Polynomial neural network for solving Caputo-conformable fractional Volterra–Fredholm integro-differential equation with three-point non-local boundary conditions

This paper investigates a new class of fractional Volterra–Fredholm integro-differential equations (FVFIDEs) involving a new Caputo-conformable operator with three-point non-local Riemann–Liouville conformable integral boundary conditions. The Banach contraction principle and Schaefer’s fixed point theorem are employed to investigate the existence and uniqueness of the solution in Banach space, and also the uniform stability is provided to prove the stability of the solution. A new generalized Gronwall inequality in the sense of the Riemann–Liouville conformable integral is established and utilized to prove the priori bounded of the solution. A hybrid technique, combining a polynomial neural network (PNN) with an extreme learning machine algorithm without using any activation functions, is developed. The Chebyshev neural network (ChebyshevNN) and Bernstein neural network (BernsteinNN) are used. To enhance the stability and control the strength of regularization numerically, L2regularization with hyperparameter x is incorporated into the error function. Feed-forward and back-propagation learning algorithms are used to initialize and update the weights, while the Adam optimization method is employed to minimize the error. The error bound, convergence, computational complexity, stability, and sensitivity for parameters for the PNN method are provided. The numerical results provide the efficiency and accuracy for the proposed method.

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