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.


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[edit]

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 ABCD are arbitrary constants.

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

External links[edit]

Bibliography[edit]

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