A study on the delay differential equations by the shifted Chebyshev neural network scheme

In this work, we present a shifted Chebyshev neural network approach based on an extreme learning machine (SCNN-ELM) algorithm for solving fractional differential equations with proportional and constant delays. Additionally, by employing this method, we are able to solve systems of delay equations and delay integro-differential equations. We choose shifted Chebyshev polynomials as activation functions for the hidden neurons. Because integral and delay terms make the numerical solution complex, we apply a Chebyshev neural network to directly handle the integral term. To solve delay integral equations, we first divide the interval into segments based on the delay term, build a single hidden-layer SCNN, and then apply the ELM algorithm to find the best solution for each segment sequentially. The best solution from the previous segment serves as the starting point for the next segment. In comparison with recent numerical algorithms, the proposed method uses fewer neurons, performs simplified computation with enhanced solution efficiency and eliminates the need for random parameter tuning to stabilise the results. We provide six numerical examples to demonstrate the advantages and efficacy of our method that effectively tackles delay-related issues.