Fractional-Order Newton-Type Iterative Methods Based on Hypergeometric Function Kernels

The paper describes the construction of 5th-order iterative methods of solving nonlinear equations through the introduction of hypergeometric function kernels to fractional derivatives. The suggested method is an extension of the classical Newton method, where the integer-order derivative is replaced with a Caputo-type fractional derivative with hypergeometric kernels to allow the development of higher memory effects and a more regular derivative action. The theoretical framework will strictly defend the well-posedness of the hypergeometric fractional operator, and the successive scheme is proved to converge in the integer order context to the classical approach of Newton. The convergence analysis demonstrates that the method approaches the solution superlinearly, and more precisely on the selection of the fractional order, and the parameters of the kernels towards a quadratic solution.Numerical experiments on representative algebraic, transcendental, and exponential nonlinear test equations prove that the proposed method is more stable as well as less sensitive to initial guess and more rapidly convergent than classical Newton and other available fractional methods, especially in situations with flat or ill-conditioned derivatives. The findings indicate the usefulness of hypergeometric-kernel-based fractional iterative techniques to nonlinear problems; the methods are flexible and can be computationally practical.