# Duhamel's principle

In mathematics, and more specifically in partial differential equations, Duhamel's principle is a general method for obtaining solutions to inhomogeneous linear evolution equations like the heat equation, wave equation, and vibrating plate equation. It is named after Jean-Marie Duhamel who first applied the principle to the inhomogeneous heat equation that models, for instance, the distribution of heat in a thin plate which is heated from beneath. For linear evolution equations without spatial dependency, such as a harmonic oscillator, Duhamel's principle reduces to the method of variation of parameters technique for solving linear inhomogeneous ordinary differential equations.[1]

The philosophy underlying Duhamel's principle is that it is possible to go from solutions of the Cauchy problem (or initial value problem) to solutions of the inhomogeneous problem. Consider, for instance, the example of the heat equation modeling the distribution of heat energy u in Rn. The initial value problem is

$\begin{cases} u_t(x,t) - \Delta u(x,t) = 0 &(x,t)\in \mathbf{R}^n\times (0,\infty)\\ u(x,0) = g(x) & x\in \mathbf{R}^n \end{cases}$

where g is the initial heat distribution. By contrast, the inhomogeneous problem for the heat equation is

$\begin{cases} u_t(x,t) -\Delta u(x,t) = f(x,t) &(x,t)\in \mathbf{R}^n\times (0,\infty)\\ u(x,0) = 0 & x\in \mathbf{R}^n \end{cases}$

corresponds to adding an external heat energy ƒ(x,t)dt at each point. Intuitively, one can think of the inhomogeneous problem as a set of homogeneous problems each starting afresh at a different time slice t = t0. By linearity, one can add up (integrate) the resulting solutions through time t0 and obtain the solution for the inhomogeneous problem. This is the essence of Duhamel's principle.

## General considerations

Formally, consider a linear inhomogeneous evolution equation for a function

$u:D\times(0,\infty)\to \mathbf{R}$

with spatial domain D in Rn, of the form

$\begin{cases} u_t(x,t) -Lu(x,t) = f(x,t) &(x,t)\in D\times (0,\infty)\\ u|_{\partial D} = 0 &\\ u(x,0) = 0 & x\in D, \end{cases}$

where L is a linear differential operator that involves no time derivatives.

Duhamel's principle is, formally, that the solution to this problem is

$u(x,t) = \int_0^t (P^sf)(x,t)\,ds$

where Psƒ is the solution of the problem

$\begin{cases} u_t - Lu = 0 & (x,t)\in D\times (s,\infty)\\ u|_{\partial D} = 0 &\\ u(x,s) = f(x,s) & x\in D. \end{cases}$

Duhamel's principle also holds for linear systems (with vector-valued functions u), and this in turn furnishes a generalization to higher t derivatives, such as those appearing in the wave equation (see below). Validity of the principle depends on being able to solve the homogeneous problem in an appropriate function space and that the solution should exhibit reasonable dependence on parameters so that the integral is well-defined. Precise analytic conditions on u and f depend on the particular application.

## Examples

### Wave equation

Given the inhomogeneous wave equation:

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

with initial conditions

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

A solution is

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

### 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, mth 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 of 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.

### 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 mth order ODE in t. Let $a_m$ be the coefficient of the highest order term of $P(\partial_t,\xi)$. Now for every $\xi$ let $G(t,\xi)$ solve

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

Define $H(t,\xi) = G(t,\xi) \chi_{[0,\infty)}(t)$. We then have

$P(\partial_t,\xi) H(t,\xi) = \delta(t)$

in the sense of distributions. Therefore

$\hat u(t,\xi) = (H(\cdot,\xi) \ast \hat F(\cdot,\xi))(t)$
$= \int_0^\infty G(\tau,\xi)F(t-\tau,\xi)\,d\tau$
$= \int_{-\infty}^t G(t-\tau,\xi)F(\tau,\xi)\, d\tau$

solves the PDE (after transforming back to x).