In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named after mathematicians Leonhard Euler and Francesco Giacomo Tricomi.

:<math>

u_{xx}+xu_{yy}=0. \,

</math>

It is elliptic in the half plane x&nbsp;>&nbsp;0, parabolic at x&nbsp;=&nbsp;0 and hyperbolic in the half plane&nbsp;x&nbsp;<&nbsp;0.

Its characteristics are

:<math> x\,dx^2+dy^2=0, \, </math>

which have the integral

:<math> y\pm\frac{2}{3}x^{3/2}=C,</math>

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

A general expression for particular solutions to the Euler–Tricomi equations is:

:<math> u_{k,p,q}=\sum_{i=0}^k(-1)^i\frac{x^{m_i}y^{n_i{c_i} \, </math>

where

:<math> k \in \mathbb{N} </math>

:<math> p, q \in \{0,1\} </math>

:<math> m_i = 3i+p </math>

:<math> n_i = 2(k-i)+q </math>

:<math> c_i = m_i!!! \cdot (m_i-1)!!! \cdot n_i!! \cdot (n_i-1)!!</math>

These can be linearly combined to form further solutions such as:

for k = 0:

:<math> u=A + Bx + Cy + Dxy \, </math>

for k = 1:

:<math> u=A(\tfrac{1}{2}y^2 - \tfrac{1}{6}x^3) + B(\tfrac{1}{2}xy^2 - \tfrac{1}{12}x^4) + C(\tfrac{1}{6}y^3 - \tfrac{1}{6}x^3y) + D(\tfrac{1}{6}xy^3 - \tfrac{1}{12}x^4y) \, </math>

etc.

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

See also

  • Burgers' equation
  • Chaplygin's equation

Bibliography

  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.
  • Tricomi and Generalized Tricomi Equations at EqWorld: The World of Mathematical Equations.