Search results
Results From The WOW.Com Content Network
Hodograph transformation is a technique used to transform nonlinear partial differential equations into linear version. It consists of interchanging the dependent and independent variables in the equation to achieve linearity.
One of the first methods used to circumvent the nonlinearity of transonic flow models was the hodograph transformation. [2] This concept was originally explored in 1923 by an Italian mathematician named Francesco Tricomi, who used the transformation to simplify the compressible flow equations and prove that they were solvable. [2]
The Cole–Hopf transformation is a change of variables that allows to transform a special kind of parabolic partial differential equations (PDEs) with a quadratic nonlinearity into a linear heat equation. In particular, it provides an explicit formula for fairly general solutions of the PDE in terms of the initial datum and the heat kernel.
However, the derivation employs a hodograph transformation [7] to produce an advection solution that does not include soil water diffusivity, wherein becomes the dependent variable and becomes an independent variable: [2]
Riemann invariants are mathematical transformations made on a system of conservation equations to make them more easily solvable. Riemann invariants are constant along the characteristic curves of the partial differential equations where they obtain the name invariant.
Synonyms include projectivity, projective transformation, and projective collineation. Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry , and the term homography , which, etymologically, roughly means "similar drawing", dates from this time.
Shock polar in the pressure ratio-flow deflection angle plane for a Mach number of 1.8 and a specific heat ratio 1.4. The minimum angle, , which an oblique shock can have is the Mach angle = (/), where is the initial Mach number before the shock and the greatest angle corresponds to a normal shock.
The equations defining the transformation in two dimensions, which rotates the xy axes counterclockwise through an angle into the x′y′ axes, are derived as follows. In the xy system, let the point P have polar coordinates ( r , α ) {\displaystyle (r,\alpha )} .