Mahler measure: Difference between revisions
Jump to navigation
Jump to search
en>Virginia-American m →See also: link |
en>JosephSilverman Added a section on multivariable Mahler measure + minor editing |
||
| Line 1: | Line 1: | ||
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]]. | |||
:<math> | |||
u_{xx}=xu_{yy}. \, | |||
</math> | |||
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. | |||
Its [[method of characteristics|characteristic]]s are | |||
:<math> x\,dx^2=dy^2, \, </math> | |||
which have the integral | |||
:<math> y\pm\frac{2}{3}x^{3/2}=C,</math> | |||
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. | |||
==Particular solutions== | |||
Particular solutions to the Euler–Tricomi equations include | |||
* <math> u=Axy + Bx + Cy + D, \, </math> | |||
* <math> u=A(3y^2+x^3)+B(y^3+x^3y)+C(6xy^2+x^4), \, </math> | |||
where ''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. | |||
{{DEFAULTSORT:Euler-Tricomi equation}} | |||
[[Category:Partial differential equations]] | |||
[[Category:Equations of fluid dynamics]] | |||
Revision as of 19:54, 5 November 2013
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.
It is hyperbolic in the half plane x > 0, parabolic at x = 0 and elliptic 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
Particular solutions to the Euler–Tricomi equations include
where A, B, C, D are arbitrary constants.
The Euler–Tricomi equation is a limiting form of Chaplygin's equation.
External links
- 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.