Developing a Novel Neural Network Method for Solving Variable-Order Fractional Partial Differential Equations with Time-Varying Delay

Subject Areas : Numerical Analysis

Farahnaz Golpour Lasaki ¹ , Hamideh Ebrahimi ^{2 *} , Mousa Ilie ³

1 - Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
2 - Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
3 - Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran

Received: 2025-02-22 Accepted : 2025-06-04 Published : 2025-01-13

Keywords: Functional link neural network, Lagrange polynomials, Variable-order fractional operators, Time-varying delay,

Abstract :

This paper presents a novel functional link neural network for solving a class of variable-order fractional partial differential equations with time-varying delay. Due to the proficiency of Lagrange polynomials in numerical approximations for fractional calculus, these polynomials serve as the foundational neuro-solutions within the neural network. Finding the right activation functions is essential for effective learning in artificial neural networks, particularly when solving variable-order fractional derivatives and time-varying delays. To reduce the computational complexity of the proposed neural network, a linear activation function is used. Numerical simulations are carried out to demonstrate the capability of the proposed method. The neural network undergoes training using a modified Newton-Raphson method instead of the traditional learning techniques. The study’s findings indicate that the suggested functional link neural network achieves greater accuracy in comparison to some traditional methods for solving fractional partial differential equations.

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