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

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 u_t - Lu = F; \quad u(0) = u_0}

for some spatial operator L, is given by

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 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 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 u } , 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 u_0 } , 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 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 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 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

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 u_{tt} - Lu = F; \quad u(0) = u_0; \quad u_t(0) = u_1}

is given (formally, at least) by

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 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 (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 {\underline y}=\begin{bmatrix} u & w \end{bmatrix}}

and

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 {\underline \Phi}=\begin{bmatrix} 0 & F(t) \end{bmatrix}}

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{bmatrix}}

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We can write the second order equation as

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 {\underline y}_{tt}-M{\underline y}={\underline \Phi(t)}}

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

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 {\underlien y}(t) = e^{tM} {\underline y}_0 + \int_0^t e^{(t-t')M} {\underline \Phi}(t')\ dt',}

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