Euler-Tricomi equation
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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. .
- uxx = xuyy.
It is hyperbolic in the half plane x > 0 and elliptic in the half plane x < 0. Its characteristics are xdx2 = dy2, which have the integral
where C is a constant of integration. The characteristics thus comprise two families of semi-cubical parabolas, with cusps on the line x = 0, the curves lying on the right hand side of the y-axis.
[edit] Particular solutions
Particular solutions to the Euler-Tricomi equations include
- u = Axy + Bx + Cy + D
- u = A(3x2 − y3) + B(x3 − xy3) + C(6yx2 − y4)
where A,B,C,D are arbitrary constants.
The Euler-Tricomi equation is a limiting form of Chaplygin's equation.
[edit] External links
- Tricomi and Generalized Tricomi Equations at EqWorld: The World of Mathematical Equations.
[edit] Bibliography
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.
