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The iteration capability in Excel can be used to find solutions to the Colebrook equation to an accuracy of 15 significant figures. [3] [4] Some of the "successive approximation" schemes used in dynamic programming to solve Bellman's functional equation are based on fixed-point iterations in the space of the return function. [5] [6]
This equation can be cast into the form of the hypergeometric differential equation.It has two linearly independent solutions, called the periods of elliptic functions. The ratio of the two periods is equal to the period ratio τ, the standard coordinate on the upper-half plane.
These equations describe boundary-value problems, in which the solution-function's values are specified on boundary of a domain; the problem is to compute a solution also on its interior. Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences.
Ackermann's formula provides a direct way to calculate the necessary adjustments—specifically, the feedback gains—needed to move the system's poles to the target locations. This method, developed by Jürgen Ackermann , [ 2 ] is particularly useful for systems that don't change over time ( time-invariant systems ), allowing engineers to ...
They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, and fluid dynamics. The basic hyperbolic functions are: [1]
In computational fluid dynamics, the MacCormack method (/məˈkɔːrmæk ˈmɛθəd/) is a widely used discretization scheme for the numerical solution of hyperbolic partial differential equations. This second-order finite difference method was introduced by Robert W. MacCormack in 1969. [1]
The idea to combine the bisection method with the secant method goes back to Dekker (1969).. Suppose that we want to solve the equation f(x) = 0.As with the bisection method, we need to initialize Dekker's method with two points, say a 0 and b 0, such that f(a 0) and f(b 0) have opposite signs.
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after F. J. Richards, who proposed the general form for the family of models in 1959.