High-accuracy solution of fractional-order corruption dynamics models using Bernstein-wavelet neural networks

The model of corruption dynamics is challenging because socio-economic systems have non-linear and memory-dependent dynamics that cannot be explained by classical integer-order models. In this paper, we propose a Stable Bernstein Wavelet Neural Network (SBWNN) for the numerical solution of a fractional-order corruption dynamics model in terms of the Caputo fractional derivative. Our model combines Bernstein polynomial basis functions with multiresolution wavelet decomposition and the physics-based collocation strategy to determine the network coefficients. The nonlocal memory term is evaluated using the matrix-based implementation of the classical L1 discretization. The approximation property, convergence, Ulam-Hyers stability, and equilibrium stability of our method are discussed in detail in the theoretical description. Our numerical experiments for fractional orders 0.3≤ξ≤1.0 demonstrate the accuracy, convergence, robustness, and computational efficiency of SBWNN. Also, the effects of polynomial degree, wavelet decomposition level, parameter sensitivity, and comparison with feed-forward neural networks are investigated. Our results agree very well with analytical solutions and we can observe that lower fractional orders have more memory and longer corruption persistence, proving that the proposed SBWNN is a very accurate, stable, and efficient approach to solve nonlinear fractional dynamics.