require low calculation times.
According to the above research, using Fourier transform to solve the analytical solution in the heat-conduction mode is feasible. In terms of the properties of the transformation of φ(t), the general theoretical solution of this kind of model is given separately. The process does not depend on the transformation of φ(t).
For model (II), the variation range of
x is 0–
+∞. Thus, the Fourier sinusoidal transformation of
x can be solved. According to the characteristics and properties of Fourier transform, the following calculation process is obtained [
60]:
where
is the Fourier transform of
u for
x,
ω is the Fourier transform operator, and
F is the transform operator.
From Equation (5) with the boundary condition (7), the results are as follows:
The upper equation is a first-order inhomogeneous linear differential equation, and its general solution is
where
C is the pending constant.
When solving the special solution of the model (II), the pending constant
C must first be determined according to the fixed solution condition Equation (6). Thus, when
u(
x,
t) = 0,
= 0.
After the pending constant
C is determined, then by Equations (13) and (14),
Substitute the boundary condition Equation (7) of the model (II) into Equation (14), and the particular solution of the model (II) is
For Equation (16), the inverse sine transform is obtained; then,
where
F−1 is the inverse conversion operator.
To solve the above equation, the relation between inverse sine and cosine transform is applied, and the order of integral exchange is focused on.
According to the known integral Equation (18) [
60],
where
b is the intermediate variable,
b =
a (
t −
ξ).
Combining with Equation (17),
u(
x,
t) can be written as
where ξ is the integral variable in time.
Equation (19) is the solution of a one-dimensional heat-conduction model in semi-infinite domain when the boundary condition is φ(t).
φ(t) is not directly engaged in the transformation throughout the calculating procedure. Thus, a solution method for the model is provided when φ(t) is complex and difficult to directly solve using Fourier transform.
According to the definition of convolution and the properties of Fourier transform, the commonly used solution can be obtained in the form of the probability density function. Thus, Equation (19) can be written as
where the asterisk is the convolution operator in Equation (19).
According to the differential properties of convolution
Note the equivalence between the third line of Equation (20) and the first item at the left side of Equation (21). When
, Equation (21) can be rearranged as follows:
Using the commutative law of convolution, the above formula can be written in integral form as
Equation (23) is the model solution obtained without direct transformation of the boundary condition φ(t), that is, the solution is valid for all boundary conditions of φ(t). Therefore, Equation (23) is the general theoretical solution of this kind of model. When the function form of φ(t) is determined, the solution can be obtained by substituting φ(t) into Equation (23).
In the transformation process,
φ(
t) is operated in the form of the operator; thus, it should meet the requirements of Fourier transform. Moreover,
φ(
t) is integrable on any interval under the Dirichlet condition [
50]. Additionally, it meets the above requirements when
φ(
t) is the exponential decay function.
For a one-dimensional heat-conduction model under the first boundary condition of homogeneous medium in a semi-infinite domain, the general theoretical solution was obtained under the complex form of the boundary condition function. Furthermore, the process applied to the general theoretical solution can provide a reference for solving similar problems in other fields, such as the contamination migration problem under the natural decay boundary conditions of source concentration in the subsoil.
4. Solutions to the Newton’s Law of Cooling Boundary
To investigate the analytical solution of a one-dimensional unsteady temperature field near the Newton’s law cooling boundary, the general theoretical solution of this kind of model is given based on Fourier transform. The integration transformation and the inverse transformation processes do not depend on the expression or function form of φ(t) = ∆T0 e−λt, where λ is the coefficient of cooling ratio, and the corresponding physical significance of λ is an indicator of the speed of temperature change under the Newton’s law of cooling boundary.
The general solution gives the form of the solution, including all solutions that satisfy the differential equation. When the initial conditions of the differential equation are given, specific values of the parameter can be determined and a unique special solution can be obtained. According to the actual situation, the special solutions under three kinds of boundary conditions are as follows: (1) λ > 0, Newton’s law of cooling boundary condition, that is, the solution of the nonlinear variation problem. (2) λ = 0, the boundary temperature remains unchanged after instantaneous change. (3) The boundary temperature changes linearly after the instantaneous change.
4.1. The Newton’s Law of Cooling Boundary
When
λ > 0, after the boundary temperature instantaneously increases by ∆
T0 and then cools naturally according to Newton’s law of cooling, the boundary temperature is
φ(
t) = ∆
T0e−λt, which is substituted into Equation (23):
By simply replacing the boundary conditions into the general theoretical solution, the model solution is produced under the Newtonian cooling boundary. This solution procedure can be used for models whose boundary conditions are other forms of function, circumventing the complicated transformation process.
4.2. The Fixed Boundary Condition
When
λ = 0, for the convenience of discussion,
e−λt is expanded using a power exponential form:
φ(t) = ∆T0 under the condition of λ = 0. The corresponding physical meaning is that when the boundary temperature remains unchanged after the instantaneous change of ∆T0, this is the general boundary condition in the classical model.
Here, the second term on the right side of Equation (25) is zero, and Equation (24) is transformed into the solution of the classical model, that is, the solution of the classical problem is a special case of Equation (23).
4.3. The Linear Boundary Condition
When
λ ≠ 0∩
φ(
t) = ∆
T0(1 −
λt), the boundary temperature changes linearly after the instantaneous change. Equation (25) is an alternating series term, for example, the first two terms of the series are taken in the process.
φ(
t) = ∆
T0(1 −
λt), which means that after the temperature instantaneously increases by ∆
T0, the boundary temperature starts to slowly decrease with a slope of
λ∆
T0. Then, from Equation (24),
At this time, the temperature at the boundary linearly decrease with time, and the speed of temperature change can be calculated using Equation (26). Then, the analytical solution of the model can be obtained. The boundary condition solution is relatively simple and is not described.