Q-Analogues of Parallel Numerical Scheme Based on Neural Networks and Their Engineering Applications

Abstract

Quantum calculus can provide new insights into the nonlinear behaviour of functions and equations, addressing problems that may be difficult to tackle by classical calculus due to high nonlinearity. Iterative methods for solving nonlinear equations can benefit greatly from the mathematical theory and tools provided by quantum calculus, e.g., using the concept of q-derivatives, which extends beyond classical derivatives. In this paper, we develop parallel numerical root-finding algorithms that approximate all distinct roots of nonlinear equations by utilizing q-analogies of the function derivative. Furthermore, we utilize neural networks to accelerate the convergence rate by providing accurate initial guesses for our parallel schemes. The global convergence of the q-parallel numerical techniques is demonstrated using random initial approximations on selected biomedical applications, and the efficiency, stability, and consistency of the proposed hybrid numerical schemes are analyzed.