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
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 , averaging method , Runge-Kutta method , Euler method
, gradient method  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 .
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