This repository contains the research and implementation of the numerical solution to the one-dimensional heat equation. The project encapsulates the mathematical derivation of the heat equation and its computational solution using the finite difference method (FDM). It serves as a confluence of applied mathematics, numerical analysis, and physical science, particularly focusing on the dynamics of heat transfer in linear media.
The one-dimensional heat equation is a cornerstone partial differential equation (PDE) in thermal physics, describing how heat diffuses through a given region over time. Despite its apparent simplicity, it encapsulates a range of phenomena and is fundamental in various scientific and engineering disciplines.
The project approaches the equation from both a theoretical and computational perspective:
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Theoretical Analysis: We start with the derivation of the 1D heat equation from first principles, elucidating the assumptions and physical laws that underpin the model.
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Numerical Implementation: We discretize the problem domain and apply the finite difference method to approximate the continuous PDE with a system of algebraic equations. This method is implemented in Python, utilizing libraries such as NumPy and Matplotlib for computation and visualization.
code/: Includes the Python scripts used for the numerical solution of the equation.report/: A comprehensive report compiled in LaTeX, discussing the methodology, results, and implications of the findings.
The repository presents a detailed discussion on the numerical method applied, along with the resulting plots that depict the temporal evolution of the temperature distribution. This visualization not only confirms the theoretical predictions but also provides practical insights relevant to thermal management applications.
Instructions on how to set up the environment, run the simulations, and interpret the results are provided. Users are encouraged to delve into the code, experiment with different parameters, and observe the resultant changes in the heat distribution.
The project lays the groundwork for extending the numerical methods to more complex PDEs and higher dimensions. Further exploration could involve adaptive mesh refinement, parallel computation, or the incorporation of more sophisticated boundary conditions to model complex systems more accurately.
Contributions to this project are welcome, whether they be in the form of suggestions, improvements, or extensions of the current work. For any inquiries or collaboration proposals, please reach out through the contact information provided.
This project was developed as part of the coursework for PHYS425: Advanced Mechanics and Computational Physics, under the guidance of Megan Pickett, at Lawrence University.
Hasif Ahmed
For more details on the project, please refer to the report and documentation provided within this repository.