Date of Award
Doctor of Philosophy (PhD)
Mathematics & Statistics
Computational and Applied Mathematics
Richard D. Noren
Volterra integral equations arise in a variety of applications in modern physics and engineering, namely in interactions that contain a memory term. Classical formulations of these problems are largely inflexible when considering non-homogeneous media, which can be problematic when considering long term interactions of real-world applications. The use of fractional derivative and integral terms naturally relax these restrictions in a natural way to consider these problems in a more general setting. One major drawback to the use of fractional derivatives and integrals in modeling is the regularity requirement for functions, where we can no longer assume that functions are as smooth or as well behaved as their classical counterparts.
This work outlines the derivation and application of a class of stable and convergent finite difference methods to discretize weakly singular integrals which occur in Volterra integral equations. This derivation is motivated by classical discretizations that arise in Caputo fractional derivatives. We present a time-fractional diffusion equation as a case study to develop the finite difference scheme, where the Laplace transform is used to pose the problem equivalently as a Volterra integral equation, which is then discretized. A generalized scheme is presented to consider a much wider class of integral equations, which allows for the consideration of applications of the Fourier transform. This ultimately allows for a natural discretization of both time- and space-fractional diffusion and differential equations. Some natural physical applications are considered to fully utilize these schemes.
The novelty of these schemes is in its simplicity and efficiency when compared to classic methods of discretization, especially for Caputo fractional derivatives. Typical discretizations in the fractional derivative form over-assume regularity to discretize a full derivative term, which subsequently restricts the admissible solution space. Other considerations from discretizing the fractional derivative form include negatively impacting the rate of convergence from the remaining fractional integration term, which is recovered by the use of non-uniform mesh partitions to recover some of the order of convergence.
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Davis, Wesley C..
"Finite Difference Schemes for Integral Equations with Minimal Regularity Requirements"
(2021). Doctor of Philosophy (PhD), Dissertation, Mathematics & Statistics, Old Dominion University, DOI: 10.25777/6e2s-6r45