= Removing a variable scalar reaction rate by an integrating factor
{title2=$u(x,t)=e^{-\int_0^ta(s)ds}v(x,t)$}
For $u_t=u_{xx}-a(t)u$, the displayed substitution converts the equation exactly into the <heat equation> $v_t=v_{xx}$. The scalar prefactor preserves homogeneous <Dirichlet boundary conditions>. Consequently each spatial <eigenfunction> $X_n$ with $X_n''=-\lambda_nX_n$ acquires temporal factor $\exp[-\lambda_nt-\int_0^ta(s)ds]$. The same algebra works with an evolution variable named $y$; calling that variable spatial does not alter the <separation of variables>. For $a(t)=1+\sin t$, the prefactor normalized to one at zero is $e^{-t+\cos t-1}$.
Back to article page
