Namespaces
Variants
Actions

Difference between revisions of "Darboux equation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
An ordinary differential equation
 
An ordinary differential 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/d030/d030130/d0301301.png" /></td> </tr></table>
+
$$\frac{\mbox{d}y}{\mbox{d}x}=\frac{P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d0301302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d0301303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d0301304.png" /> are integral polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d0301305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d0301306.png" />. This equation was first studied by G. Darboux [[#References|[1]]]. The [[Jacobi equation|Jacobi equation]] is a special case of the Darboux equation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d0301307.png" /> be a highest degree of the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d0301308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d0301309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013010.png" />; if the Darboux equation has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013011.png" /> known particular algebraic solutions, then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013012.png" />, its general solution is found without quadratures, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013013.png" />, an integrating factor can be found [[#References|[2]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013015.png" /> are homogeneous functions of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013016.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013017.png" /> is a homogeneous function of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013018.png" /> then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013019.png" />, the Darboux equation is a homogeneous differential equation; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013020.png" />, the Darboux equation may be reduced to a [[Bernoulli equation|Bernoulli equation]] by substituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013021.png" />.
+
where $P$, $Q$ and $R$ are integral polynomials in $x$ and $y$. This equation was first studied by G. Darboux [[#References|[1]]]. The [[Jacobi equation|Jacobi equation]] is a special case of the Darboux equation. Let $n$ be the highest degree of the polynomials $P$, $Q$, $R$; if the Darboux equation has $s$ known particular algebraic solutions, then if $s\geq 2+n(n+1)/2$, its general solution is found without quadratures, and if $s=1+n(n+1)/2$, an integrating factor can be found [[#References|[2]]]. If $P$ and $Q$ are homogeneous functions of degree $m$, and $R$ is a homogeneous function of degree $k$ then, if $k=m-1$, the Darboux equation is a homogeneous differential equation; if $k\neq m-1$, the Darboux equation may be reduced to a [[Bernoulli equation|Bernoulli equation]] by substituting $y=zx$.
  
 
====References====
 
====References====
Line 11: Line 12:
  
 
The hyperbolic equation
 
The hyperbolic equation
 +
$$u_{tt}-\Delta u+\frac{\lambda(t,x)}{t}u_t=0,\,\,\,t\neq0,$$
  
<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/d030/d030130/d03013022.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013023.png" /> is a non-negative continuously-differentiable function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013024.png" />. The following uniqueness theorem is valid both for the solution of the Darboux equation and for the solution of the wave equation. If some twice continuously-differentiable solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013025.png" /> of the Darboux equation vanishes together with its derivative on the base of the characteristic cone lying in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013026.png" />, it vanishes inside the entire domain bounded by this cone. The form of the characteristic cone is the same as for the [[Wave equation|wave equation]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013027.png" />, the solution of the Darboux equation satisfying the initial conditions
+
where $\lambda(t,x)$ is a non-negative continuously-differentiable function of $x=(x_1,\ldots,x_n)$. The following uniqueness theorem is valid both for the solution of the Darboux equation and for the solution of the wave equation. If some twice continuously-differentiable solution $u(x,y)$ of the Darboux equation vanishes together with its derivative on the base of the characteristic cone lying in the plane $t=0$, it vanishes inside the entire domain bounded by this cone. The form of the characteristic cone is the same as for the [[Wave equation|wave equation]]. If $\lambda(t,x)=n-1>0$, the solution of the Darboux equation satisfying the initial conditions
  
<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/d030/d030130/d03013028.png" /></td> </tr></table>
+
$$u(t,x)\bigg|_{t=0}=\phi(x),\,\,\,u_t(t,x)\bigg|_{t=0}=0,$$
  
with a twice continuously-differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013029.png" />, is the function
+
with a twice continuously-differentiable function $\phi(x)$, is the function
  
<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/d030/d030130/d03013030.png" /></td> </tr></table>
+
$$u(x,t)=\frac{\Gamma(n/2)}{2\pi^{n/2}t^{n-1}}\int_{\lvert x-y\rvert=t}\phi(y)\,\mbox{d}S_y,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013031.png" /> is the gamma-function. This solution of the Darboux equation and the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030130/d03013032.png" /> of the wave equation satisfying the conditions
+
where $\Gamma(z)$ is the gamma-function. This solution of the Darboux equation and the solution $v(x,t)$ of the wave equation satisfying the conditions
  
<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/d030/d030130/d03013033.png" /></td> </tr></table>
+
$$v(t,x)\bigg|_{t=0}=\phi(x),\,\,\,v_t(t,x)\bigg|_{t=0}=0,$$
  
 
are connected by the relation
 
are connected by the relation
 +
$$u(t,x)=2\frac{\Gamma(n/2)}{\Gamma((n-1)/2)\sqrt{\pi}}\int_0^1v(t\beta,x)(1-\beta^2)^{(n-3)}/2\,\,\mbox{d}\beta.$$
  
<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/d030/d030130/d03013034.png" /></td> </tr></table>
 
  
 
The equation was named after G. Darboux.
 
The equation was named after G. Darboux.

Revision as of 08:50, 20 April 2012

An ordinary differential equation

$$\frac{\mbox{d}y}{\mbox{d}x}=\frac{P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)},$$

where $P$, $Q$ and $R$ are integral polynomials in $x$ and $y$. This equation was first studied by G. Darboux [1]. The Jacobi equation is a special case of the Darboux equation. Let $n$ be the highest degree of the polynomials $P$, $Q$, $R$; if the Darboux equation has $s$ known particular algebraic solutions, then if $s\geq 2+n(n+1)/2$, its general solution is found without quadratures, and if $s=1+n(n+1)/2$, an integrating factor can be found [2]. If $P$ and $Q$ are homogeneous functions of degree $m$, and $R$ is a homogeneous function of degree $k$ then, if $k=m-1$, the Darboux equation is a homogeneous differential equation; if $k\neq m-1$, the Darboux equation may be reduced to a Bernoulli equation by substituting $y=zx$.

References

[1] G. Darboux, "Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré" Bull. Sci. Math. , 2 (1878) pp. 60–96
[2] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)

N.Kh. Rozov

The hyperbolic equation $$u_{tt}-\Delta u+\frac{\lambda(t,x)}{t}u_t=0,\,\,\,t\neq0,$$


where $\lambda(t,x)$ is a non-negative continuously-differentiable function of $x=(x_1,\ldots,x_n)$. The following uniqueness theorem is valid both for the solution of the Darboux equation and for the solution of the wave equation. If some twice continuously-differentiable solution $u(x,y)$ of the Darboux equation vanishes together with its derivative on the base of the characteristic cone lying in the plane $t=0$, it vanishes inside the entire domain bounded by this cone. The form of the characteristic cone is the same as for the wave equation. If $\lambda(t,x)=n-1>0$, the solution of the Darboux equation satisfying the initial conditions

$$u(t,x)\bigg|_{t=0}=\phi(x),\,\,\,u_t(t,x)\bigg|_{t=0}=0,$$

with a twice continuously-differentiable function $\phi(x)$, is the function

$$u(x,t)=\frac{\Gamma(n/2)}{2\pi^{n/2}t^{n-1}}\int_{\lvert x-y\rvert=t}\phi(y)\,\mbox{d}S_y,$$

where $\Gamma(z)$ is the gamma-function. This solution of the Darboux equation and the solution $v(x,t)$ of the wave equation satisfying the conditions

$$v(t,x)\bigg|_{t=0}=\phi(x),\,\,\,v_t(t,x)\bigg|_{t=0}=0,$$

are connected by the relation $$u(t,x)=2\frac{\Gamma(n/2)}{\Gamma((n-1)/2)\sqrt{\pi}}\int_0^1v(t\beta,x)(1-\beta^2)^{(n-3)}/2\,\,\mbox{d}\beta.$$


The equation was named after G. Darboux.

References

[1] F. John, "Plane waves and spherical means applied to partial differential equations" , Interscience (1955)
How to Cite This Entry:
Darboux equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_equation&oldid=17940
This article was adapted from an original article by A.K. Gushchin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article