Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.

1
% LaTeX document was generated by GPT-4o
2
% Created 2024-10-20 19:24:52
3

4
\documentclass[letterpaper, 10pt, conference]{IEEEtran}
5

6
% Packages
7
\usepackage{amsmath, amssymb, amsfonts}
8
\usepackage{graphicx}
9
\usepackage{pythontex}
10
\usepackage{cite}
11

12
% Title and Author
13
\title{Title of Research Article}
14
\author{Author Name}
15

16
\begin{document}
17
\maketitle
18

19
\begin{abstract}
20
    This abstract summarizes the research article. It provides an overview of the findings, methodology, and significance of the heat equation solved using finite differences in 2-D.
21
\end{abstract}
22

23
\section{Introduction}
24
This section introduces the historical context of the heat equation and its significance in mathematical physics and engineering.
25

26
\section{Mathematical Background}
27
The heat equation is a parabolic partial differential equation given by:
28

29
\begin{equation}
30
    \frac{\partial u}{\partial t} = \alpha \nabla^2 u
31
\end{equation}
32

33
where \( u = u(x, y, t) \) represents the temperature distribution, \( t \) is time, and \( \alpha \) is the thermal diffusivity constant.
34

35
\section{Numerical Solution Using Finite Differences}
36
The finite difference method approximates derivatives by differences and provides a way to solve partial differential equations numerically.
37

38
\subsection{Discretization}
39
The 2-D spatial domain is discretized into a grid and finite difference approximations are applied.
40

41
\begin{equation}
42
    u_{i, j}^{n+1} = u_{i, j}^n + \frac{\alpha \Delta t}{(\Delta x)^2} \left( u_{i+1, j}^n - 2u_{i, j}^n + u_{i-1, j}^n \right) + \frac{\alpha \Delta t}{(\Delta y)^2} \left( u_{i, j+1}^n - 2u_{i, j}^n + u_{i, j-1}^n \right)
43
\end{equation}
44

45
\section{Implementation in Python}
46
The problem is implemented using Python and visualized with PythonTeX.
47

48
\begin{pycode}
49
import numpy as np
50
import matplotlib.pyplot as plt
51

52
# Parameters
53
nx, ny = 50, 50
54
nt = 100
55
alpha = 0.1
56
dx, dy = 1/nx, 1/ny
57
dt = (dx*dy)/(2*alpha*(dx+dy))
58

59
# Initial Condition
60
u = np.zeros((ny, nx))
61
u[20:30, 20:30] = 100  # Initial hot spot
62

63
# Time-stepping
64
for n in range(nt):
65
    u_new = u.copy()
66
    for i in range(1, nx-1):
67
        for j in range(1, ny-1):
68
            u_new[j, i] = u[j, i] + alpha * dt * (
69
                (u[j+1, i] - 2*u[j, i] + u[j-1, i]) / dy**2 +
70
                (u[j, i+1] - 2*u[j, i] + u[j, i-1]) / dx**2 )
71
    u = u_new
72

73
plt.imshow(u, cmap='hot', interpolation='none')
74
plt.title('Temperature Distribution')
75
plt.colorbar(label='Temperature')
76
plt.xlabel('x')
77
plt.ylabel('y')
78

79
plt.savefig('heat_equation.png')
80
\end{pycode}
81

82
\begin{figure}[h]
83
    \centering
84
    \includegraphics[width=\linewidth]{heat_equation.png}
85
    \caption{Heat distribution after time evolution}
86
    \label{fig:heat_dist}
87
\end{figure}
88

89
\section{Conclusion}
90
Summarizes the findings and discusses implications and future work in solving PDEs using finite differences.
91

92
\section*{Acknowledgments}
93
Acknowledgment of any assistance or funding.
94

95
\bibliographystyle{IEEEtran}
96
\bibliography{references}
97

98
\end{document}
99