4.4 Sources: Duhamel’s principle
Theorem 4.4.1. If \(F\) is \(C^1\), the solution of \(u_{tt} = c^2u_{xx} + F(x, t)\) with \(u(x, 0) = u_t(x, 0) = 0\) is \[ u(x, t) = \frac 1{2c}\int _0^t\!\!\int _{x - c(t - s)}^{x + c(t - s)}F(y, s)\,dy\,ds , \] the integral of \(F\) over the characteristic triangle below \((x, t)\), divided by \(2c\).
Proof. For each \(s\), let \(v(\cdot , \cdot ; s)\) solve the homogeneous equation for \(t > s\) with \(v = 0\) and \(v_t = F(\cdot , s)\) at \(t = s\); by d’Alembert, \(v(x, t; s) = \frac 1{2c}\int _{x - c(t - s)}^{x + c(t - s)}F(y, s)\,dy\). Put \(u = \int _0^tv(x, t; s)\,ds\). Then \(u_t = v(x, t; t) + \int _0^tv_t\,ds = \int _0^tv_t\,ds\), since \(v(x, s; s) = 0\), and \[ u_{tt} = v_t(x, t; t) + \int _0^tv_{tt}\,ds = F(x, t) + c^2\int _0^tv_{xx}\,ds = F + c^2u_{xx} . \] The initial conditions hold since both integrals vanish at \(t = 0\). □
The idea, Duhamel’s principle, is that a source acting over time is a succession of impulses, each of which sets off a solution of the homogeneous problem, as for the transport equation in Section 1. With non-zero data, add the d’Alembert solution.
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