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Cnoidal wave solution to the Korteweg–De Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter m = 0.9). Numerical solution of the KdV equation u t + uu x + δ 2 u xxx = 0 (δ = 0.022) with an initial condition u(x, 0) = cos(πx).
The idea of cointegration may be demonstrated in a simple macroeconomic setting. Suppose, consumption and disposable income are macroeconomic time series that are related in the long run (see Permanent income hypothesis).
To find the length of the gradually varied flow transitions, iterate the “step length”, instead of height, at the boundary condition height until equations 4 and 5 agree. (e.g. For an M1 Profile, position 1 would be the downstream condition and you would solve for position two where the height is equal to normal depth.)
The application of MacCormack method to the above equation proceeds in two steps; a predictor step which is followed by a corrector step. Predictor step: In the predictor step, a "provisional" value of at time level + (denoted by ) is estimated as follows
"New high-order Runge-Kutta formulas with step size control for systems of first and second-order differential equations". Zeitschrift für Angewandte Mathematik und Mechanik . 44 (S1): T17–T29.
The method is as follows: Suppose you have p-values, sorted into order lowest-to-highest , …,, and their corresponding hypotheses , …, (null hypotheses). You want the FWER to be no higher than a certain pre-specified significance level.
When considering the numerical solution of ordinary differential equations (ODEs), a predictor–corrector method typically uses an explicit method for the predictor step and an implicit method for the corrector step.
Tukey's range test, also known as Tukey's test, Tukey method, Tukey's honest significance test, or Tukey's HSD (honestly significant difference) test, [1] is a single-step multiple comparison procedure and statistical test.