Fractional Singular Coupled System: Existence Theory via Extended Leggett-Williams Theorem

Abstract

In this paper, a very detailed existence theory is proposed for a coupled system of fractional differential equations with two competing cubic-root singularities. The system has two opposite nonlinear terms x^(-1/3) (repulsive) and x^(1/3) (restorative). It is a perfect example of a very delicate balance between two extremes that may even render analytical methods useless. We prove that there exist positive solutions with coupled fractional boundary conditions by combining three new and innovative methods: (1) a new version of the Leggett-Williams fixed point theorem for nondifferentiable cubic-root nonlinearities, (2) a weighted cone factorization with concave-convex function pairs designed to model the singularities, and (3) a green's function with well-estimated error bounds. The analytical framework has been supported by a very exact numerical verification (error is below 0.1% ). Besides, the non-Newtonian fluid, plasma physics, and fractional-order reaction-diffusion systems are the examples where this work can be beneficial. First of all, this research settled a difficult issue raised by singular differential equations and, secondly, it sets a new standard of managing competing singularities in nonlinear analysis.