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.
Indicating by ut (x, t) the time derivative of u(x, t), the initial value problem is
By contrast, the inhomogeneous problem for the heat equation,
corresponds to adding an external heat energy f (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.
Formally, consider a linear inhomogeneous evolution equation for a function
with spatial domain D in Rn, of the form
where L is a linear differential operator that involves no time derivatives.
Duhamel's principle is, formally, that the solution to this problem is
for the Cauchy problem with initial condition
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.
The linear wave equation models the displacement u of an idealized dispersionless one-dimensional string, in terms of derivatives with respect to time t and space x:
The function f (x, t), in natural units, represents an external force applied to string at the position (x, t).
In order to be a suitable physical model for nature, it should be possible to solve it for any initial state that the string is in, specified by its initial displacement and velocity:
More generally, we should be able to solve the equation with data specified on any t = constant slice:
That contribution comes from changing the velocity of the string by f (x, T) dT.
A solution to this equation is achieved by straightforward integration:
So a solution of the original initial value problem is obtained by starting with a solution to the problem with the same prescribed initial values problem but with zero initial displacement, and adding to that (integrating) the contributions from the added force in the time intervals from T to T+dT:
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.
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.
More generally, suppose we have a constant coefficient inhomogeneous partial differential equation
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.
solves the PDE (after transforming back to x).