The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

Abstract

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods.

Keywords

• orthogonal neural network
• simulated annealing optimization technique
• fractional differential equations
• Caputo derivative