Euler-Tricomi equation

In mathematics, the Euler-Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named for Leonhard Euler and Francesco Giacomo Tricomi.

:$displaystyleu\_\{xx\}=xu\_\{yy\}.$

It is hyperbolic in the half plane "x > 0" and elliptic in the half plane "x < 0".Its characteristics are

:$displaystyle\; x;dx^2=dy^2,$

which have the integral

:$ypmfrac\{2\}\{3\}x^\{frac\{3\}\{2=C,$

where "C" is a constant of integration. The characteristics thus comprise two families of semicubical parabolas, with cusps on the line "x = 0", the curves lying on the right hand side of the "y"-axis.

Particular solutions

Particular solutions to the Euler-Tricomi equations include
* $displaystyle\; u=Axy\; +\; Bx\; +\; Cy\; +\; D,$
* $displaystyle\; u=A(3y^2+x^3)+B(y^3+x^3y)+C(6xy^2+x^4),$where $displaystyle\; A,\; B,\; C,\; D$ are arbitrary constants.

The Euler-Tricomi equation is a limiting form of Chaplygin's equation.

External links

* [http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc4.pdf Tricomi and Generalized Tricomi Equations] at EqWorld: The World of Mathematical Equations.

Bibliography

* A. D. Polyanin, "Handbook of Linear Partial Differential Equations for Engineers and Scientists", Chapman & Hall/CRC Press, 2002.