- Euler-Tricomi equation
In
mathematics , the Euler-Tricomi equation is alinear partial differential equation useful in the study oftransonic flow. It is named forLeonhard Euler andFrancesco Giacomo Tricomi .:displaystyleu_{xx}=xu_{yy}.
It is hyperbolic in the half plane "x > 0" and elliptic in the half plane "x < 0".Its characteristics are
:displaystyle x;dx^2=dy^2,
which have the integral
:ypmfrac{2}{3}x^{frac{3}{2=C,
where "C" is a constant of integration. The characteristics thus comprise two families of
semicubical parabola s, 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^3y)+C(6xy^2+x^4),where displaystyle A, B, C, D are arbitrary constants.The Euler-Tricomi equation is a limiting form of
Chaplygin's equation .External links
* [http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc4.pdf Tricomi and Generalized Tricomi Equations] at EqWorld: The World of Mathematical Equations.
Bibliography
* A. D. Polyanin, "Handbook of Linear Partial Differential Equations for Engineers and Scientists", Chapman & Hall/CRC Press, 2002.
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