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% LaTeX document was generated by GPT-4 Omni 8k
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% Created 2024-10-11 15:34:09
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\documentclass[letterpaper, 10pt, conference]{IEEEtran}
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\usepackage{amsmath, amssymb}
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\usepackage{graphicx}
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\usepackage{pythontex}
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\usepackage{cite}
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\title{3-D Plotting of a Convection-Diffusion Equation}
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\author{Author Name \\ Institution Name \\ City, Country \\ Email: author@example.com}
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\begin{document}
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\maketitle
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\begin{abstract}
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  This document discusses the Numerical Solutions of the Convection-Diffusion Equation, which models various physical phenomena. Using Python for simulations, we demonstrate a 3-D plot of the solution.
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\end{abstract}
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\section{Introduction}
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Convection-diffusion equations (\textit{CDE}) describe transport phenomena in various fields such as fluid dynamics, heat transfer, and more. The historical context of these equations dates back to the works on diffusion by \textit{Fick} in 1855 and convection analyzed by \textit{Darcy} in the 19th century. The general form of a convection-diffusion equation is:
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\begin{equation}
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  \frac{\partial u}{\partial t} + \mathbf{v} \cdot \nabla u = D \nabla^2 u + S
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\end{equation}
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where $u$ is the scalar field (e.g., concentration or temperature), $\mathbf{v}$ is the velocity vector, $D$ is the diffusion coefficient, and $S$ represents sources or sinks.
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\section{Methodology}
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In this paper, we use numerical methods to solve the convection-diffusion equation in three dimensions. We demonstrate the result using Python for plotting.
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\section{PythonTeX Implementation}
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Below is the Python code using PythonTeX to plot the result of the convection-diffusion equation:
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\begin{pycode}
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  import numpy as np
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  import matplotlib.pyplot as plt
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  from mpl_toolkits.mplot3d import Axes3D
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  # Define the equation parameters
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  x = np.linspace(0, 1, 100)
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  y = np.linspace(0, 1, 100)
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  X, Y = np.meshgrid(x, y)
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  Z = np.sin(np.pi * X) * np.sin(np.pi * Y)  # Placeholder function
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  fig = plt.figure()
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  ax = fig.add_subplot(111, projection='3d')
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  ax.plot_surface(X, Y, Z, cmap='viridis')
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  ax.set_title('3D plot of the solution')
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  plt.xlabel('X-axis')
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  plt.ylabel('Y-axis')
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  plt.savefig('cde_solution.png')
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\end{pycode}
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Please refer to the generated plot in Fig. \ref{fig:solution}.
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\begin{figure}[h]
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  \centering
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  \includegraphics[width=\linewidth]{cde_solution.png}
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  \caption{3D plot of the convection-diffusion equation solution.}
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  \label{fig:solution}
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\end{figure}
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\section{Results and Discussion}
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The solution obtained provides insights into the behavior of the scalar field under given conditions. By observing the plotted surface, we note the interaction of convective and diffusive transport within the domain.
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\section{Conclusion}
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Incorporating numerical methods and Python enhances the understanding and visualization of the convection-diffusion phenomena. This framework allows further exploration into complex systems governed by such equations.
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\bibliographystyle{IEEEtran}
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\bibliography{references}
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\end{document}
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