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.
\[ u_{xx}=xu_{yy}. \, \]
It is hyperbolic in the half plane x > 0, parabolic at x = 0 and elliptic in the half plane x < 0. Its characteristics are
\[ x\,dx^2=dy^2, \, \]
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
Particular solutions to the Euler–Tricomi equations include
- \( u=Axy + Bx + Cy + D, \, \)
- \( u=A(3y^2+x^3)+B(y^3+x^3y)+C(6xy^2+x^4), \, \)
where A, B, C, D are arbitrary constants.
The Euler–Tricomi equation is a limiting form of Chaplygin's equation.
External links
- Tricomi and Generalized Tricomi Equations at EqWorld: The World of Mathematical Equations.
Bibliography
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.uk:Рівняння Ейлера-Трікомі
