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A representation of the solution of the [[Cauchy problem|Cauchy problem]] (or of a mixed problem) for an inhomogeneous linear partial differential equation with homogeneous boundary conditions by means of the solution of the corresponding problem for the homogeneous equation. Consider the equation
 
A representation of the solution of the [[Cauchy problem|Cauchy problem]] (or of a mixed problem) for an inhomogeneous linear partial differential equation with homogeneous boundary conditions by means of the solution of the corresponding problem for the homogeneous equation. Consider the equation
 +
$$
 +
(1) \qquad
 +
\frac{\partial^{2} u(t,x)}{\partial t^{2}} + L[u(t,x)] = f(t,x), \qquad x \in \mathbf{R}^{n}, ~ t > 0,
 +
$$
 +
where $ L $ is a linear differential operator with coefficients independent of $ t $, containing derivatives in $ t $ of order not exceeding $ 1 $. The Cauchy problem for (1) is posed with the initial conditions
 +
$$
 +
(2) \qquad
 +
u(t,x)|_{t = 0} = 0 \qquad \text{and} \qquad
 +
\frac{\partial u(t,x)}{\partial t} \Bigg|_{t = 0} = 0.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d0341601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
Let the sufficiently smooth function $ v(t,x;\tau) $, where $ t \geq \tau \geq 0 $ and $ x \in \mathbf{R}^{n} $, for $ t > \tau $, be a solution of the homogeneous equation
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d0341602.png" /> is a linear differential operator with coefficients independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d0341603.png" />, containing derivatives in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d0341604.png" /> of order not exceeding 1. The Cauchy problem for (1) is posed with the initial conditions:
+
\frac{\partial^{2} v(t,x;\tau)}{\partial t^{2}} + L[v(t,x;\tau)] = 0,
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d0341605.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
and let it satisfy, for $ t = \tau $, the initial conditions
 
+
$$
Let the sufficiently smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d0341606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d0341607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d0341608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d0341609.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416010.png" />, be a solution of the homogeneous equation
+
v(t,x;\tau)|_{t = \tau} = 0 \qquad \text{and} \qquad
 
+
\frac{\partial v(t,x;\tau)}{\partial t} \Bigg|_{t = \tau} = f(\tau,x).
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416011.png" /></td> </tr></table>
+
$$
 
+
Then the solution of the Cauchy problem (1) + (2) is given by the Duhamel integral
and let it satisfy, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416012.png" />, the initial conditions
+
$$
 
+
u(t,x) = \int_{0}^{t} v(t,x;\tau) ~ \mathrm{d}{\tau}.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416013.png" /></td> </tr></table>
+
$$
 
+
This theorem, known as '''Duhamel’s principle''', is an analogue of the method of variation of constants.
Then the solution of the Cauchy problem (1), (2) is given by the Duhamel integral
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416014.png" /></td> </tr></table>
 
 
 
This theorem, known as Duhamel's principle, is an analogue of the method of variation of constants.
 
  
 
A similar construction can be used for the Cauchy problem with a homogeneous initial condition for the equation
 
A similar construction can be used for the Cauchy problem with a homogeneous initial condition for the equation
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416015.png" /></td> </tr></table>
+
\frac{\partial u(t,x)}{\partial t} + M[u(t,x)] = f(t,x), \qquad x \in \mathbf{R}^{n}, ~ t > 0,
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416016.png" /> is a linear differential operator with coefficients independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416017.png" />, containing derivatives with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416018.png" /> only.
+
where $ M $ is a linear differential operator with coefficients independent of $ t $, containing derivatives with respect to the variable $ x $ only.
  
 
The solution of the Cauchy problem with homogeneous initial conditions for the inhomogeneous heat equation is expressed by the Duhamel integral
 
The solution of the Cauchy problem with homogeneous initial conditions for the inhomogeneous heat equation is expressed by the Duhamel integral
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416019.png" /></td> </tr></table>
+
u(t,x) = \int_{0}^{t} \int_{\mathbf{R}^{n}} [4 \pi (t - \tau)]^{- n / 2} e^{- \| x - \xi \|^{2} / 4(t - \tau)} f(\tau,\xi) ~ \mathrm{d}{\xi} ~ \mathrm{d}{\tau}.
 
+
$$
For the wave equation if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416020.png" /> one has:
+
For the wave equation, if $ n = 1 $, one has
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034160/d03416021.png" /></td> </tr></table>
+
u(t,x) = \int_{0}^{t} \int_{x - (t - \tau)}^{x + (t - \tau)} f(\tau,\xi) ~ \mathrm{d}{\xi}.
 +
$$
  
 
The integral is named after J. Duhamel.
 
The integral is named after J. Duhamel.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. John,  "Planar waves and spherical means as applied to partial differential equations" , Interscience  (1955)</TD></TR></table>
 
  
 +
<table>
 +
<TR><TD valign="top">[1]</TD><TD valign="top">
 +
A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, “Differentialgleichungen der mathematischen Physik”, Deutsch. Verlag Wissenschaft (1959). (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD><TD valign="top">
 +
F. John, “Planar waves and spherical means as applied to partial differential equations”, Interscience (1955).</TD></TR>
 +
</table>
  
 +
====References====
  
====Comments====
+
<table>
 
+
<TR><TD valign="top">[a1]</TD><TD valign="top">
 
+
H.S. Carslaw, J.C. Jaeger, “Conduction of heat in solids”, Clarendon Press (1959).</TD></TR>
====References====
+
</table>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.S. Carslaw,   J.C. Jaeger,   "Conduction of heat in solids" , Clarendon Press (1959)</TD></TR></table>
 

Latest revision as of 06:54, 3 March 2017

A representation of the solution of the Cauchy problem (or of a mixed problem) for an inhomogeneous linear partial differential equation with homogeneous boundary conditions by means of the solution of the corresponding problem for the homogeneous equation. Consider the equation $$ (1) \qquad \frac{\partial^{2} u(t,x)}{\partial t^{2}} + L[u(t,x)] = f(t,x), \qquad x \in \mathbf{R}^{n}, ~ t > 0, $$ where $ L $ is a linear differential operator with coefficients independent of $ t $, containing derivatives in $ t $ of order not exceeding $ 1 $. The Cauchy problem for (1) is posed with the initial conditions $$ (2) \qquad u(t,x)|_{t = 0} = 0 \qquad \text{and} \qquad \frac{\partial u(t,x)}{\partial t} \Bigg|_{t = 0} = 0. $$

Let the sufficiently smooth function $ v(t,x;\tau) $, where $ t \geq \tau \geq 0 $ and $ x \in \mathbf{R}^{n} $, for $ t > \tau $, be a solution of the homogeneous equation $$ \frac{\partial^{2} v(t,x;\tau)}{\partial t^{2}} + L[v(t,x;\tau)] = 0, $$ and let it satisfy, for $ t = \tau $, the initial conditions $$ v(t,x;\tau)|_{t = \tau} = 0 \qquad \text{and} \qquad \frac{\partial v(t,x;\tau)}{\partial t} \Bigg|_{t = \tau} = f(\tau,x). $$ Then the solution of the Cauchy problem (1) + (2) is given by the Duhamel integral $$ u(t,x) = \int_{0}^{t} v(t,x;\tau) ~ \mathrm{d}{\tau}. $$ This theorem, known as Duhamel’s principle, is an analogue of the method of variation of constants.

A similar construction can be used for the Cauchy problem with a homogeneous initial condition for the equation $$ \frac{\partial u(t,x)}{\partial t} + M[u(t,x)] = f(t,x), \qquad x \in \mathbf{R}^{n}, ~ t > 0, $$ where $ M $ is a linear differential operator with coefficients independent of $ t $, containing derivatives with respect to the variable $ x $ only.

The solution of the Cauchy problem with homogeneous initial conditions for the inhomogeneous heat equation is expressed by the Duhamel integral $$ u(t,x) = \int_{0}^{t} \int_{\mathbf{R}^{n}} [4 \pi (t - \tau)]^{- n / 2} e^{- \| x - \xi \|^{2} / 4(t - \tau)} f(\tau,\xi) ~ \mathrm{d}{\xi} ~ \mathrm{d}{\tau}. $$ For the wave equation, if $ n = 1 $, one has $$ u(t,x) = \int_{0}^{t} \int_{x - (t - \tau)}^{x + (t - \tau)} f(\tau,\xi) ~ \mathrm{d}{\xi}. $$

The integral is named after J. Duhamel.

References

[1] A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, “Differentialgleichungen der mathematischen Physik”, Deutsch. Verlag Wissenschaft (1959). (Translated from Russian)
[2] F. John, “Planar waves and spherical means as applied to partial differential equations”, Interscience (1955).

References

[a1] H.S. Carslaw, J.C. Jaeger, “Conduction of heat in solids”, Clarendon Press (1959).
How to Cite This Entry:
Duhamel integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Duhamel_integral&oldid=40213
This article was adapted from an original article by A.K. Gushchin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article