- #1

- 101

- 25

$$

K(x,t) = \frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})

$$

is a solution to the heat equation:

$$

\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}

$$

I would like to ask how to prove:

$$

u(x,t) = \int_{-\infty}^{\infty} K(x-y,t)f(y)dy

$$

is also the solution to the equation, and also:

$$

\lim_{t\to 0^+} u(x,t) = f(x)

$$