Duhamel integral

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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).
How to Cite This Entry:
Duhamel integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Duhamel_integral&oldid=40213
This article was adapted from an original article by A.K. Gushchin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article