Applications of the method of lines for modeling the physical phenomenon of heat conduction

It is claimed that most of the fundamental principles that govern the physical phenomena of interest in engineering applications can be described by differential equations. Therefore, the ability to analyze, solve and understand differential equations is essential for decision-making in the applied areas. In this sense, studying efficient methods to solve differential equations is a fundamental contribution in advancing the understanding of relevant physical models in engineering applications. The purpose of this research allowed to study the differential equation associated with Fourier’s heat transfer law and calculate its solution by two different methods than the traditional method. From this context, the study is related to the phenomenon of heat conduction in a metal bar under ideal conditions to later carry out its application in a particular case. First, the solution of the differential equation that models the physical phenomenon derived from the use of Fourier’s physical heat law is calculated with the use of statistical tools; then, a solution scheme is implemented using method of lines. Given the nature of the investigation, the solutions by both methods are compared about what is expected in the physical interpretation of Fourier’s law.


Introduction
Mathematics is playing an increasingly important role in physics and the life sciences, showing a resurgence of interest in modern and classical applied mathematical techniques [1]. The application of mathematical models in understanding the physical phenomenon of heat conduction has a major impact on applications throughout the world [2]. An important conceptual place to start the study of mathematical models related to heat transfer is directly related to Fourier theory [3]. The study of heat conduction phenomena in kilns has had a wide interest in the research [4], the mechanical engineering department of Universidad Francisco de Paula Santander, Ocaña, Colombia, has devoted research to the study of heat transfer processes in industrial and artisanal kils [5].
The search for numerical techniques to solve mathematical models associated with heat conduction processes is important because of their computational implementation. In contrast, analytical solutions have elaborate developments that are difficult to process algorithmically [6]. The main contribution of this article is the use of the method of lines as a numerical tool to solve the heat equation that models the heat conduction phenomenon. In addition, it is verified that the method of lines coincides with a  [7]. It is important to mention that this article is directly related to the study of thermal phenomena in the region of influence, Ocaña, Colombia [8,9].

Mathematical modeling of heat conduction
The process of mathematical modeling describes several stages. The definition of the physical phenomenon and the empirical relationships it describes, this generates a mathematical model which must be solved by different mathematical techniques. Finally, the interpretation and verification of the data.

Formulation of the physical phenomena
A metal bar 100 cm long, whose sides maintain constant temperatures of 0 ℃ and 100 ℃, is put in contact with another metal bar that has a temperature of 0 ℃. The problem is to calculate the temperature across the bar after the cooling process starts.
Since this problem describes a phenomenon of heat conduction in one dimension [10], the appropriate mathematical model for this situation is the partial derivative Equation (1).
The function T(x, t) represents the temperature in the bar, where x is the position and t the time. The constant α represents the thermal diffusivity of the bar. The initial and boundary conditions are defined in complete Equation (2) and Equation (3).
Equation (1) and Equation (3) represent the mathematical model to calculate the temperature in the bar.

Analytical solution for the heat conduction equation
Initially it is assumed that the temperature function T(x, t) = g(x)h(t), this method is called separation of variables [11]. When calculating the partial derivatives with respect to the variables x and t, in the previous equality, it is possible to conclude complete Equation (4) and Equation (5).
where the constant β $ is the result of equating two functions that depend on different variables. The Equation (6) allows the definition of two ordinary differential equations whose solutions [11] are expressed in complete Equation (7) and Equation (8).
When applying the Fourier integral technique [12] it is natural to assume the Equation (9).
Using the complete Equation (9) by the Fourier integral method [7], it is possible to compute the general solution Equation (10). (10)

Method of lines
The numerical method chosen to solve the heat conduction model Equation (1) and Equation (3) is method of lines [13]. The first step to use the lines method consists in discretizing the equation Equation (1), for this aim it is possible to apply the Taylor's theorem Equation (11).
Substituting the complete Equation (12) in Equation (1) produces a set of recursive complete Equation (13).
Using Equation (13), the boundary conditions Equation (2) and the initial condition Equation (3), the mathematical model for heat conduction Equation (1) and Equation (3) can be represented in the matrix form Equation (14).
Is a tridiagonal matrix and Equation (16).

Results and discussion
In this section, the temperature along the 100 cm long metal bar will be calculated after 900 seconds have already elapsed. We assume that α $ = 0.173, the estimate of the temperature along the bar is calculated to x = 0. 25 cm, 50 cm, 75 cm and 100 cm.

Lines method
By applying complete Equations (14), Equation (15) and Equation (16) to the study of the change in temperature of the metal bar with the initial and boundary conditions of Equation (2) and Equation (3), for the case n = 3, has the matrix form, show in the Equation (21). The numerical method for solving the system of ordinary is the Rungen Kutta method [15].

Analytical method and method of lines
Applying the Lagrange interpolation method to the temperature values the complete Equation (19) and Equation (20) generates the temperature curve (red) that is illustrated in Figure 1. When solving the Equation (21) using a computer program, a set of curves are generated that represent the temperature of the bar as time passes. The curve in Figure 1 (blue) represents the temperature when the time is 900 seconds.   [7], it is possible to conclude that the analytical method fits the expected result for the physical behavior of the temperature variation in the bar. Therefore, from the qualitative behavior shown by Figure 1 it is possible to conclude that the method of line is a suitable candidate for modeling heat conduction on homogeneous media.

Conclusion
The study of the cooling of the metal bar can be calculated in two different ways. First, a solution of the heat equation is proposed by using the techniques of Fourier analysis to calculate the temperature function. Second, the solution by method of lines of the conduction phenomena is proposed by the numerical scheme. The advantage of the solution by the numerical method of lines is the possibility of the calculation by means of a computer program. As a general conclusion to the work, for the modeling of thermal processes, where the diffusion constant is homogeneous, method of lines is an efficient alternative for this type of phenomena.