Generalized fractional approach to solving partial differential equations with arbitrary dispersion relations

Fractional calculus (FC) has shown promise for the study of highly nonlinear and chaotic dynamics [1]. The application of this field has produced wide-reaching benefits where nonlocal, or even global, effects impact the evolution of the system [2]. Our utilization of FC involves the generalized transformation of partial differential equations (PDEs) to conjugate fractional partial differential equations [3]. This modifies the dispersion relation to encapsulate the nonlocal perturbations, thus allowing the nonlinear dynamics to be simulated in a new and more computationally efficient way. We show two examples of our approach; the Landau-Lifshitz-Gilbert (LLG) for magnetization dynamics is modified to contain the atomic-level dispersion of magnons, and the Korteweg-De Vries (KdV) equation is modified to support an Euler-like dispersion relation, similar to the Whitham equation. Furthermore, we present the necessary conditions to meet to apply our method for the study of other physical systems, or for cases of greater complexity.