NUMERICAL ALGORITHMS FOR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS WITH TIME-DEPENDENT BOUNDARY CONDITIONS

Abstract

This dissertation focuses on developing and analyzing numerical schemes for fractional partial differential equations (PDEs). The development is important because several models involving fractional derivatives exhibit non-locality and memory dependencies, making them difficult to solve. Moreover, many of such models do not have analytical solutions due to the non-linearity involved in their formulation. In the first part of the study, we develop numerical schemes for space-fractional reaction-diffusion equations with time-dependent boundary conditions. The methods are based on using the matrix transfer technique (MTT) for spatial discretization, and rational approximations to the matrix exponential function are used in time. In particular, predictor-corrector schemes based on $(1,1)$- and $(0,2)$-Pad\'e, and a real distinct pole approximation to the exponential function are developed. We observe that the solutions produced by the $(1,1)$-Pad\'e scheme incur oscillatory behavior for some time steps. These oscillations are due to high-frequency components present in the solution and diminish as the order of the space-fractional derivative decreases (slow diffusion). A priori reliability constraint is proposed to avoid these unwanted oscillations. Furthermore, the constraints are generalized for all $(m,m)$-Pad\'e approximants, $m \in \Z^+$, to the matrix exponential functions. In the second part of the study, a novel numerical scheme for time-space fractional PDEs is developed. The developed scheme is similar to the Crank-Nicholson scheme for integer-order PDEs and is shown to be of order $1 + \alpha$ in time, where $\alpha$ is the order of the time derivative described in the Caputo sense. We implement the algorithms in parallel using the shared memory systems (OpenMP) and the distributed memory systems (MPI). We discuss the merits and demerits of each of the parallel versions of the algorithms. Error and stability analysis of the scheme is also discussed. Unlike the Crank-Nicholson scheme for integer-order PDEs, the derived scheme has a lower order ($1 + \alpha$). This lower order is due to the singular kernel (as a result of the Caputo derivative) involved in the scheme's formulation. We used the time-graded mesh to improve the scheme's accuracy from $1 + \alpha$ to two. The last part of the study focuses on applying fractional derivatives and, in particular, the derived schemes to a scientific domain. We propose a time-fractional compartmental model comprising the susceptible, exposed, infected, hospitalized, recovered, and dead population for the COVID-19 epidemic. The properties and dynamics of the proposed model are discussed. We run several model simulations and estimate parameters using the Center for Systems and Science Engineering data at John Hopkins University for some selected states in the US. Furthermore, the efficacy of contact tracing (CT) is investigated by linking the disease model dynamics with actions of contact tracers such as monitoring and tracking. CT's impact on the reproduction number $\mathcal{R}_0$ of COVID-19 is described. In particular, the importance and relevance of the model parameters such as the number of reported cases, effectiveness of tracking and monitoring policy, and the transmission rates to CT are discussed.