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)