Duhamel's formula

Duhamel's formula expresses the solution to a general inhomogeneous linear equation as a superposition of free solutions arising from both the initial data and the forcing term. For instance, the solution to the inhomogeneous initial value problem

$u_{t}-Lu=F;\quad u(0)=u_{0}$

for some spatial operator L, is given by

$u(t)=e^{tL}u_{0}+\int _{0}^{t}e^{(t-t')L}F(t')\ dt',$

provided that L has enough of a functional calculus, and u, u_0, F have enough regularity, to justify all computations. (If L is constant coefficient, then the Fourier transform can usually be used to justify everything so long as one works in the category of tempered distributions.) Note that the case L=0 is simply the fundamental theorem of calculus, indeed one can view Duhamel's formula as the fundamental theorem of calculus twisted (conjugated) by the free propagator $e^{tL}$.

For equations which are second order in time, the formula is slightly more complicated. For instance, the solution to the inhomogeneous initial value problem

$u_{tt}-Lu=F;\quad u(0)=u_{0};\quad u_{t}(0)=u_{1}$

is given (formally, at least) by

$u(t)=\cos(t{\sqrt {L}})u_{0}+{\frac {\sin(t{\sqrt {L}})}{\sqrt {L}}}u_{1}+\int _{0}^{t}{\frac {\sin((t-t'){\sqrt {L}})}{\sqrt {L}}}F(t')\ dt'.$