Euler–Tricomi equation

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]\displaystyle{ u_{xx}+xu_{yy}=0. \, }[/math]

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

[math]\displaystyle{ x\,dx^2+dy^2=0, \, }[/math]

which have the integral

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

where

[math]\displaystyle{ k \in \mathbb{N} }[/math]

[math]\displaystyle{ p, q \in \{0,1\} }[/math]

[math]\displaystyle{ m_i = 3i+p }[/math]

[math]\displaystyle{ n_i = 2(k-i)+q }[/math]

[math]\displaystyle{ 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]\displaystyle{ u=A + Bx + Cy + Dxy \, }[/math]

for k = 1:

[math]\displaystyle{ 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.

External links

• Tricomi and Generalized Tricomi Equations at EqWorld: The World of Mathematical Equations.