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.

u

x

x

+

x

u

y

y

=

0.

{\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

x

d

x

2

+

d

y

2

=

0

,

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

which have the integral

y

±

2

3

x

3

/

2

=

C

,

{\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:

u

k

,

p

,

q

=

∑

i

=

0

k

(

−

1

)

i

x

m

i

y

n

i

c

i

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

where

k

∈

N

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

p

,

q

∈

{

0

,

1

}

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

m

i

=

3

i

+

p

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

n

i

=

2

(

k

−

i

)

+

q

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

c

i

=

m

i

!

!

!

⋅

(

m

i

−

1

)

!