A Weak Fractional Calculus Theory and Numerical Methods for Fractional Differential Equations

This dissertation is comprised of four integral parts. The first part comprises a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions.

The second part of this work presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the well-established integer order Sobolev spaces and theory. In particular, two new families of one-sided fractional Sobolev spaces are introduced and analyzed. Many key theorems/properties, such as density/approximation theorem, extension theorems, one-sided trace theorem, and various embedding theorems and Sobolev inequalities in those Sobolev spaces are established. Moreover, a few relationships with existing fractional Sobolev spaces are also discovered.

The third part presents two new families of fractional PDEs obtained as Euler-Lagrange equations from the fractional calculus of variations. Several new fractional differential operators are introduced, including the fractional $p$-Laplacian, Laplacian, and Neumann boundary operator. The first family of problems connects minimization problems with essential boundary conditions to fractional PDEs with Dirichlet boundary data via the calculus of variations. The second family of problems establishes the connection between minimization problems with natural boundary conditions and fractional PDEs with Neumann boundary data. Existence, uniqueness, and regularity of weak solutions in the newly developed fractional Sobolev space(s) are proven.

The final section presents a new finite element method for approximating solutions to fractional PDEs introduced in the third part of this dissertation. Much effort is given to developing a linear finite element interpolation theory in the newly introduced fractional Sobolev spaces as well as introducing a numerical fractional derivative concept. These serve as the primary tools for constructing and analyzing the new finite element method. Additionally, many computations and numerical experiments are presented to validate the convergence theories presented in this section.