Advanced neural network approaches for coupled equations with fractional derivatives

Abstract

We investigate numerical solutions and compare them with Fractional Physics-Informed Neural Network (FPINN) solutions for a coupled wave equation involving fractional partial derivatives. The problem explores the evolution of functions u and v over time t and space x. We employ two numerical approximation schemes based on the finite element method to discretize the system of equations. The effectiveness of these schemes is validated by comparing numerical results with exact solutions. Additionally, we introduce the FPINN method to tackle the coupled equation with fractional derivative orders and compare its performance against traditional numerical methods. Key findings reveal that both numerical approaches provide accurate solutions, with the FPINN method demonstrating competitive performance in terms of accuracy and computational efficiency. Our study highlights the significance of employing FPINNs in solving fractional differential equations and underscores their potential as alternatives to conventional numerical methods. The novelty of this work lies in its comparative analysis of traditional numerical techniques and FPINNs for solving coupled wave equations with fractional derivatives, offering insights into advancing computational methods for complex physical systems.