Show simple item record

dc.contributor.advisorBaeumer, Boris
dc.contributor.advisorKovács, Mihály
dc.contributor.advisorFrauendiener, Jörg
dc.contributor.authorSankaranarayanan, Harish
dc.date.available2014-11-13T03:52:07Z
dc.date.copyright2014
dc.identifier.citationSankaranarayanan, H. (2014). Grünwald-type approximations and boundary conditions for one-sided fractional derivative operators (Thesis, Doctor of Philosophy). University of Otago. Retrieved from http://hdl.handle.net/10523/5216en
dc.identifier.urihttp://hdl.handle.net/10523/5216
dc.description.abstractThe 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.
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.publisherUniversity of Otago
dc.rightsAll items in OUR Archive are provided for private study and research purposes and are protected by copyright with all rights reserved unless otherwise indicated.
dc.subjectFractional Derivatives
dc.subjectGrunwald formula
dc.subjectFourier Multipliers
dc.subjectCarlson's Inequality
dc.subjectFractional Differential Equations
dc.subjectFractional Powers of Operators
dc.subjectGrunwald-type Approximations
dc.subjectFractional Partial Differential Equations
dc.subjectFractional Derivatives on Bounded Domains
dc.titleGrünwald-type approximations and boundary conditions for one-sided fractional derivative operators
dc.typeThesis
dc.date.updated2014-11-13T02:34:02Z
dc.language.rfc3066en
thesis.degree.disciplineMathematics and Statistics
thesis.degree.nameDoctor of Philosophy
thesis.degree.grantorUniversity of Otago
thesis.degree.levelDoctoral
otago.openaccessOpen
 Find in your library

Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record