Solving 1D Heat Equation Using Fourier Transform

Use the Fourier Transform to solve the 1D heat equation $U_T = \alpha^2 U_{XX}$, assuming an infinite 1D piece of metal with an initial temperature distribution $U(X, 0)$ that dies out to zero at $\pm\infty$.

The 1D heat equation is a fundamental partial differential equation that describes how heat diffuses through a medium. In this problem, the Fourier Transform is utilized to solve the heat equation on an infinite domain, which is a powerful technique due to its ability to transform differential equations into algebraic equations in the frequency domain. This approach simplifies the equation as it allows us to work with functions in terms of their frequency components rather than their spatial or temporal ones. By applying the Fourier Transform, boundary conditions at infinity become much easier to handle since we assume the initial temperature distribution diminishes as you approach positive or negative infinity.

This means we're essentially transforming a problem from the space-time domain into a more manageable one in the frequency domain, solving it there, and then converting back to the original domain with the inverse Fourier Transform. Conceptually, the Fourier Transform helps us analyze the heat diffusion problem by breaking down the initial temperature distribution into sinusoidal components, each of which can be independently evolved over time based on their frequency characteristics.This approach not only provides insight into how heat diffusion operates in different modes of frequency but also exemplifies the elegance and utility of Fourier analysis in solving linear partial differential equations, particularly on infinite or unbounded domains.