High-accuracy power series solutions with arbitrarily large radius of convergence for fractional nonlinear differential equations

TL;DR

This paper introduces a new power series method with arbitrarily large convergence radius for solving fractional nonlinear differential equations, enabling accurate solutions for complex physical models.

Contribution

The authors develop a systematic iterative power series approach that overcomes divergence issues, providing controlled accuracy for fractional nonlinear differential equations.

Findings

Derived fractional generalizations of cnoidal waves and solitons.

Achieved solutions with large convergence radius, reducing divergence problems.

Revealed localized solutions with power law tails in fractional nonlinear diffusion.

Abstract

Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These effective descriptions thus appear commonly in physical and mathematical modeling. We present a new series method providing systematic controlled accuracy for solutions of fractional nonlinear differential equations. The method relies on spatially iterative use of power series expansions. Our approach permits an arbitrarily large radius of convergence and thus solves the typical divergence problem endemic to power series approaches. We apply our method to the fractional nonlinear Schr\"odinger equation and its imaginary time rotation, the fractional nonlinear diffusion equation. For the fractional nonlinear Schr\"odinger equation we find fractional generalizations of cnoidal waves of Jacobi elliptic functions as well as a fractional bright soliton. For the fractional nonlinear diffusion equation we find the combination of fractional and nonlinear effects results in a more strongly localized solution which nevertheless still exhibits power law tails, albeit at a much lower density.