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

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

Anyhow, we note that in this case the solution can be cast in the standard form. So, let us introduce the vectors

Failed to parse (unknown function "\begin{bmatrix}"): {\displaystyle y=\begin{bmatrix} u & w \end{pmatrix}}

and

Failed to parse (unknown function "\begin{bmatrix}"): {\displaystyle \Phi=\begin{bmatrix} 0 & F(t) \end{pmatrix}}

with the matrix

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M=\begin{bmatrix} 0 & 1 \\ L & 0 \end{pmatrix}}

.

We can write the second order equation as

$y_{tt}-My=\Phi (t)$

and write the solution as expected in the original Duhamel's formula, that is

$u(t)=e^{tM}y_{0}+\int _{0}^{t}e^{(t-t')M}\Phi (t')\ dt',$

.

Useful applications of this approach can be found for systems having a Hamiltonian flow.