Analytical Solution of Homogeneous One-Dimensional Heat Equation with Neumann Boundary Conditions

A partial differential equation is an equation which includes derivatives of an unknown function with respect to two or more independent variables. The analytical solution is needed to obtain the exact solution of partial differential equation. To solve these partial differential equations, the appropriate boundary and initial conditions are needed. The general solution is dependent not only on the equation, but also on the boundary conditions. In other words, these partial differential equations will have different general solution when paired with different sets of boundary conditions. In the present study, the homogeneous one-dimensional heat equation will be solved analytically by using separation of variables method. Our main objective is to determine the general and specific solution of heat equation based on analytical solution. To verify our objective, the heat equation will be solved based on the different functions of initial conditions on Neumann boundary conditions. The results have been compared with different values of initial conditions but the boundary condition remain the same. Based on the results obtained, it can be concluded that increase the number of n will reduce the heat temperature and the time taken. For short length of the rod, the heat temperature quickly converges to zero and take less time to release or reduced the heat temperature when compared to the long length of the rod.


Introduction
Most of mathematical physics are described by partial differential equations. Typically, a given partial differential equation will be solved by using numerical solution [1][2][3] and analytical solution [4]. However, it is vital to understand the general theory of partial differential equations to ensure the numerical solution is valid. Thus, the analytical solution is needed to obtain the exact solution of partial differential equation.
A partial differential equation is an equation which includes derivatives of an unknown function with respect to two or more independent variables. The partial differential equation can be classified into three types, which are parabolic [5][6], hyperbolic and elliptic [7][8]. A parabolic partial differential equation describing a large family of problems in science such as ocean acoustic propagation and heat diffusion. Hyperbolic partial differential equation describing the wave transformation and vibrations of an elastic string, while elliptic partial differential equation describing the Laplace equation.
To solve these partial differential equations, the appropriate boundary and initial conditions are needed. The general solution is dependent not only on the equation, but also on the boundary conditions. In other words, these partial differential equations will have different general solution when paired with different sets of boundary conditions. Heat equation propagates energy at infinite speed, which is strongly non-physical. However, the validity of the heat equation as a model of temperature evolution is still extremely good for all classical physics and engineering applications. One of the major effects of heat transfer is temperature change, where the heating process will increase the temperature, while cooling process decrease the temperature [9]. In this process, here is assume that no phase change and that no work is done on or by the system [10]. Javed [11] studies about dry or moist heat sources. Dry applications include hot water bottles, radiant heat and electric pads. Moist heat is considered more penetrating than dry heat, but this is due more to the fact that water-soaked materials lose heat slower than dry ones.
In Islamic perspective, Sabaeian et al. [12] stated that the temperature distribution function is essential in calculation, simulation, and prediction of thermal effects. Temperature are specific to heat capacity, or the amount of energy required to change the temperature of a substance. The measurement of changes in heat as a result of physical or chemical changes [13].
According to As-Suyuti [14] and Al-Mahalli [15], the verse in Quran (Surah Yassin: 80) tells us about the production of fire from green trees. On the other words, the fire can be produced by using green plants. In that life, fire is generated from the friction of two surface objects [16]. The heat will flow from one area of high heat to low. The rate of heat velocity depends on the degree of friction speed between the two objects. In this research, the velocity of heat from a high heat area to a low heat area would be calculated.
In the present study, the homogeneous one-dimensional heat equation will be solved analytically by using separation of variables method. Our main objective is to determine the general and specific solution of heat equation based on analytical solution. To verify our objective, the heat equation will be solved based on the Neumann boundary conditions by using this separation of variables method.

Mathematical Formulation
The mathematical models are used to describe the one-dimensional homogeneous heat boundary value problems with Neumann boundary conditions are presented below. The heat equation is used to determine the change in the function of temperature, u over time, t. The simplified diagram of a physical model of the heat equation problem is shown in Fig. 1.

Boundary Value Problem
The partial differential equation of one-dimensional homogeneous heat conduction equation is given by: where u is defined as heat temperature, x is space and t is time.

Boundary Conditions
The Neumann boundary conditions at the initial point, 0  x and at the end point

Initial Conditions
The initial conditions at 0  t is: These mathematical models of equations (1)-(3) described the heat conduction in a one dimensional uniform rod of length one unit with no internal heat sources, thermal diffusivity one, perfect lateral insulation and initial condition, x when Both left end and right end is insulated and kept at . 0 

Analytical Solution
Most of the heat equation will be solved numerically by using Crank-Nicolson [17], finite different method [18][19][20] or finite element method [21][22]. However, the analytical solution is needed to obtain the exact solution of partial differential equation. To solve analytically the partial differential equation (1), Separation of Variables (SOV) is used. The single partial differential equation can be separated into two ordinary differential equations, where there is only one independent variable for each equation.

Transform the Partial Differential Equation into Separable Method
The partial differential equation (1) can be written in the form: Then, substituting equations (5a) and (5b) into equation (1) yields: Now, the two ordinary differential equations become:

Solve the X-Problem by Strum-Liouville
To solve the heat equation (1), the boundary conditions (2) will be used. From equation (4), we have: By using boundary conditions (2), we get: Then, solve the X-problem. To solve the boundary value problem (BVP), there are three cases need to be considered: . Thus, the solution becomes: By applying the boundary conditions (8a) and (8b) into equation (9) yields:

Solve the T-Problem
An integrating the T-problem we have:

Find the Fundamental Solution
An integrating the T-problem we have: By substituting equations (10) and (11) into equation (4), the fundamental solution can be written as: Therefore, the general solution of homogeneous one-dimensional heat equation with Neumann boundary condition can be written as:

Find the Specific Solution
An integrating the T-problem we have:  (14) and Therefore, the complete solution of homogeneous one-dimensional heat equation (1)     0  Based on the Fig. 2, the heat temperature clearly show that it is quickly converge to zero when compared to Figs. 3 and 4. In other word, the short length of rod take less time to release or reduce the heat temperature, compared with the long rod. Table 1 shows the analytical solution of different functions of initial conditions on Neumann boundary conditions. The heat equation (1)

Conclusion
The solution of heat equation is dependent not only on the equation, but also on the boundary conditions or initial conditions. These partial differential equations will have different general solution when paired with different sets of boundary conditions or initial conditions. In the present study, the homogeneous one-dimensional heat equation will be solved analytically by using separation of variables method. To verify our objective, the heat equation will be solved based on the different function of initial conditions on Neumann boundary conditions. For short length of rod, the heat temperature quickly reduced compared to the long length of rod. For , the values of heat temperature at that values tends to zero. Increase the number of n will reduce the heat temperature and the time taken. For short length of the rod, the heat temperature quickly converges to zero and take less time to release or reduced the heat temperature when compared to the long length of the rod.