Heat Equation Solver using Fourier Series (1D)

This repository presents a numerical simulation of the one-dimensional heat conduction in a rod, based on the analytical solution of the heat equation using a Fourier series. The project is implemented in Python using NumPy and Matplotlib.

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📘 Description

This project simulates the temperature distribution T(x, t) in a one-dimensional rod of length L, given an initial temperature profile. The solution is based on the analytical solution of the 1D heat equation with Dirichlet boundary conditions (T=0 at both ends), using the classical Fourier sine series expansion.

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🔍 Governing Equation

The heat equation (1D) is given by:

$$ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} $$

With:

• Initial condition: ( T(x, 0) = T_0 )
• Boundary conditions: ( T(0, t) = T(L, t) = 0 )

The analytical solution using the Fourier sine series is implemented.

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🔢 Parameters

• L = 1 : Length of the rod
• T0 = 100 : Initial temperature of the rod
• alpha = 0.01 : Thermal diffusivity
• terms = 100 : Number of terms in Fourier series
• x = np.linspace(0, L, 200)
• t = 0 to 10 seconds with step 0.25

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📈 Output

The result is a dynamic plot of the temperature distribution over time, showing how the rod cools down from its initial temperature distribution.

Example:

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🚀 Requirements

Install the required Python packages using:

pip install numpy matplotlib