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In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
This is the Euler method (or forward Euler method, in contrast with the backward Euler method, to be described below). The method is named after Leonhard Euler who described it in 1768. The Euler method is an example of an explicit method. This means that the new value y n+1 is defined in terms of things that are already known, like y n.
In Itô calculus, the Euler–Maruyama method (also simply called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations named after Leonhard Euler and Gisiro Maruyama. The ...
In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation, is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential equation can be solved ...
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. They are named in honour of Leonhard Euler. Their general vector form is
This differs from the (forward) Euler method in that the forward method uses (,) in place of (+, +). The backward Euler method is an implicit method: the new approximation y k + 1 {\displaystyle y_{k+1}} appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown y k + 1 {\displaystyle y_{k+1}} .
Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt ...
Forward-Backward Euler method The result of applying both the Forward Euler method and the Forward-Backward Euler method for a = 5 {\displaystyle a=5} and n = 30 {\displaystyle n=30} . In order to apply the IMEX-scheme, consider a slightly different differential equation: