Duhamel's principle as a Green's-function statement
/ doo-ah-MEL /
Suppose a system with no external forcing already has a known way of evolving from any starting state — you know the solution to the homogeneous problem. Now turn on a continuous source that keeps pushing over time. Duhamel's principle says you do not need a new method: treat the ongoing source as an unbroken stream of tiny instantaneous kicks, let each kick evolve forward freely from the moment it is applied, and add up all their contributions. It is the time-dependent face of the point-source idea — the Green's function in the time direction.
Concretely, for an inhomogeneous evolution equation u_t = L u + f(t) with zero initial data, Duhamel's formula is u(t) = integral from 0 to t of S(t - s) f(s) ds, where S(tau) is the solution operator (the propagator) of the homogeneous equation run for time tau. Read the recipe in plain steps: the source f(s) ds delivered in the small interval at time s acts like an initial condition switched on at time s; from then on it evolves freely under the homogeneous flow for the remaining time t - s, contributing S(t - s) f(s); integrate over all firing times s from 0 to t and you have the full forced solution. The kernel S(t - s) is exactly the causal Green's function in time — it vanishes for s > t, encoding that only past sources matter. For the heat equation S(t - s) acts by convolving with the heat kernel; for the wave equation it acts through the wave propagator. This is the PDE generalization of variation of parameters from ODEs.
Why it is powerful: it reduces every inhomogeneous (forced) linear evolution problem to the homogeneous one you already solved, separating the easy part (free evolution) from the bookkeeping (integrating the source over time). It underlies the mild-solution and semigroup theory of evolution equations, where S(t) is the C0-semigroup generated by L. The honest scope: Duhamel's principle is a linearity statement — it relies on superposition of the source's instantaneous responses — so it applies to linear equations; a nonlinear forcing breaks the clean superposition (though Duhamel-type integral equations are still a key fixed-point tool for nonlinear problems, where you iterate). It also assumes the homogeneous flow S is well-defined and well-posed, which for the backward heat equation, for instance, it is not.
For the heated rod u_t = k u_xx + f(x, t) with zero initial temperature, Duhamel gives u(x, t) = integral from 0 to t of [ integral of K(x - y, t - s) f(y, s) dy ] ds, where K is the heat kernel: each instant's heat input diffuses freely for the remaining time t - s.
Each instant's source becomes an initial condition that evolves freely.
Duhamel relies on superposition, so it is a linear-equation principle; nonlinear forcing breaks the clean sum (though Duhamel-type integral equations remain a key fixed-point tool for nonlinear problems), and it presumes the homogeneous flow is well-posed — false, e.g., for the backward heat equation.