Euler-Tricomi equation

From Wikipedia, the free encyclopedia

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 x x = x u y y .

It is hyperbolic in the half plane $x > 0$ and elliptic in the half plane $x < 0$ . Its characteristics are $x d x 2 = d y 2$ , which have the integral

$y\pm\frac{2}{3}x^{\frac{3}{2}}=C$

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

$u = A x y + B x + C y + D$
$u = A (3 x 2 - y 3) + B (x 3 - x y 3) + C (6 y x 2 - y 4)$

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.

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Categories: Partial differential equations | Equations of fluid dynamics

Euler-Tricomi equation

From Wikipedia, the free encyclopedia

[edit] Particular solutions

[edit] External links

[edit] Bibliography

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