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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 \begin u_t(x,t) - \Delta u(x,t) = 0 &(x,t)\in \R^n\times (0,\infty)\\ u(x,0) = g(x) & x\in \R^n \end where ''g'' is the initial heat distribution. By contrast, the inhomogeneous problem for the heat equation, \begin u_t(x,t) -\Delta u(x,t) = f(x,t) &(x,t)\in \R^n\times (0,\infty)\\ u(x,0) = 0 & x\in \R^n \end 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 u:D\times(0,\infty)\to \R with spatial domain in , of the form \begin u_t(x,t) -Lu(x,t) = f(x,t) &(x,t)\in D\times (0,\infty)\\ u, _ = 0 &\\ u(x,0) = 0 & x\in D, \end 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 is the solution of the problem \begin v_t - Lv = 0 & (x,t)\in D\times (s,\infty)\\ v, _ = 0 &\\ v(x,s) = f(x,s) & x\in D. \end The integrand is the retarded solution P^sf, evaluated at time , representing the effect, at the later time , of an infinitesimal force f(x,s)\,ds applied at time . (The operator P^s can be thought of as an inverse of the operator \partial_t - L for the Cauchy problem with initial condition f(x,s).) 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 : \frac-c^2\frac=f(x,t). 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: u(x,0)=u_0(x),\qquad \frac(x,0) = v_0(x). More generally, we should be able to solve the equation with data specified on any slice: u(x,T)=u_T(x),\qquad \frac(x,T)=v_T(x). 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 \frac-c^2\frac = 0 with the initial conditions U(x,T)=0,\qquad \frac(x,T)=f(x,T) dT. A solution to this equation is achieved by straightforward integration: U(x,t) = \left(\frac\int_^ f(\xi,T)\,d\xi\right)\,dT (The expression in parentheses is just P^Tf(x,t) 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'': u(x,t) = \frac\left _0(x+ct)+ u_0(x-ct)\right \frac\int_^ v_0(y) dy + \frac\int_0^t\int_^ f(\xi,T)\,d\xi\,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 ...
. 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^ G(0) = 1/a_m. Define H = G \chi_ , with \chi_ 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 P(\partial_t) H = \delta in the sense of Distribution (mathematics)">distributions. Therefore \begin u(t) &= (H \ast F)(t) \\ &= \int_0^\infty G(\tau)F(t-\tau)\,d\tau \\ &= \int_^t G(t-\tau)F(\tau)\, d\tau \end 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 \frac. 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 mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
in we have P(\partial_t,\xi)\hat u(t,\xi) = \hat F(t,\xi). Assume that P(\partial_t,\xi) is an -th order ODE in . 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 \; \text 0\leq j \leq m-2, \; \partial_t^ G(0,\xi) = 1/a_m. Define H(t,\xi) = G(t,\xi) \chi_(t) . We then have P(\partial_t,\xi) H(t,\xi) = \delta(t) in the sense of distributions. Therefore \begin \hat u(t,\xi) &= (H(\cdot,\xi) \ast \hat F(\cdot,\xi))(t) \\ &= \int_0^\infty G(\tau,\xi) \hat F(t-\tau,\xi)\,d\tau \\ &= \int_^t G(t-\tau,\xi) \hat F(\tau,\xi)\, d\tau \end solves the PDE (after transforming back to ).


See also

*
Retarded potential In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so t ...
*
Propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. I ...
*
Impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
*
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 ...


References

{{reflist Wave mechanics Partial differential equations Mathematical principles