Publication:
Solutions to differential equations via fixed point approaches: new mathematical foundations and applications

dc.contributor.advisor Tisdell, Chris en_US
dc.contributor.advisor Goodrich, Chris en_US
dc.contributor.author Almuthaybiri, Saleh en_US
dc.date.accessioned 2022-03-23T16:04:20Z
dc.date.available 2022-03-23T16:04:20Z
dc.date.issued 2021 en_US
dc.description.abstract The central aim of this thesis is to construct a fuller and firmer mathematical foundation for the solutions to various classes of nonlinear differential equations than is currently available in the literature. This includes boundary value problems (BVPs) that involve ordinary differential equations, and initial value problems (IVPs) for fractional differential equations. In particular, we establish new conditions that guarantee the existence, uniqueness and approximation of solutions to second-order BVPs, third-order BVPs, and fourth-order BVPs for ordinary differential equations. The results enable us, in turn, to shed new light on problems from applied mathematics, engineering and physics, such as: the Emden and Thomas-Fermi equations; the bending of elastic beams through an application of our general theories; and laminar flow in channels with porous walls. We also ensure the existence, uniqueness and approximation of solutions to some IVPs for fractional differential equations. An understanding of the existence, uniqueness and approximation of solutions to these problems is fundamental from both pure and applied points of view. Our methods involve an analysis of nonlinear operators through fixed-point theory in new and interesting ways. Part of the novelty involves generating new conditions under which these operators are contractive, invariant and/or establishing new a priori bounds on potential solutions. As such, we draw on: Banach fixed- point theorem, Schauder fixed-point theorem, Rus's contraction mapping theorem, and a continuation theorem due to A. Granas and its constructive version known as continuation method for contractive maps. The ideas in this thesis break new ground at the intersection of pure and applied mathematics. Thus, this work will be of interest to those who are researching the theoretical aspects of differential equations, and those who are interested in better understanding their applications. en_US
dc.identifier.uri http://hdl.handle.net/1959.4/71186
dc.language English
dc.language.iso EN en_US
dc.publisher UNSW, Sydney en_US
dc.rights CC BY-NC-ND 3.0 en_US
dc.rights.uri https://creativecommons.org/licenses/by-nc-nd/3.0/au/ en_US
dc.subject.other initial value problems en_US
dc.subject.other fixed point approaches en_US
dc.subject.other boundary value problems en_US
dc.subject.other fractional differential equations en_US
dc.subject.other Rus's contraction mapping theorem en_US
dc.subject.other continuation theorem en_US
dc.subject.other existence en_US
dc.subject.other uniqueness en_US
dc.subject.other approximation en_US
dc.subject.other Thomas-Fermi equations en_US
dc.subject.other laminar flow en_US
dc.subject.other Banach fixed-point theorem en_US
dc.subject.other Schauder fixed-point theorem en_US
dc.title Solutions to differential equations via fixed point approaches: new mathematical foundations and applications en_US
dc.type Thesis en_US
dcterms.accessRights open access
dcterms.rightsHolder Almuthaybiri, Saleh
dspace.entity.type Publication en_US
unsw.accessRights.uri https://purl.org/coar/access_right/c_abf2
unsw.identifier.doi https://doi.org/10.26190/unsworks/22790
unsw.relation.faculty Science
unsw.relation.originalPublicationAffiliation Almuthaybiri, Saleh, School of Mathematics & Statistics, Science, UNSW en_US
unsw.relation.originalPublicationAffiliation Tisdell, Chris, School of Mathematics & Statistics, Science, UNSW en_US
unsw.relation.originalPublicationAffiliation Goodrich, Chris, School of Mathematics & Statistics, Science, UNSW en_US
unsw.relation.school School of Mathematics & Statistics *
unsw.thesis.degreetype PhD Doctorate en_US
Files
Original bundle
Now showing 1 - 1 of 1
No Thumbnail Available
Name:
public version.pdf
Size:
1.66 MB
Format:
application/pdf
Description:
Resource type