Euler–Tricomi equation - Alchetron, The Free Social 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 = 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 d x 2 + d y 2 = 0 ,

which have the integral

y ± 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 = A x y + B x + C y + D ,

u = A ( 3 y 2 + x 3 ) + B ( y 3 + x 3 y ) + C ( 6 x y 2 + x 4 ) ,

where A, B, C, D are arbitrary constants.

The Euler–Tricomi equation is a limiting form of Chaplygin's equation.

References

Euler–Tricomi equation Wikipedia

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