In

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, and more specifically in partial differential equations,

**Duhamel's principle** is a general method for obtaining solutions to

inhomogeneousThe term homogeneous differential equation has several distinct meanings.One meaning is that a first-order ordinary differential equation is homogeneous if it has the formwhere F is a homogeneous function of degree zero; that is to say, that F = F.In a related, but distinct, usage, the term linear...

linear evolution equations like the

heat equationThe heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

,

wave equationThe wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...

, and

vibrating plateIn solid mechanics, a branch of mathematics and physics, a plate is modeled as a two-dimensional elastic body whose potential energy depends on how it is bent from a planar configuration, rather than how it is stretched . A vibrating plate can be modeled in a manner analogous to a vibrating drum...

equation. It is named for

Jean-Marie DuhamelJean-Marie Constant Duhamel was a noted French mathematician and physicist. His studies were affected by troubles of the Napoleonic era. He went on to form his own school École Sainte-Barbe. Duhamel's principle is named for him. He was primarily a mathematician but did studies on the mathematics...

who first applied the principle to the inhomogeneous heat equation that models, for instance, the distribution of heat in a thin plate heated from beneath. For linear evolution equations without spatial dependency, such as a

harmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

, 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 problemA Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain. Cauchy problems are an extension of initial value problems and are to be contrasted with boundary value problems...

(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

**R**^{n}. 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 set of homogeneous problems each starting afresh at a different time slice

*t* =

*t*_{0}. By linearity, one can add up (integrate) the resulting solutions through time

*t*_{0} 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

with spatial domain

*D* in

**R**^{n}, 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

where

*P*^{s}*ƒ* is the solution of the problem

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.

### Wave equation

Given the inhomogeneous wave equation:

with initial conditions

A solution is

### 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 equationIn mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

.

where

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

Define

, with

being the

characteristic functionIn mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* In probability theory, the characteristic function of any probability distribution on the real line is given by...

on the interval

. Then we have

in the sense of

distributionsIn mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

. Therefore

solves the ODE.

### Constant-coefficient linear PDE

More generally, suppose we have a constant coefficient inhomogeneous

partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

where

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 transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

in

*x* we have

Assume that

is an m

^{th} order ODE in

*t*. Let

be the coefficient of the highest order term of

.

Now for every

let

solve

Define

. We then have

in the sense of

distributionsIn mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

. Therefore

solves the PDE (after transforming back to

*x*).