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.

\displaystyle u_{xx}=xu_{yy}.

It is hyperbolic in the half plane x > 0 and elliptic in the half plane x < 0. Its characteristics are

\displaystyle x\;dx^2=dy^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 semicubical 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

  • \displaystyle u=Axy + Bx + Cy + D,
  • \displaystyle u=A(3y^2+x^3)+B(y^3+x^3y)+C(6xy^2+x^4),

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

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

[edit] External links

[edit] Bibliography

  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.