Fractional calculus introduces tractable mathematical tools for accurate description of anomalous phenomena, regarded as a manifestation of self-similar structures and memory effects. Fractional approaches in modeling anomalous dynamics have been increasingly employed over the last decade in a disparate range of applications, where the standard methods have been found experimentally to be inadequate. The physical significance of employing fractional operators in such models is indicated by their potentials in describing the inherent abnormal-exponential or heavy-tailed processes due to their power-law or logarithmic kernels of convolution type. However, it is also inevitably challenging to provide efficient and yet accurate numerical methods for fractional differential equations. More concretely, history-dependent and nonlocal characteristics of fractional operators impose further numerical complexities, ranging from theoretical analysis to large memory requirement. Therefore, to establish a robust framework for developing fractional models, a great deal of attention is required in theoretical and numerical studies of fractional calculus. In particular, developing a fractional model depends on some key elements, given as: stochastic interpretation, theoretical analysis, numerical methods, and optimization of model parameters. In this dissertation, we provide a robust framework for fractional modeling of anomalous features in the applied areas of research such as solute transport in underground waters and homogeneous isotropic turbulent flows. Inspired by the stochastic interpretation of fractional diffusion problems in terms of time-changed Markov processes, we develop a unified Petrov-Galerkin (PG) spectral method for a class of fractional partial differential equations (FPDEs) with two-sided derivatives and constant coefficients. We employ the eigenfunctions of the fractional Sturm-Liouville eigen-problems, called Jacobi poly-fractonomials, as temporal bases and test functions. Furthermore, we formulate a novel unified fast linear solver for the resulting high-dimensional linear system based on the solution of generalized eigenproblem of spatial mass matrices with respect to the corresponding stiffness matrices. On the theoretical side, we prove existence and uniqueness of the solution to the corresponding bilinear form of the problem and subsequently, provide the corresponding stability and error analyses. Moreover, we compare the theoretical and computational rates of spectral convergence by performing several numerical simulations. Motivated by the multifractal characteristics of anomalous phenomena, we extend the unified PG method to the generalized form of FPDEs with distributed-order two-sided derivatives and carry out the corresponding stability and error analyses. In the large-eddy simulation of homogeneous turbulence, presence of nonlocal triad interactions and intermittent structures urges developing new nonlocal closure models. Such anomalous effects become even more pronounced when the filter-width enlarges. Within the proposed framework, we start modeling of turbulent effects at the Boltzmann transport using fractional calculus. In practice, we approximate the filtered collision operator with a power-law function which contributes to the corresponding fractional subgrid-scale (SGS) model in the filtered NS equations through a meticulous derivation. Inspired by the self-similar cascading of energy from large to small scales and the exponential decay in the dissipation range, we adopt tempered \textit{L\'evy}-stable distribution as a statistically-consistent choice at the kinetic level and subsequently, we derive the corresponding operator in terms of the tempered fractional Laplacian, $(\Delta+\lambda)^{\alpha}(\cdot)$, $\alpha \in (0,1]$, $\alpha \neq \frac{1}{2}$, and $\lambda>0$ in the filtered NS equations, termed as tempered fractional SGS (TFSGS) model. The model parameters appear to be strictly depending on the filter-width and the flow Reynolds number. Such a tractable fractional operator offers a great flexibility in characterizing nonlocal structures in the turbulent inertial and dissipation ranges through fractional and tempering parameters. To infer optimum values of the model parameters, we develop a robust optimization algorithm, involving two-point structure functions and conventional correlation coefficients. In \textit{a priori} / \textit{a posteriori} statistical analyses, we evaluate capabilities of the TFSGS model in fulfilling the closed essential requirements and also study the numerical stability of LES solutions in time.
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