<p style='text-indent:20px;'>Ordinary differential equations are converted into a constrained optimization problems to find their approximate solutions. In this work, an algorithm is proposed by applying particle swarm optimization (PSO) to find an approximate solution of ODEs based on an expansion approximation. Since many cases of linear and nonlinear ODEs have singularity point, Padé approximant which is fractional expansion is employed for more accurate results compare to Fourier and Taylor expansions. The fitness function is obtained by adding the discrete least square weighted function to a penalty function. The proposed algorithm is applied to 13 famous ODEs such as Lane Emden, Emden-Fowler, Riccati, Ivey, Abel, Thomas Fermi, Bernoulli, Bratu, Van der pol, the Troesch problem and other cases. The proposed algorithm offer fast and accurate results compare to the other methods presented in this paper. The results demonstrate the ability of proposed approach to solve linear and nonlinear ODEs with initial or boundary conditions.</p>

A numerical algorithm for solving linear and non-linear fractional differential equations is proposed based on the Bees algorithm and Chebyshev polynomials. The proposed algorithm was applied to a set of numerical examples. Faster results are obtained compared to the wavelet methods.