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Difference between revisions of "Tricomi equation"

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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]]]).
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Tricomi,  "On second-order linear partial differential equations of mixed type" , Moscow-Leningrad  (1947)  (In Russian; translated from Italian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.A. Chaplygin,  "On gas-like structures" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F.I. Frankl',  "Selected work on gas dynamics" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Bitsadze,  "Some classes of partial differential equations" , Gordon &amp; Breach  (1988)  (Translated from Russian)</TD></TR></table>
 
 
  
  
 
====Comments====
 
====Comments====
See also [[Tricomi problem|Tricomi problem]] and [[Mixed-type differential equation|Mixed-type differential equation]], for additional references.
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See also [[Tricomi problem]] and [[Mixed-type differential equation]], for additional references.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  F. Tricomi,  "On second-order linear partial differential equations of mixed type" , Moscow-Leningrad  (1947)  (In Russian; translated from Italian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  S.A. Chaplygin,  "On gas-like structures" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  F.I. Frankl',  "Selected work on gas dynamics" , Moscow  (1973)  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Bitsadze,  "Some classes of partial differential equations" , Gordon &amp; Breach  (1988)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR>
 +
</table>

Latest revision as of 06:00, 30 May 2023

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]).


Comments

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

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)
[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