Duhamel integral

A representation of the solution of the Cauchy problem (or of a mixed problem) for an inhomogeneous linear partial differential equation with homogeneous boundary conditions by means of the solution of the corresponding problem for the homogeneous equation. Consider the equation $$ (1) \qquad \frac{\partial^{2} u(t,x)}{\partial t^{2}} + L[u(t,x)] = f(t,x), \qquad x \in \mathbf{R}^{n}, ~ t > 0, $$ where $ L $ is a linear differential operator with coefficients independent of $ t $, containing derivatives in $ t $ of order not exceeding $ 1 $. The Cauchy problem for (1) is posed with the initial conditions $$ (2) \qquad u(t,x)|_{t = 0} = 0 \qquad \text{and} \qquad \frac{\partial u(t,x)}{\partial t} \Bigg|_{t = 0} = 0. $$

Let the sufficiently smooth function $ v(t,x;\tau) $, where $ t \geq \tau \geq 0 $ and $ x \in \mathbf{R}^{n} $, for $ t > \tau $, be a solution of the homogeneous equation $$ \frac{\partial^{2} v(t,x;\tau)}{\partial t^{2}} + L[v(t,x;\tau)] = 0, $$ and let it satisfy, for $ t = \tau $, the initial conditions $$ v(t,x;\tau)|_{t = \tau} = 0 \qquad \text{and} \qquad \frac{\partial v(t,x;\tau)}{\partial t} \Bigg|_{t = \tau} = f(\tau,x). $$ Then the solution of the Cauchy problem (1) + (2) is given by the Duhamel integral $$ u(t,x) = \int_{0}^{t} v(t,x;\tau) ~ \mathrm{d}{\tau}. $$ This theorem, known as Duhamel’s principle, is an analogue of the method of variation of constants.

A similar construction can be used for the Cauchy problem with a homogeneous initial condition for the equation $$ \frac{\partial u(t,x)}{\partial t} + M[u(t,x)] = f(t,x), \qquad x \in \mathbf{R}^{n}, ~ t > 0, $$ where $ M $ is a linear differential operator with coefficients independent of $ t $, containing derivatives with respect to the variable $ x $ only.

The solution of the Cauchy problem with homogeneous initial conditions for the inhomogeneous heat equation is expressed by the Duhamel integral $$ u(t,x) = \int_{0}^{t} \int_{\mathbf{R}^{n}} [4 \pi (t - \tau)]^{- n / 2} e^{- \| x - \xi \|^{2} / 4(t - \tau)} f(\tau,\xi) ~ \mathrm{d}{\xi} ~ \mathrm{d}{\tau}. $$ For the wave equation, if $ n = 1 $, one has $$ u(t,x) = \int_{0}^{t} \int_{x - (t - \tau)}^{x + (t - \tau)} f(\tau,\xi) ~ \mathrm{d}{\xi}. $$

The integral is named after J. Duhamel.

References

[1]	A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, “Differentialgleichungen der mathematischen Physik”, Deutsch. Verlag Wissenschaft (1959). (Translated from Russian)
[2]	F. John, “Planar waves and spherical means as applied to partial differential equations”, Interscience (1955).

References

[a1]	H.S. Carslaw, J.C. Jaeger, “Conduction of heat in solids”, Clarendon Press (1959).