Английская Википедия: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.[1] It is also an indispensable tool in the study of nonlinear partial differential equations such as the Navier–Stokes equations and nonlinear Schrödinger equation 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 Шаблон:Math in Шаблон:Math. Indicating by Шаблон:Math the time derivative of Шаблон:Math, the initial value problem is <math display="block">\begin{cases} 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{cases}</math> where g is the initial heat distribution. By contrast, the inhomogeneous problem for the heat equation, <math display="block">\begin{cases} 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{cases}</math> corresponds to adding an external heat energy Шаблон:Math 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 Шаблон:Math. By linearity, one can add up (integrate) the resulting solutions through time Шаблон:Math 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 <math display="block">u:D\times(0,\infty)\to \R</math> with spatial domain Шаблон:Mvar in Шаблон:Math, of the form <math display="block">\begin{cases} u_t(x,t) -Lu(x,t) = f(x,t) &(x,t)\in D\times (0,\infty)\\ u|_{\partial D} = 0 &\\ u(x,0) = 0 & x\in D, \end{cases}</math> where L is a linear differential operator that involves no time derivatives.
Duhamel's principle is, formally, that the solution to this problem is <math display="block">u(x,t) = \int_0^t (P^sf)(x,t)\,ds</math> where Шаблон:Math is the solution of the problem <math display="block">\begin{cases} u_t - Lu = 0 & (x,t)\in D\times (s,\infty)\\ u|_{\partial D} = 0 &\\ u(x,s) = f(x,s) & x\in D. \end{cases}</math> The integrand is the retarded solution <math>P^sf</math>, evaluated at time Шаблон:Mvar, representing the effect, at the later time Шаблон:Mvar, of an infinitesimal force <math>f(x,s)\,ds</math> applied at time Шаблон:Mvar.
Duhamel's principle also holds for linear systems (with vector-valued functions Шаблон:Math), 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 Шаблон:Math and Шаблон:Math depend on the particular application.
Examples
Wave equation
The linear wave equation models the displacement Шаблон:Math of an idealized dispersionless one-dimensional string, in terms of derivatives with respect to time Шаблон:Mvar and space Шаблон:Mvar:
<math display="block">\frac{\partial^2 u}{\partial t^2}-c^2\frac{\partial^2 u}{\partial x^2}=f(x,t).</math>
The function Шаблон:Math, in natural units, represents an external force applied to string at the position Шаблон:Math. 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:
<math display="block">u(x,0)=u_0(x),\qquad \frac{\partial u}{\partial t}(x,0) = v_0(x).</math>
More generally, we should be able to solve the equation with data specified on any Шаблон:Math slice:
<math display="block">u(x,T)=u_T(x),\qquad \frac{\partial u}{\partial t}(x,T)=v_T(x).</math>
To evolve a solution from any given time slice Шаблон:Mvar to Шаблон:Math, the contribution of the force must be added to the solution. That contribution comes from changing the velocity of the string by Шаблон:Math. That is, to get the solution at time Шаблон:Math from the solution at time Шаблон:Mvar, we must add to it a new (forward) solution of the homogeneous (no external forces) wave equation
<math display="block">\frac{\partial^2 U}{\partial t^2}-c^2\frac{\partial^2 U}{\partial x^2} = 0</math>
with the initial conditions
<math display="block">U(x,T)=0,\qquad \frac{\partial U}{\partial t}(x,T)=f(x,T) dT.</math>
A solution to this equation is achieved by straightforward integration:
<math display="block"> U(x,t) = \left(\frac{1}{2c}\int_{x-c(t-T)}^{x+c(t-T)} f(\xi,T)\,d\xi\right)\,dT</math>
(The expression in parentheses is just <math>P^Tf(x,t)</math> 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:
<math display="block">u(x,t) = \frac{1}{2}\left[u_0(x+ct)+ u_0(x-ct)\right]+ \frac{1}{2c}\int_{x-ct}^{x+ct} v_0(y) dy + \frac{1}{2c}\int_0^t\int_{x-c(t-T)}^{x+c(t-T)} f(\xi,T)\,d\xi\,dT.</math>
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, Шаблон:Mvar-th order inhomogeneous ordinary differential equation. <math display="block"> P(\partial_t)u(t) = F(t) </math> <math display="block"> \partial_t^j u(0) = 0, \; 0 \leq j \leq m-1 </math> where <math display="block"> P(\partial_t) := a_m \partial_t^m + \cdots + a_1 \partial_t + a_0,\; a_m \neq 0. </math>
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 <math display="block"> P(\partial_t)G = 0, \; \partial^j_t G(0) = 0, \quad 0\leq j \leq m-2, \; \partial_t^{m-1} G(0) = 1/a_m. </math>
Define <math> H = G \chi_{[0,\infty)} </math>, with <math>\chi_{[0,\infty)}</math> being the characteristic function of the interval <math>[0,\infty)</math>. Then we have
<math display="block"> P(\partial_t) H = \delta </math>
in the sense of distributions. Therefore
<math display="block">\begin{align}
u(t) &= (H \ast F)(t) \\
&= \int_0^\infty G(\tau)F(t-\tau)\,d\tau \\
&= \int_{-\infty}^t G(t-\tau)F(\tau)\, d\tau
\end{align}</math>
solves the ODE.
Constant-coefficient linear PDE
More generally, suppose we have a constant coefficient inhomogeneous partial differential equation
<math display="block"> P(\partial_t,D_x)u(t,x) = F(t,x) </math>
where <math display="block"> D_x = \frac{1}{i} \frac{\partial}{\partial x}. </math>
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 Шаблон:Mvar we have <math display="block"> P(\partial_t,\xi)\hat u(t,\xi) = \hat F(t,\xi).</math>
Assume that <math> P(\partial_t,\xi) </math> is an Шаблон:Mvar-th order ODE in Шаблон:Mvar. Let <math>a_m </math> be the coefficient of the highest order term of <math> P(\partial_t,\xi) </math>. Now for every <math>\xi </math> let <math>G(t,\xi) </math> solve <math display="block"> P(\partial_t,\xi)G(t,\xi) = 0, \; \partial^j_t G(0,\xi) = 0 \; \text{ for } 0\leq j \leq m-2, \; \partial_t^{m-1} G(0,\xi) = 1/a_m. </math>
Define <math>H(t,\xi) = G(t,\xi) \chi_{[0,\infty)}(t) </math>. We then have <math display="block"> P(\partial_t,\xi) H(t,\xi) = \delta(t) </math> in the sense of distributions. Therefore <math display="block">\begin{align}
\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_{-\infty}^t G(t-\tau,\xi) \hat F(\tau,\xi)\, d\tau
\end{align}</math> solves the PDE (after transforming back to Шаблон:Mvar).
See also
References
- ↑ Fritz John, "Partial Differential Equations', New York, Springer-Verlag, 1982, 4th ed., 0387906096
