A numerical scheme based on Chebyshev-improved extreme learning machines for fractal-fractional reaction-diffusion equations

Abstract In this work, we present the Chebyshev-Improved Extreme Learning Machine (ChIELM), a neural network-based numerical technique for solving fractal-fractional reaction-diffusion equations (FFRDEs). The proposed approach extends the improved extreme learning machine (IELM) by incorporating shifted Chebyshev polynomials as hidden-layer basis functions, thereby enhancing approximation accuracy and numerical stability. The effectiveness of the ChIELM is demonstrated through two benchmark problems, and its performance is compared with several existing numerical methods. Numerical results demonstrate that the proposed method achieves superior accuracy, with global error measures reaching the order of 10 −9 , 10 −11 , and 10 −12 , outperforming the compared approaches. These findings indicate that the ChIELM provides a robust and accurate computational framework for solving fractal-fractional reaction-diffusion equations.