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| In [[mathematics]], the '''Euler–Tricomi equation''' is a [[linear]] [[partial differential equation]] useful in the study of [[transonic]] [[fluid mechanics|flow]]. It is named for [[Leonhard Euler]] and [[Francesco Giacomo Tricomi]].
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| :<math>
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| u_{xx}=xu_{yy}. \,
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| </math>
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| It is [[hyperbolic partial differential equation|hyperbolic]] in the half plane ''x'' > 0, [[parabolic partial differential equation|parabolic]] at ''x'' = 0 and [[elliptic partial differential equation|elliptic]] in the half plane ''x'' < 0.
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| Its [[method of characteristics|characteristic]]s are
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| :<math> x\,dx^2=dy^2, \, </math>
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| which have the integral
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| :<math> y\pm\frac{2}{3}x^{3/2}=C,</math> | |
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| where ''C'' is a constant of [[Integral|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.
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| ==Particular solutions==
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| Particular solutions to the Euler–Tricomi equations include
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| * <math> u=Axy + Bx + Cy + D, \, </math>
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| * <math> u=A(3y^2+x^3)+B(y^3+x^3y)+C(6xy^2+x^4), \, </math>
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| where ''A'', ''B'', ''C'', ''D'' are arbitrary constants.
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| The Euler–Tricomi equation is a limiting form of [[Chaplygin's equation]].
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| == External links==
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| * [http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc4.pdf Tricomi and Generalized Tricomi Equations] at EqWorld: The World of Mathematical Equations.
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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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| {{DEFAULTSORT:Euler-Tricomi equation}}
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| [[Category:Partial differential equations]]
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| [[Category:Equations of fluid dynamics]]
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