- #1

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## Homework Statement

Hi all,

I came across these steps in my notes, relating to a step whereby,

$$\hat{G} (k, t - t') = \int_{-\infty}^{\infty} e^{-ik(x - x')}G(x-x' , t-t')dx$$

and performing the following operation on ##\hat{G}## gives the following expression,

$$[\frac{\partial}{\partial t} - D\frac{\partial ^2}{\partial x^2}] \hat{G} = \frac{\partial \hat{G}}{\partial t} + Dk^2 \hat{G}$$

These steps are in the context of a more complex problem of solving the Heat Equation using the Green's function.

## Homework Equations

## The Attempt at a Solution

I can partially understand the writing of ##\frac{\partial \hat{G}}{\partial t}## at the LHS as the same thing, ##\frac{\partial \hat{G}}{\partial t}## at the RHS; we do not know how ##G## (found in the first integral) depends on time exactly.

But the other terms that imply ##- D\frac{\partial ^2 \hat{G}}{\partial x^2} = Dk^2 \hat{G}## I can't follow. First of all, we don't know how ##G## varies with ##x##. Add to that that the fact that the integral isn't independent of ##x##, I don't see how we can apply differentiation under the integral sign to bring down the factor of ##(-ik)^2## down from ##e^{-ik(x-x')}##

Could someone explain the logic behind the steps? Help is greatly appreciated!