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.
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.
January 29, 2023
euler, tricomi, equation, mathematics, linear, partial, differential, equation, useful, study, transonic, flow, named, after, mathematicians, leonhard, euler, francesco, giacomo, tricomi, displaystyle, elliptic, half, plane, parabolic, hyperbolic, half, plane,. 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 u x x x u y y 0 displaystyle u xx xu yy 0 It is elliptic in the half plane x gt 0 parabolic at x 0 and hyperbolic in the half plane x lt 0 Its characteristics are x d x 2 d y 2 0 displaystyle x dx 2 dy 2 0 which have the integral y 2 3 x 3 2 C 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 Contents 1 Particular solutions 2 See also 3 Bibliography 4 External linksParticular solutions EditA general expression for particular solutions to the Euler Tricomi equations is u k p q i 0 k 1 i x m i y n i c i displaystyle u k p q sum i 0 k 1 i frac x m i y n i c i where k N displaystyle k in mathbb N p q 0 1 displaystyle p q in 0 1 m i 3 i p displaystyle m i 3i p n i 2 k i q displaystyle n i 2 k i q c i m i m i 1 n i n i 1 displaystyle c i m i cdot m i 1 cdot n i cdot n i 1 These can be linearly combined to form further solutions such as for k 0 u A B x C y D x y displaystyle u A Bx Cy Dxy for k 1 u A 1 2 y 2 1 6 x 3 B 1 2 x y 2 1 12 x 4 C 1 6 y 3 1 6 x 3 y D 1 6 x y 3 1 12 x 4 y displaystyle u A tfrac 1 2 y 2 tfrac 1 6 x 3 B tfrac 1 2 xy 2 tfrac 1 12 x 4 C tfrac 1 6 y 3 tfrac 1 6 x 3 y D tfrac 1 6 xy 3 tfrac 1 12 x 4 y etc The Euler Tricomi equation is a limiting form of Chaplygin s equation See also EditBurgers equation Chaplygin s equationBibliography EditA D Polyanin Handbook of Linear Partial Differential Equations for Engineers and Scientists Chapman amp Hall CRC Press 2002 External links EditTricomi and Generalized Tricomi Equations at EqWorld The World of Mathematical Equations Retrieved from https en wikipedia org w index php title Euler Tricomi equation amp oldid 1131061223, wikipedia, wiki, book, books, library,