In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named for Leonhard Euler and Francesco Giacomo Tricomi.
\( {\displaystyle u_{xx}+xu_{yy}=0.\,} \)
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
Particular solutions to the Euler–Tricomi equations include
\( {\displaystyle u=Axy+Bx+Cy+D,\,} \)
\({\displaystyle u=A(3y^{2}+x^{3})+B(y^{3}+x^{3}y)+C(6xy^{2}+x^{4})+D(2xy^{3}+x^{4}y),\,} \)
where A, B, C, D are arbitrary constants.
A general expression for these solutions is:
\( {\displaystyle u=\sum _{i=0}^{k}{\frac {x^{m_{i}}\cdot y^{n_{i}}}{c_{i}}}\,} \)
where
\( {\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)!!}
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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