# Euler method

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## Euler method

[′oi·lər ‚meth·əd]
(mathematics)
A method of obtaining an approximate solution of an ordinary differential equation of the form dy / dx = f (x, y), where f is a specified function of x and y. Also known as Eulerian description.
(mechanics)
A method of studying fluid motion and the mechanics of deformable bodies in which one considers volume elements at fixed locations in space, across which material flows; the Euler method is in contrast to the Lagrangian method.
References in periodicals archive ?
As we all know, there are many solutions corresponding to the second-order ODEs, including the Euler method, the improved Euler method and the Runge-Kutta method et al.
There are several methods for the approximate solution of the swing equations such as: Euler's method, semi-implicit Euler method (trapezoidal), collocation, Runge Kutta, shampine, Radeau, among many more.
For a single small time step, this approach is much easier to implement and more efficient than the implicit method; at the same time, the stress is return-mapped effectively onto the subsequent yield surface by enforcing the consistency condition, and the problem of drift in the explicit forward Euler method is overcome.
After years of accumulation and development, there are too many methods to solve the numerical solution of nonlinear dynamic systems; the main methods are as follows: perturbation method [3], averaging method [4], Runge-Kutta method [5], Euler method [6], gradient method [7] and so on.
Further, it has been shown that the implicit Euler method is more stable than explicit one (see [9, 11, 14]).
Since the Euler method is only a first order method we also tried the trapezoidal scheme which is a second order scheme [7].
Among the topics are first-order scalar equations, the implicit Euler method, two-step and multistep methods, 1-periodic solutions of time-dependent partial differential equations with constant coefficients, linear initial boundary value problems, and nonlinear problems.
Between separate steps numerical methods we remember: Taylor series development method, Euler method (polygonal lines method), Runge-Kutta method, improved Euler method and from those which linked steps: Adams-Moulton, Milne etc.
In this study, the finite approximates equations in addition to non-explicit Euler method were used for numerical solving of these equations.
In Section 2, the explicit Euler method for an ordinary differential equation (ODE) initial value problem (IVP) is investigated under a parameter transformation.
We are applying the Euler method for time derivative and the application of Galerkin method to diffusion term only over the entire domain [OMEGA] to the equation (28).
In this paper we apply Runge-Kutta Heun method of order 3 to solve fuzzy differential equations and establish that this method is better than Euler method.

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