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 after mathematicians Leonhard Euler and Francesco Giacomo Tricomi.

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

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

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

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

A general expression for particular solutions to the Euler–Tricomi equations is:

$u_{k,p,q}=\sum _{i=0}^{k}(-1)^{i}{\frac {x^{m_{i}}y^{n_{i}}}{c_{i}}}\,$

where

$k\in \mathbb {N}$

$p,q\in \{0,1\}$

$m_{i}=3i+p$

$n_{i}=2(k-i)+q$

$c_{i}=m_{i}!!!\cdot (m_{i}-1)!!!\cdot n_{i}!!\cdot (n_{i}-1)!!$

These can be linearly combined to form further solutions such as:

for k = 0:

$u=A+Bx+Cy+Dxy\,$

for k = 1:

$u=A({\tfrac {1}{2}}y^{2}-{\tfrac {1}{6}}x^{3})+B({\tfrac {1}{2}}xy^{2}-{\tfrac {1}{12}}x^{4})+C({\tfrac {1}{6}}y^{3}-{\tfrac {1}{6}}x^{3}y)+D({\tfrac {1}{6}}xy^{3}-{\tfrac {1}{12}}x^{4}y)\,$

etc.

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.