In mathematics, the EulerâÂÂTricomi 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.
It is elliptic in the half plane x > 0, parabolic at x = 0 and hyperbolic in the half plane x < 0. Its characteristics are
which have the integral
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.
A general expression for particular solutions to the EulerâÂÂTricomi equations is:
where
These can be linearly combined to form further solutions such as:
for k = 0:
for k = 1:
etc.
The EulerâÂÂTricomi equation is a limiting form of Chaplygin's equation.