§ 09 / diffusion

The Diffusion Equation

Heat equation derivation, solutions via Gaussians, uniqueness, and maximum principles.

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Section 1 / 6

The Diffusion Equation — Overview

The diffusion equation is an example of the remarkable efficiency of mathematics in describing very different natural phenomena. The diffusion equation

∂u∂t=λ∂2u∂x2,λ>0,\boxed{ \frac{\partial u}{\partial t} = \lambda\frac{\partial^2u}{\partial x^2}, \qquad \lambda>0, }

is a partial differential equation that occurs in the theory of heat conduction, the spreading of gases, and the description of the movement of microorganisms.

A related differential equation, with a negative sign in front of λ\lambda, already appeared in Chapter 5 in connection with Itô's formula. Before discussing the connections with stochastic analysis in the next chapter, we first study the diffusion equation from an analytical perspective.