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.

:$u_+xu_=0. \,$

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

:$y\pm\fracx^=C,$

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:

:$u_=\sum_^k(-1)^i\frac \,$

where

:$k \in \mathbb$
:$p, q \in \$
:$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'':
:$u=A(\tfracy^2 - \tfracx^3) + B(\tfracxy^2 - \tfracx^4) + C(\tfracy^3 - \tfracx^3y) + D(\tfracxy^3 - \tfracx^4y) \,$
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.

{{DEFAULTSORT:Euler-Tricomi equation
Partial differential equations
Equations of fluid dynamics
Leonhard Euler