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Fractional nonlinear differential equations present an interplay between twocommon and important effective descriptions used to simplify high dimensionalor more complicated theories: nonlinearity and fractional derivatives. Theseeffective descriptions thus appear commonly in physical and mathematicalmodeling. We present a new series method providing systematic controlledaccuracy for solutions of fractional nonlinear differential equations. Themethod relies on spatially iterative use of power series expansions. Ourapproach permits an arbitrarily large radius of convergence and thus solves thetypical divergence problem endemic to power series approaches. We apply ourmethod to the fractional nonlinear Schr\"odinger equation and its imaginarytime rotation, the fractional nonlinear diffusion equation. For the fractionalnonlinear Schr\"odinger equation we find fractional generalizations of cnoidalwaves of Jacobi elliptic functions as well as a fractional bright soliton. Forthe fractional nonlinear diffusion equation we find the combination offractional and nonlinear effects results in a more strongly localized solutionwhich nevertheless still exhibits power law tails, albeit at a much lowerdensity.

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