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In this paper, a hybrid method between Variational Iteration Method (VIM) and gray wolf optimization algorithm (GWO) was proposed to solve the fractional differential equations (FDE), where the optimal parameter value ([Formula: see text] for the VIM was estimated by the GWO. The solutions in the proposed method, GWO-VIM, demonstrated the efficiency and reliability compared to the default method VIM, by calculating the maximum absolute errors (MAE) and mean square error (MSE).

In this work, a combined technique the Variation Iteration Method (VIM) with the Trapezoidal Rule (TR) was recommend to solve linear and nonlinear fractional ordinary differential equations (F.O.D.E.), where the results obtained from the Variation Iteration method were improved, and numerical results were obtained by determining the maximum absolute errors (MAE) and mean square error (MSE) for the given examples. As the results It is proved that that the proposed method is better than the default method.

In this paper, Lyapunov's artificial small parameter method (LASP-M) with continuous particle swarm optimization (CPSO) is presented and used for solving nonlinear differential equations. The proposed method, LASPM-CPSO, is based on estimating the ε parameter in LASPM through a PSO algorithm and based on a proposed objective function. Three different examples are used to evaluate the proposed method LASPM-CPSO, and compare it with the classical method LASPM through different intervals of the domain. The results from the maximum absolute error (MAE) and mean squared error (MSE) obtained through the given examples show the reliability and efficiency of the proposed LASPM-CPSO method, compared to the classical method LASPM.

In this paper, ordinary differential equations (ODEs) of two types, linear and non-linear will be solved by using the homotopy analytical method (HAM), and the sine cosine algorithm (SCA) will be used to modify the parameter h and obtain a better approximate solution than what was in the previous method HAM. The suggested method, HAM-SCA, covers a solution that reveals the reliability and the efficiency correspondingly to the default method HAM by computing the maximum absolute errors (MAE) and mean square error (MSE).