Duhamel's principle

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In mathematics, and more specifically in partial differential equations, Duhamel's principle is the solution to the inhomogeneous wave equation. It is named after Jean-Maria-Constant Duhamel (1797–1872), a French applied mathematician.

Given the inhomogeneous wave equation:

$u_{tt}-c^2u_{xx}=f(x,t)\,$

with initial conditions

$u(x,0)=u_t(x,0)=0.\,$

The solution is

$u(x,t) = \frac{1}{2c}\int_0^t\int_{x-c(t-s)}^{x+c(t-s)} f(\xi,s)\,d\xi\,ds.\,$

[edit] Constant-coefficient linear ODE

Duhamel's principle is the result that the solution to an inhomogeneous, linear, partial differential equation can be solved by first finding the solution for a step input, and then superposing using Duhamel's integral. Suppose we have a constant coefficient, m^th order inhomogeneous ordinary differential equation.

$P(\partial_t)u(t) = F(t) \,$

$\partial_t^j u(0) = 0, \; 0 \leq j \leq m-1$

where

$P(\partial_t) := a_m \partial_t^m + \cdots + a_1 \partial_t + a_0,\; a_m \neq 0.$

We can reduce this to the solution of a homogeneous ODE using the following method. All steps are done formally, ignoring necessary requirements for the solution to be well defined.

First let G solve

$P(\partial_t)G = 0, \; \partial^j_t G(0) = 0, \quad 0\leq j \leq m-2, \; \partial_t^{m-1} G(0) = 1/a_m.$

Define $H = G \chi_{[0,\infty)}$ , with $\chi_{[0,\infty]}$ being the characteristic function on the interval $[0,\infty)$ . Then we have

$P(\partial_t) H = \delta$

in the sense of distributions. Therefore

$u(t) = (H \ast F)(t)$

$= \int_0^\infty G(\tau)F(t-\tau)\,d\tau$

$= \int_{-\infty}^t G(t-\tau)F(\tau)\, d\tau$

solves the ODE.

[edit] Constant-coefficient linear PDE

More generally, suppose we have a constant coefficient inhomogeneous partial differential equation

$P(\partial_t,D_x)u(t,x) = F(t,x) \,$

where

$D_x = \frac{1}{i} \frac{\partial}{\partial x} \,$

We can reduce this to the solution of a homogeneous ODE using the following method. All steps are done formally, ignoring necessary requirements for the solution to be well defined.

First, taking the Fourier transform in x we have

$P(\partial_t,\xi)\hat u(t,\xi) = \hat F(t,\xi).$

Assume that $P(\partial_t,\xi)$ is an m^th order ODE in t. Let $a m$ be the coefficient of the highest order term of $P(\partial_t,\xi)$ . Now for every $ξ$ let $G (t,ξ)$ solve