# Tricomi equation

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