In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and more specifically in
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how ...
, Duhamel's principle is a general method for obtaining solutions to
inhomogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
linear evolution equations like the
heat equation
In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...
,
wave equation
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...
, 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
In 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 const ...
, Duhamel's principle reduces to the method of
variation of parameters
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous differential equation, inhomogeneous linear ordinary differential equations.
For first-order inhomogeneous linear differenti ...
technique for solving linear inhomogeneous
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives ...
.
[Fritz John, "Partial Differential Equations', New York, Springer-Verlag, 1982, 4th ed., 0387906096] It is also an indispensable tool in the study of nonlinear partial differential equations such as the
Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
and
nonlinear Schrödinger equation
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonli ...
where one treats the nonlinearity as an inhomogeneity.
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 in . Indicating by the time derivative of , the initial value problem is
where ''g'' is the initial heat distribution. By contrast, the inhomogeneous problem for the heat equation,
corresponds to adding an external heat energy 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 . By linearity, one can add up (integrate) the resulting solutions through time and obtain the solution for the inhomogeneous problem. This is the essence of Duhamel's principle.
General considerations
Formally, consider a
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
inhomogeneous evolution equation for a function
with spatial domain in , 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 is the solution of the problem
The integrand is the retarded solution
, evaluated at time , representing the effect, at the later time , of an infinitesimal force
applied at time . (The operator
can be thought of as an inverse of the operator
for the Cauchy problem with initial condition
.)
Duhamel's principle also holds for linear systems (with vector-valued functions ), 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 and depend on the particular application.
Examples
Wave equation
The linear wave equation models the displacement of an idealized dispersionless one-dimensional string, in terms of derivatives with respect to time and space :
The function , in natural units, represents an external force applied to string at the position . 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 slice:
To evolve a solution from any given time slice to , the contribution of the force must be added to the solution. That contribution comes from changing the velocity of the string by . That is, to get the solution at time from the solution at time , we must add to it a new (forward) solution of the ''homogeneous'' (no external forces) wave equation
with the initial conditions
A solution to this equation is achieved by straightforward integration:
(The expression in parentheses is just
in the notation of the general method above.) 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'':
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, -th order inhomogeneous
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
.
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 function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
of the interval