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Difference between revisions of "Tricomi equation"

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A differential equation of the form
 
A differential equation of the form
  
<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/t/t094/t094180/t0941801.png" /></td> </tr></table>
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$$yu_{xx}+u_{yy}=0,$$
  
which is a simple model of a second-order partial differential equation of mixed elliptic-hyperbolic type with two independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094180/t0941802.png" /> and one open non-characteristic interval of parabolic degeneracy. The Tricomi equation is elliptic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094180/t0941803.png" />, hyperbolic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094180/t0941804.png" /> and degenerates to an equation of parabolic type on the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094180/t0941805.png" /> (see [[#References|[1]]]). The Tricomi equation is a prototype of the Chaplygin equation
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which is a simple model of a second-order partial differential equation of mixed elliptic-hyperbolic type with two independent variables $x,y$ and one open non-characteristic interval of parabolic degeneracy. The Tricomi equation is elliptic for $y>0$, hyperbolic for $y<0$ and degenerates to an equation of parabolic type on the line $y=0$ (see [[#References|[1]]]). The Tricomi equation is a prototype of the Chaplygin 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/t/t094/t094180/t0941806.png" /></td> </tr></table>
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$$k(y)u_{xx}+u_{yy}=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094180/t0941807.png" /> is the stream function of a plane-parallel steady-state gas flow, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094180/t0941808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094180/t0941809.png" /> are functions of the velocity of the flow, which are positive at subsonic and negative at supersonic speeds, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094180/t09418010.png" /> is the angle of inclination of the velocity vector (see [[#References|[2]]] [[#References|[3]]]).
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where $u=u(x,y)$ is the stream function of a plane-parallel steady-state gas flow, $k(y)$ and $y$ are functions of the velocity of the flow, which are positive at subsonic and negative at supersonic speeds, and $x$ is the angle of inclination of the velocity vector (see [[#References|[2]]] [[#References|[3]]]).
  
 
Many important problems in the mechanics of continuous media reduce to a boundary value problem for the Tricomi equation, in particular, mixed flows involving the formation of local subsonic zones (see [[#References|[3]]], [[#References|[4]]]).
 
Many important problems in the mechanics of continuous media reduce to a boundary value problem for the Tricomi equation, in particular, mixed flows involving the formation of local subsonic zones (see [[#References|[3]]], [[#References|[4]]]).

Revision as of 07:58, 3 October 2014

A differential equation of the form

$$yu_{xx}+u_{yy}=0,$$

which is a simple model of a second-order partial differential equation of mixed elliptic-hyperbolic type with two independent variables $x,y$ and one open non-characteristic interval of parabolic degeneracy. The Tricomi equation is elliptic for $y>0$, hyperbolic for $y<0$ and degenerates to an equation of parabolic type on the line $y=0$ (see [1]). The Tricomi equation is a prototype of the Chaplygin equation

$$k(y)u_{xx}+u_{yy}=0,$$

where $u=u(x,y)$ is the stream function of a plane-parallel steady-state gas flow, $k(y)$ and $y$ are functions of the velocity of the flow, which are positive at subsonic and negative at supersonic speeds, and $x$ is the angle of inclination of the velocity vector (see [2] [3]).

Many important problems in the mechanics of continuous media reduce to a boundary value problem for the Tricomi equation, in particular, mixed flows involving the formation of local subsonic zones (see [3], [4]).

References

[1] F. Tricomi, "On second-order linear partial differential equations of mixed type" , Moscow-Leningrad (1947) (In Russian; translated from Italian)
[2] S.A. Chaplygin, "On gas-like structures" , Moscow-Leningrad (1949) (In Russian)
[3] F.I. Frankl', "Selected work on gas dynamics" , Moscow (1973) (In Russian)
[4] A.V. Bitsadze, "Some classes of partial differential equations" , Gordon & Breach (1988) (Translated from Russian)


Comments

See also Tricomi problem and Mixed-type differential equation, for additional references.

References

[a1] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
How to Cite This Entry:
Tricomi equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tricomi_equation&oldid=33467
This article was adapted from an original article by A.M. Nakhushev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article