Euler–Tricomi equation

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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.

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

which have the integral

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

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

 

where

 
 
 
 
 


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

for k = 0:

 

for k = 1:

 

etc.


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

See also

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Bibliography

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  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.
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