Laplace Equation 1D for Steady-State Heat Conduction Model in BVP PDE

Topic: Steady-State Heat Conduction

Subject: Heat Transfer and Numerical Methods

Tool: Scilab

By: Gani Comia, Jan. 2025

The steady-state heat conduction model, as represented by the Laplace equation in one dimension (1D), is the simplest partial differential equation (PDE) you may encounter solving. This equation explains the temperature characteristics or profile between the boundaries of the given domain. Presented in this article is the application of the Laplace equation in 1D using the Finite Difference Method (FDM) for the solution.

The Laplace equation in 1D for the steady-state heat conduction is shown in Equation (1).

$$\frac{\partial^2 u}{\partial x^2}\;=\;0 \tag{1}$$

The second-order PDE in 1D can also be written in ordinary differential equations (ODE). This can be approximated using the FDM shown in Equation (2).

$$\frac{d^2 u}{d x^2}\; \approx \; \frac{u_{i+1} - 2 u_i + u_{i-1}}{\left(\Delta x\right)^2}\;=\;0 \tag{2}$$

This leads to the solution in Equation (3),

$$u_i\;=\; \frac{u_{i+1} + u_{i-1}}{2} \quad for \quad i\;=\;2,3, \ldots ,N-1 \tag{3}$$

With boundary conditions,

$$u_1\;=\;u_1, \quad u_N\;=\;u_L \tag{4}$$

Let us illustrate an example of steady-state heat conduction as shown in Figure (1) and apply the FDM solution to understand its temperature profile between boundaries. The Scilab script to implement the solution will be explained in detail in relation to the Laplace equation in 1D.

Figure 1. Sample Domain under Steady-State Heat Conduction

The solution using the Scilab script is shown below. A short explanation is provided for each code snippet.

Scilab Script.

clear;clc;clf;

// (1) primary & derived parameters
L = 100;                            // mm, length of workpiece
Nx = 11;                            // unit, number of nodes
dx = L / (Nx - 1);                  // mm, spatial step

Equation (5) is the boundary condition for this problem.

$$u_1\;=\;u_1, \quad u_{2,3, \dots ,N-1,N}\;=\;u_L \tag{5}$$

// (2) boundary temperature conditions
u = zeros(Nx, 1) + 25;              // node temp cond @ 25 deg C
u(1) = 100;                         // boundary temperature

The temperature condition at node points between boundaries can be solved using the "for-end” function of the Scilab.

// (3) temperature profile at node points using FDM
for i = 2:Nx-1
    u(i) = (u(i+1) + u(i-1))/2;
end

Below is the script for plotting the temperature profile.

// (4) plotting of temperature at node points
f = gcf()
f.figure_size = [600,600]

len = linspace(0,L,Nx);
plot(len, u, 'r-o');
title("Temperature Profile @ Eleven (11) Node Points","fontsize",3)
xlabel("Length, x [ mm ]","fontsize",3)
ylabel("Temperature, T [ C ]","fontsize",3)
xstring(58,73,"https://gani-mech-toolbox.blogspot.com")
eqn = ["$\LARGE \;\frac{\partial^2\,T}{\partial x^2}\,=\,0$"]
legend(eqn,with_box=%f)
xgrid(3,1)

a = gca()
a.data_bounds = [0 0; L max(u)]
a.children.children.mark_background = 9;

Figure (2) is the resulting plot from executing the whole script.

Figure 2. Graphical Solution for Laplace 1D using FDM with 11 Node Points

The given boundary conditions in Equation (5) will result in a curved-like temperature profile. If Equation (4) is considered as boundary conditions with the corresponding change in primary and derived parameters, specifically reducing the number of node points to 3, a linear temperature profile will become the solution.

Below is the modified script for 3 node points.

// (1) primary & derived parameters
L = 100;                            // mm, length of workpiece
Nx = 3;                             // unit, number of nodes
dx = L / (Nx - 1);                  // mm, spatial step

Figures (3) and (4) are the illustration and graphical solution for 3 node points leading to a linear temperature profile.

Figure 3. Sample Domain with 3 Node Points under Steady-State Heat Conduction

Figure 4. Graphical Solution to Laplace 1D using FDM for 3 Node Points

The number of node points taken into consideration in solving the Laplace equation using FDM dictates the type of temperature profile, linear or curve-like. You may send a comment about the consideration for a basis as to how many node points are to be used for certain conditions of the steady-state heat conduction model.

Feel free to comment for inquiry, clarification, or suggestion for improvement. Drop your email to request the soft copy of the file.