Euler–Tricomi equation explained

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 _{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

xdx ^{2+dy} ^{2=0,}

which have the integral

y\pm (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} = \sum ^{k( -1 )} _{i =0} ^{i (x ^{m _{i}} y ^{n _{i}})/(c _{i})}

where

k \in N p, q \in \{ 0 , 1 \} m _{i} = 3 i + p n _{i} = 2 (k - i) + q c _{i} = m _{i!!!} ⋅ (m _{i-1)!!!} ⋅ n _{i!!} ⋅ (n _{i-1)!!}

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

for k = 0:

u = A + Bx + Cy + Dxy for k = 1: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)} 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.