Boundary elements approach for solving stochastic nonlinear problems with fractional Laplacian terms

Abstract

Models involving stochastic diffusion equations are utilized for describing the evolution of a number of natural phenomena and are widely discussed in the open literature. In recent years, these models have been revisited in light of experimental observations in which “anomalous” diffusion processes were identified, such as in the propagation of acoustic waves in random media. In this context, a critical characteristic of the theoretical models is the introduction of fractional derivative operators in the associated governing equations. Specifically, anomalous diffusion involves a fractional Laplacian operator replacing the classical Laplacian. Currently, solutions to equations with fractional Laplacians are available for a quite limited numbers of cases. Further, to the authors’ knowledge, no solutions are available for nonlinear equations involving fractional Laplacians. This fact creates the need of developing adequate numerical methods for estimating the response of this kind of systems. This paper proposes a Boundary Element Method (BEM)-based approach to determine the response of a system governed by a nonlinear fractional diffusion equation involving a random excitation. The approach is constructed by utilizing the integral representation of the classical Poisson equation solution, in which unknown constants are determined by the BEM. Then, based on a recently proposed representation of the fractional Laplacian operator, the value of the fractional Laplacian of the response is updated progressively by matrix transformation of these constants. Numerical results pertaining to a system exposed to white noise are presented to elucidate the mechanization of the approach. Further, parameter studies are done for examining the influence of the fractional Laplacian order on the system response.