Analysis and Approximation of a Fractional Dierential Equation

A differential equation is fractional if it involves an operator that can be considered to be between a (k − 1)th and kth order differential operator, for some positive integer k, and it is said to be of fractional-order if this operator is the highest order operator in the equation. The diffusion equation is of order 2, because its highest order operator is the Laplacian, a 2nd order differential operator, but we can consider an analogous equation of order 2s, where s ∈ (0, 1), involving the so-called fractional Laplacian operator. Such fractional-order equations appear in a surprising number of real world models. For example, a diffusion model used for cardiac tissue is what is known as anomalous, or non-Fickian, because the diffusion does not satisfy Fick’s law of diffusion and is not modelled accurately by the diffusion equation, but actually by a differential equation of fractional order. The diffusion is also anisotropic (directionally dependent) because diffusion along fibers happens at a different rate to that across fibers in the tissue; the mathematical models of are harder to work with. This thesis covers some analysis for the study of fractional-order advection-diffusion equations relevant to this anisotropic cardiac tissue model. The study of fractional-order equations is difficult: Firstly, fractional-order operators are nonlocal, i.e. the value of a fractional derivative of a function at a point in the domain depends on values of the function throughout the domain; and secondly, boundary conditions (traces) do not make sense in fractional Sobolev spaces of order s ≤ 1/2, so constraints must be defined on a region of non-zero volume. We review and derive some relevant results on fractional Sobolev spaces, fractional-order operators and the nonlocal calculus developed by Du, Gunzburger, Lehoucq, Zhou (2011). We prove well-posedness of a general class of fractional-order elliptic problems and develop Galerkin approximations, focusing on the derivation of a-priori error bounds.