View source for Tricomi equation

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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 [[#References|[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 [[#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]]]).


====Comments====
See also [[Tricomi problem]] and [[Mixed-type differential equation]], for additional references.

====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>
<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>