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.

$u_{xx}+x{u}_{yy}=0.$

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

$xd{x}^2+d{y}^2=0,$

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

$k\in \mathbb{\mathbb{N}}$

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

$m_i=3i+p$

$n_i=2(k-i)+q$

$c_i=m_i!!!\cdot (m_i-1)!!!\cdot n_i!!\cdot (n_i-1)!!$

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

for k = 0:

$u=A+Bx+Cy+Dxy$

for k = 1:

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

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.