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