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.

${\displaystyle u_{xx}+xu_{yy}=0.\,}$

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

${\displaystyle x\,dx^{2}+dy^{2}=0,\,}$

which have the integral

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

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

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

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

A general expression for these solutions is:

• ${\displaystyle u=\sum _{i=0}^{k}{\frac {x^{m_{i}}\cdot y^{n_{i}}}{c_{i}}}\,}$

where

• ${\displaystyle p,q\in [0,1]}$
• ${\displaystyle m_{i}=3i+p}$
• ${\displaystyle n_{i}=2(k-i)+q}$
• ${\displaystyle c_{i}=m_{i}!!!\cdot (m_{i}-1)!!!\cdot n_{i}!!\cdot (n_{i}-1)!!}$

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

See also

• Burgers equation
• Chaplygin's equation

Bibliography

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

External links

• Tricomi and Generalized Tricomi Equations at EqWorld: The World of Mathematical Equations.