non homogeneous heat equation?

Question

consider the non homogeneous heat equation :

let $f:\mathbb{R}^n\times \mathbb{R}^+ \rightarrow \mathbb{R}$ be a smooth function

$\frac{\partial u}{\partial t}-\Delta u = f(x,t)$ , $x\in\mathbb{R}^n,t\in\mathbb{R}$

$u(x,0)=0$

so can someone tell me how to verify that the solution of this equation is given by :

$u(x,t)=\int_0^t(\int_{\mathbb{R}^n}\phi_n(x-y,t-s)f(y,s)dy)ds$

where $\phi_n$ is the fundamentale solution of the homogeneous heat equation.

thank you very much.

try with $f(x,t) = e^{i r x +i \omega t}$ you'll see if it gives the correct result — reuns, Feb 02 '16 at 21:40
HINT: Look for the keywords "Duhamel's formula for the heat equation". — Giuseppe Negro, Feb 02 '16 at 21:41

score 0 · Answer 1 · answered Jan 28 '18 at 22:04

0

See Evans PDE book, Theorem 2, p. 50. or see page 18 of PDF.

answered Jan 28 '18 at 22:04

All

1 Answers1