Heat Kernel

PDEs · Axiom Academy

The fundamental solution to the heat equation The heat kernel (or fundamental solution) is the temperature distribution that evolves from a single point source of heat. It's denoted K(x,t) and is the building block for all heat equation solutions. For the 1D heat equation u t = ku xx , the heat kernel is: where k is the thermal diffusivity and t > 0 . Initial Condition: The Delta Function As t → 0 + , the heat kernel approaches a Dirac delta function: This means all the heat starts concentrated at a single point (x = 0). The heat kernel is a Gaussian (bell curve) that starts infinitely narrow and spreads out over time. Watch as the heat kernel evolves from a concentrated spike to a broad, flat distribution! Width grows as √t: The standard deviation is σ = √(2kt), so the spread increases with the square root of time. Height decreases as 1/√t: As the Gaussian spreads, it must get shorter to preserve the total amount of heat. Area is conserved: The integral ∫ K(x,t) dx = 1 for all t > 0 (conservation of heat). Any solution to the heat equation can be expressed as a convolution of the initial temperature distribution f(x) with the heat kernel. This is the superposition principle in action! Given initial condition u(x,0) = f(x), the solution at time t is: This integral "smears out" the initial data using the heat kernel as a smoothing function.

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