P(t,x) is in $c^2$, and f(t) is a time dependent polynomial. Here we just consider $x \in R$ does the following heat equation have closed formula? Thank you.
$P_t$ = $P_{xx}$ + $f(t)P_x$
If yes, could you tell me where I can find it? Thank you.
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Introduce new independent variables $T=t$, $X=x+\int f(t)dt$. Then your equation becomes nothing but the standard heat equation $$ P_T=P_{XX}. $$ You can take the formula for general solution of this equation and transform it back to your case using the inverse change of variables $x=X-\int f(T)dT$, $t=T$. Of course, all of this works for an arbitrary smooth $f$ (not necessarily a polynomial). |
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