Mahler measure

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.

$u_{xx}=xu_{yy}.\,$

It is hyperbolic in the half plane x > 0, parabolic at x = 0 and elliptic in the half plane x < 0. Its characteristics are

$x\,dx^{2}=dy^{2},\,$

which have the integral

$y\pm {\frac {2}{3}}x^{3/2}=C,$

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

• $u=Axy+Bx+Cy+D,\,$
• $u=A(3y^{2}+x^{3})+B(y^{3}+x^{3}y)+C(6xy^{2}+x^{4}),\,$

where A, B, C, D are arbitrary constants.

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

External links

• 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.