Euler–Tricomi equation

In

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

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

${\displaystyle x\,dx^{2}+dy^{2}=0,\,}$

which have the integral

${\displaystyle y\pm {\frac {2}{3}}x^{3/2}=C,}$

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

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

${\displaystyle u_{k,p,q}=\sum _{i=0}^{k}(-1)^{i}{\frac {x^{m_{i}}y^{n_{i}}}{c_{i}}}\,}$

where

${\displaystyle k\in \mathbb {N} }$

${\displaystyle p,q\in \{0,1\}}$

${\displaystyle m_{i}=3i+p}$

${\displaystyle n_{i}=2(k-i)+q}$

${\displaystyle c_{i}=m_{i}!!!\cdot (m_{i}-1)!!!\cdot n_{i}!!\cdot (n_{i}-1)!!}$

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

for k = 0:

${\displaystyle u=A+Bx+Cy+Dxy\,}$

for k = 1:

${\displaystyle u=A({\tfrac {1}{2}}y^{2}-{\tfrac {1}{6}}x^{3})+B({\tfrac {1}{2}}xy^{2}-{\tfrac {1}{12}}x^{4})+C({\tfrac {1}{6}}y^{3}-{\tfrac {1}{6}}x^{3}y)+D({\tfrac {1}{6}}xy^{3}-{\tfrac {1}{12}}x^{4}y)\,}$

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.