The fractional Laplacian is an integro-differential operator that is currently widely used in nonlocal models, such as the anomalous diffusion, which arises when a particle moves randomly in the space involving a random process that allows long jumps. Determining the response of such dynamic systems is a daunting task, as general analytical solutions are not available. This thesis proposes approximate and numerical methods for determining the response of dynamic systems containing fractional Laplacian terms. Based on the Riesz-Marchaud and Caputo-type representations of the fractional Laplacian, two Boundary Element Methods (BEM) are introduced to treat fractional dynamic systems. Further, a modal expansion is proposed as a novel expression of the fractional Laplacian. Furthermore, based on the proposed eigenfunctions, statistical linearization procedures are developed to approximate the response statistics.

A BEM-based numerical algorithm is first introduced to estimate the solution of the fractional Poisson equation based on the Riesz-Marchaud definition of the fractional Laplacian. Further, the algorithm is applied to a fractional diffusion equation. The properties of the Caputo-type fractional Laplacian are next investigated. Then, a different BEM-based algorithm is developed for time domain simulation of the response of dynamic systems with Caputo-type fractional Laplacian terms. The analog equation is constructed with the unknown load, which is used in calculation of the fractional Laplacian of the response. A discretization and numerical integration scheme are then employed for estimating the response.

It is shown that a frequency domain analysis of a nonlinear fractional diffusion equation with stochastic excitation can be conducted by a statistical linearization procedure. The approach is implemented by introducing non-orthogonal eigenfunctions of the fractional Laplacian of the response, which are transformed from the linear modes of the classical diffusion equation solution. Such a representation allows deriving an MDOF nonlinear ordinary differential equation, which is linearized in the mean square sense. Further, a simplified statistical linearization approximation method is proposed. The variance and the power spectral density of the response are then calculated by an iterative procedure.

Numerical results pertaining to linear and nonlinear systems exposed to periodic and stochastic excitation are provided to demonstrate the effectiveness of the proposed methods.