Grünwald-type approximations and boundary conditions for one-sided fractional derivative operators

Abstract

The focus of this thesis is two-fold. The first part investigates higher order numerical schemes for one-dimensional fractional-in-space partial differential equations in 𝐿₁(ℝ). The approximations for the (space) fractional derivative operators are constructed using a shifted Grünwald-Letnikov fractional difference formula. Rigorous error and stability analysis of the Grünwald-type numerical schemes for space-time discretisations of the associated Cauchy problem are carried out using (Fourier) multiplier theory and semigroup theory. The use of a transference principle facilitates the generalisation of the results from the 𝐿₁-setting to any function space where the translation (semi) group is strongly continuous. Furthermore, the results extend to the case when the fractional derivative operator is replaced by the fractional power of a (semi) group generator on an arbitrary Banach space. The second part is dedicated to the study of certain fractional-in-space partial differential equations associated with (truncated) Riemann-Liouville and first degree Caputo fractional derivative operators on Ω:= [(0, 1)]. The boundary conditions encoded in the domains of the fractional derivative operators dictate the inclusion or exclusion of the end points of Ω. Elaborate technical constructions and detailed error analysis are carried out to show convergence of Grünwald-type approximations to fractional derivative operators on 𝑋 = C₀(Ω) and L₁[0, 1]. The wellposedness of the associated Cauchy problem on 𝑋 is established using the approximation theory of semigroups. The culmination of the thesis is the result which shows convergence in the Skorohod topology of the well understood stochastic processes associated with Grünwald-type approximations to the processes governed by the corresponding fractional-in-space partial differential equations.