- #1

lriuui0x0

- 101

- 25

- TL;DR Summary
- Trying to understand heat equation general solution through Green's function

Given a 1D heat equation on the entire real line, with initial condition ##u(x, 0) = f(x)##. The general solution to this is:

$$

u(x, t) = \int \phi(x-y, t)f(y)dy

$$

where ##\phi(x, t)## is the heat kernel.

The integral looks a lot similar to using Green's function to solve differential equation. The fact that ##\phi(x, 0) = \delta(x)## also signals something related to Green's function. This wikipedia page talks about Green's function related to heat equation as well.

However after searching on the internet, I don't get how do I understand the Green's function in the context of heat equation. As I understand, Green's function is related to a particular ordinary linear differential operator. What's the differential operator for heat equation?

$$

u(x, t) = \int \phi(x-y, t)f(y)dy

$$

where ##\phi(x, t)## is the heat kernel.

The integral looks a lot similar to using Green's function to solve differential equation. The fact that ##\phi(x, 0) = \delta(x)## also signals something related to Green's function. This wikipedia page talks about Green's function related to heat equation as well.

However after searching on the internet, I don't get how do I understand the Green's function in the context of heat equation. As I understand, Green's function is related to a particular ordinary linear differential operator. What's the differential operator for heat equation?