Heat Equation via Fourier Transform

PDEs · Axiom Academy

LESSON Heat Equation via Fourier Transform A comprehensive approach to solving the heat equation on the infinite line using Fourier analysis 1. The Heat Equation on the Infinite Line We begin with the one-dimensional heat equation on the entire real line, which governs the evolution of temperature u(x,t) over space x and time t. u(x,t) : Temperature at position x and time t k : Thermal diffusivity constant (k > 0) u(x,0) = f(x) : Initial temperature distribution 2. Taking the Fourier Transform in x We apply the Fourier transform with respect to the spatial variable x, treating t as a parameter. This converts our PDE into an ODE. Using the property that the Fourier transform of u_xx is -(k^2)u-hat, we obtain a simple first-order ODE for each Fourier mode k. Solving the ODE with initial condition u-hat(k,0) = f-hat(k) gives us an explicit solution for each Fourier mode. 5. Inverse Transform and the Heat Kernel Taking the inverse Fourier transform reveals the solution as a convolution with the fundamental solution, known as the heat kernel or Gaussian kernel. 6. Solution as Convolution with Gaussian The complete solution is obtained by convolving the initial condition with the heat kernel, showing how initial temperature distributions spread and smooth out over time. Infinite speed of propagation (Gaussian has infinite support) Smoothing effect: Solutions become C-infinity for any t > 0 Maximum principle: Temperature stays bounded by initial extrema

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