An Iterative Method Based on Fractional Derivatives for Solving Nonlinear Equations Béla J. Szekeres and Ferenc Izsák Abstract In this work, we showed a fractional derivative based iterative method for solving nonlinear time-independent equation, where the operator is affecting on a Hilbert space. We assumed that it is equally monotone and Lipschitz-continuous. We proved that the algorithm is convergent. We also have tested our method numerically previously on a fluid dynamical problem and the results showed that the algorithm is stable. 1 Introduction The theory of fractional order derivatives are almost as old as the integer-order [5]. There are many applications, for example in physics [1, 2, 6], finance [8, 9] or biology [3]. Our aim is to prove theoretical mathematical statements. In this work our goal is to find a solution numerically for the equation A(u) = f . If we assume that u is time-dependent, then one can do this by finding a stationary solution of the equation ∂ t u(t) =−(A(u(t)) − f). The numerical solution of this problem can be highly inaccurate. To avoid this we propose to replace the time derivative with a fractional one. Since the fractional order time derivative is a non- local operator, we expect that this stabilizes the time integration in the numerical solutions. Since the fractional order derivative here is defined as a limit of linear combination of past values, the time discretization will be simple. We also tested our method numerically in a fluid dynamical problem [10]. B. J. Szekeres () Department of Numerical Analysis, Eötvös Loránd University, Faculty of Informatics, Budapest, Hungary e-mail: szekeres@inf.elte.hu F. Izsák Department of Applied Analysis and Computational Mathematics & ELTE-MTA Numnet Research Group, Eötvös Loránd University, Budapest, Hungary e-mail: izsakf@cs.elte.hu © Springer Nature Switzerland AG 2019 I. Faragó et al. (eds.), Progress in Industrial Mathematics at ECMI 2018, Mathematics in Industry 30, https://doi.org/10.1007/978-3-030-27550-1_42 337