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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 ...
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes.
Numerical methods for ordinary differential equations approximate solutions to initial value problems of the form ′ = (,), =.. The result is approximations for the value of () at discrete times : = +, where is the time step (sometimes referred to as ) and is an integer.
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 ...
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}} .
A simple predictor–corrector method (known as Heun's method) can be constructed from the Euler method (an explicit method) and the trapezoidal rule (an implicit method). Consider the differential equation ′ = (,), =, and denote the step size by .